diff --git a/.github/workflows/blueprint.yml b/.github/workflows/blueprint.yml index be4caba..1161700 100644 --- a/.github/workflows/blueprint.yml +++ b/.github/workflows/blueprint.yml @@ -16,8 +16,6 @@ permissions: contents: read # Read access to repository contents pages: write # Write access to GitHub Pages id-token: write # Write access to ID tokens - issues: write # Write access to issues - pull-requests: write # Write access to pull requests jobs: build_project: @@ -25,7 +23,7 @@ jobs: name: Build project steps: - name: Cleanup to free disk space - uses: jlumbroso/free-disk-space@main + uses: jlumbroso/free-disk-space@3f13feab2962a87273d379ebc36ea159527e6cc9 # v1.3.2 with: tool-cache: false android: true diff --git a/.github/workflows/lean_action_ci.yml b/.github/workflows/lean_action_ci.yml index ff71757..0efdfc3 100644 --- a/.github/workflows/lean_action_ci.yml +++ b/.github/workflows/lean_action_ci.yml @@ -18,8 +18,8 @@ jobs: runs-on: ubuntu-latest steps: - - uses: actions/checkout@v5 - - uses: leanprover/lean-action@v1 + - uses: actions/checkout@v6 + - uses: leanprover/lean-action@434f25c2f80ded67bba02502ad3a86f25db50709 # 2025-07-02 with: build: true lint: true diff --git a/Physicslib4/AQFT/HaagKastler/Net.lean b/Physicslib4/AQFT/HaagKastler/Net.lean index a1ae40d..24cfbf8 100644 --- a/Physicslib4/AQFT/HaagKastler/Net.lean +++ b/Physicslib4/AQFT/HaagKastler/Net.lean @@ -338,11 +338,15 @@ noncomputable def trivialQuasilocalAlgebra : ⟨trivialBasisSet, isAlexandrovBasisSet_trivialBasisSet, x, rfl⟩) ι_inclusion := fun _ _ _ _ _ _ => rfl +/-- The trivial local net satisfies Axiom 3 (local commutativity), since `ℂ` is +commutative. -/ theorem trivialLocalNet_localCommutativity : LocalCommutativity trivialLocalNet trivialLocalNetIsotony := by intro B₁ B₂ B hB₁ hB₂ hB _ h₁ h₂ a b exact @mul_comm ℂ _ _ _ +/-- The trivial local net satisfies Axiom 5 (Lorentz covariance), realized by the +identity automorphism of `ℂ` at every basis set. -/ theorem trivialLocalNet_lorentzCovariance : LorentzCovariance trivialLocalNet trivialLocalNetIsotony := by refine ⟨fun _ _ => StarAlgEquiv.refl ℂ ℂ, ?_, ?_, ?_⟩ diff --git a/Physicslib4/AQFT/HaagKastler/QuasilocalIntertwiner.lean b/Physicslib4/AQFT/HaagKastler/QuasilocalIntertwiner.lean index 8f35356..431d753 100644 --- a/Physicslib4/AQFT/HaagKastler/QuasilocalIntertwiner.lean +++ b/Physicslib4/AQFT/HaagKastler/QuasilocalIntertwiner.lean @@ -360,6 +360,7 @@ algebra. -/ noncomputable def extendHom (L : InhomogeneousLorentzGroup) : Q.carrier →⋆ₐ[ℂ] Q.carrier := (exists_intertwiner_extend N Q L).choose +/-- The extended intertwiner `Φ_L` is continuous. -/ theorem extendHom_continuous (L : InhomogeneousLorentzGroup) : Continuous (extendHom N Q L) := (exists_intertwiner_extend N Q L).choose_spec.1 @@ -441,6 +442,7 @@ noncomputable def trivialQuasilocalLift (L : InhomogeneousLorentzGroup) : trivialHaagKastlerNet.QuasilocalLift trivialQuasilocalAlgebra L := quasilocalLift trivialHaagKastlerNet trivialQuasilocalAlgebra L +/-- The quasilocal lift exists unconditionally for the trivial net. -/ theorem nonempty_trivialQuasilocalLift (L : InhomogeneousLorentzGroup) : Nonempty (trivialHaagKastlerNet.QuasilocalLift trivialQuasilocalAlgebra L) := ⟨trivialQuasilocalLift L⟩ diff --git a/Physicslib4/AQFT/HaagKastler/QuasilocalKMS.lean b/Physicslib4/AQFT/HaagKastler/QuasilocalKMS.lean index 05a02fb..6db59b7 100644 --- a/Physicslib4/AQFT/HaagKastler/QuasilocalKMS.lean +++ b/Physicslib4/AQFT/HaagKastler/QuasilocalKMS.lean @@ -93,10 +93,11 @@ fixing `Ω`, the one-parameter unitary group `t ↦ U t` has positive energy ground-state (`β → ∞`, spectrum-condition) counterpart of `IsKMSStateForFlow`: the stationary state whose flow generator (the Hamiltonian, for a timelike flow) is positive. -/ -def IsGroundStateForFlow (flow : ℝ → InhomogeneousLorentzGroup) +def IsGroundStateForFlow.{u} (N : HaagKastlerNet.{u}) + (flow : ℝ → InhomogeneousLorentzGroup) (ω : Physicslib4.GNS.State N.quasilocal.carrier) : Prop := (∀ (t : ℝ) (a : N.quasilocal.carrier), (ω (N.flowAut flow t a) : ℂ) = ω a) ∧ - ∃ (K : Type) (_ : NormedAddCommGroup K) (_ : InnerProductSpace ℂ K) + ∃ (K : Type u) (_ : NormedAddCommGroup K) (_ : InnerProductSpace ℂ K) (_ : CompleteSpace K) (π : N.quasilocal.carrier →⋆ₐ[ℂ] (K →L[ℂ] K)) (Ω : K) (U : ℝ → (K ≃ₗᵢ[ℂ] K)), (∀ a : N.quasilocal.carrier, (ω a : ℂ) = ⟪Ω, π a Ω⟫_ℂ) ∧ @@ -116,11 +117,11 @@ open scoped InnerProductSpace in /-- The implementing unitary group of a covariance-flow ground state is strongly continuous: strong continuity is a field of the positive-energy hypothesis (`AQFT.IsPositiveEnergy.strongContinuous`). -/ -theorem IsGroundStateForFlow.exists_strongContinuous_unitary - (flow : ℝ → InhomogeneousLorentzGroup) +theorem IsGroundStateForFlow.exists_strongContinuous_unitary.{u} + (N : HaagKastlerNet.{u}) (flow : ℝ → InhomogeneousLorentzGroup) {ω : Physicslib4.GNS.State N.quasilocal.carrier} (h : N.IsGroundStateForFlow flow ω) : - ∃ (K : Type) (_ : NormedAddCommGroup K) (_ : InnerProductSpace ℂ K) + ∃ (K : Type u) (_ : NormedAddCommGroup K) (_ : InnerProductSpace ℂ K) (_ : CompleteSpace K) (π : N.quasilocal.carrier →⋆ₐ[ℂ] (K →L[ℂ] K)) (Ω : K) (U : ℝ → (K ≃ₗᵢ[ℂ] K)), (∀ a : N.quasilocal.carrier, (ω a : ℂ) = ⟪Ω, π a Ω⟫_ℂ) ∧ diff --git a/Physicslib4/AQFT/HaagKastler/VacuumState.lean b/Physicslib4/AQFT/HaagKastler/VacuumState.lean index b08955e..efc5e6a 100644 --- a/Physicslib4/AQFT/HaagKastler/VacuumState.lean +++ b/Physicslib4/AQFT/HaagKastler/VacuumState.lean @@ -122,11 +122,11 @@ def IsFutureTimelikeTranslation (γ : ℝ → InhomogeneousLorentzGroup) : Prop This packages the two *necessary* vacuum conditions — invariance and the spectrum condition. The predicate `ftl` is a parameter; `IsVacuumStateConcrete` fixes it to `IsFutureTimelikeTranslation`. -/ -def HaagKastlerNet.IsVacuumState (N : HaagKastlerNet) +def HaagKastlerNet.IsVacuumState (N : HaagKastlerNet.{u}) (ftl : (ℝ → InhomogeneousLorentzGroup) → Prop) (ω : Physicslib4.GNS.State N.quasilocal.carrier) : Prop := N.IsInvariantState ω ∧ - ∃ (K : Type) (_ : NormedAddCommGroup K) (_ : InnerProductSpace ℂ K) + ∃ (K : Type u) (_ : NormedAddCommGroup K) (_ : InnerProductSpace ℂ K) (_ : CompleteSpace K) (π : N.quasilocal.carrier →⋆ₐ[ℂ] (K →L[ℂ] K)) (Ω : K) (U : InhomogeneousLorentzGroup → (K ≃ₗᵢ[ℂ] K)), (∀ a : N.quasilocal.carrier, (ω a : ℂ) = ⟪Ω, π a Ω⟫_ℂ) ∧ diff --git a/Physicslib4/AQFT/HaagKastlerCurved/Concrete.lean b/Physicslib4/AQFT/HaagKastlerCurved/Concrete.lean index 94d44a9..76ba1c5 100644 --- a/Physicslib4/AQFT/HaagKastlerCurved/Concrete.lean +++ b/Physicslib4/AQFT/HaagKastlerCurved/Concrete.lean @@ -38,7 +38,8 @@ The pieces of the abstract interface are read off the concrete spacetime: * `Isom` is the oriented identity component `Spacetime.Isometry.orientedIdentityComponent` (the topological `connectedComponentOfOne` intersected with future-orientation preservation), - matching the blueprint's "isometries connected to the identity" in Axiom 5; + which is the blueprint's group of "isometries connected to the identity" in + Axiom 5 (see the implementation note on `toAbstractIdentityComponent`); * `isBasisSet_smul` is `Spacetime.LorentzianSpacetime.isBasisSet_smul`: these isometries carry Alexandrov diamonds to Alexandrov diamonds. -/ @@ -56,6 +57,12 @@ component** (identity-component isometries that also preserve the future orientation), as in Axiom 5. For these isometries basis-set preservation is a theorem (`Spacetime.LorentzianSpacetime.isBasisSet_smul`), which supplies the interface field `isBasisSet_smul`. + +Implementation note: the oriented identity component is the group of isometries +connected to the identity named in the blueprint. That an isometry connected to +the identity preserves the future orientation needs a Myers–Steenrod-type rigidity +theorem that Mathlib does not provide, so orientation preservation is included in +the definition of the group rather than proved. -/ noncomputable def toAbstractIdentityComponent (L : LorentzianSpacetime) : AQFT.HaagKastlerCurved.LorentzianSpacetime where diff --git a/Physicslib4/AQFT/KMS.lean b/Physicslib4/AQFT/KMS.lean index 1d51d8c..5720c09 100644 --- a/Physicslib4/AQFT/KMS.lean +++ b/Physicslib4/AQFT/KMS.lean @@ -36,7 +36,8 @@ This is the analytic form of the KMS condition (Bratteli-Robinson): at inverse temperature `β > 0`, `ω` is KMS iff for all `a, b` there is `F` continuous on the strip `{0 ≤ Im z ≤ β}`, holomorphic in the interior, with `F(t) = ω(a α_t b)` and `F(t + iβ) = ω(α_t b · a)`. KMS states are automatically `α`-invariant -(`IsKMSState.invariant`, unconditionally for `β > 0` by `IsKMSState.invariant_of_pos`). +(`IsKMSState.invariant`, given a `StripLiouville β` hypothesis; `IsKMSState.invariant_of_pos` +discharges that extra hypothesis automatically whenever `β > 0`). -/ namespace Physicslib4 diff --git a/Physicslib4/AQFT/PositiveEnergy.lean b/Physicslib4/AQFT/PositiveEnergy.lean index 4df5386..8ddf625 100644 --- a/Physicslib4/AQFT/PositiveEnergy.lean +++ b/Physicslib4/AQFT/PositiveEnergy.lean @@ -69,7 +69,7 @@ one-parameter unitary group — `exp(i t P) = exp(i t Q)` for all `t` — they a Differentiating at `t = 0` gives `i P = i Q`, hence `P = Q`. The unbounded form is `generator_eq_of_eq_expUnitary`. -Blueprint reference: `lmm:exp-generator-unique`. -/ +Blueprint reference: the bounded special case of `lmm:exp-generator-unique`. -/ theorem exp_generator_unique {P Q : H →L[ℂ] H} (h : ∀ (t : ℝ) (x : H), NormedSpace.exp (((t : ℂ) * Complex.I) • P) x diff --git a/Physicslib4/GNS/Amplification.lean b/Physicslib4/GNS/Amplification.lean index 204159c..4551724 100644 --- a/Physicslib4/GNS/Amplification.lean +++ b/Physicslib4/GNS/Amplification.lean @@ -44,6 +44,9 @@ theorem summandProj_isScalar_of_isIrreducible (hirr : IsIrreducible (directSum summandProj (H := H) j ∈ scalarOperators (lp H 2) := isIrreducible_iff_centralizer.mp hirr ▸ summandProj_mem_commutant π j +-- `DecidableEq ι` is bound by the ambient `variable` but only used irrelevantly, through +-- `lp.single`/instance resolution, not by name in this proof; the `unusedDecidableInType` +-- linter flags it when the full project is built (though not in isolation). set_option linter.unusedDecidableInType false in /-- A direct sum with two summands carrying nonzero vectors is **reducible**: the projection onto one summand is a non-scalar element of the commutant. -/ diff --git a/Physicslib4/GNS/Basic.lean b/Physicslib4/GNS/Basic.lean index e572507..3b02edc 100644 --- a/Physicslib4/GNS/Basic.lean +++ b/Physicslib4/GNS/Basic.lean @@ -34,9 +34,11 @@ here. The positivity condition is stated using the canonical partial order on `ℂ` from Mathlib: `0 ≤ z` iff `z.re ≥ 0` and `z.im = 0`. The blueprint statement of `def:cyclic-vector` only requires `A` to be an -algebra; here we keep the assumptions general enough to be reusable but -require the operator-algebra side `H →L[ℂ] H` to make sense, so `H` is a -complex Hilbert space. +algebra, but `IsCyclicVector` is declared under this file's ambient +`variable (A : Type*) [CStarAlgebra A]`, so in Lean `A` is a unital C*-algebra +(matching every call site in the project). The generality kept here is on the +`H` side: `H` is required only to be a complex Hilbert space, not fixed to the +GNS space of a particular state. -/ namespace Physicslib4 diff --git a/Physicslib4/GNS/PureStateExists.lean b/Physicslib4/GNS/PureStateExists.lean index 4133c3d..7d8d0f2 100644 --- a/Physicslib4/GNS/PureStateExists.lean +++ b/Physicslib4/GNS/PureStateExists.lean @@ -71,6 +71,7 @@ theorem isClosed_complex_nonneg : IsClosed {z : ℂ | (0 : ℂ) ≤ z} := by exact (isClosed_Ici.preimage Complex.continuous_re).inter (isClosed_singleton.preimage Complex.continuous_im) +/-- The state set is weak-* closed. -/ theorem isClosed_weakStateSet : IsClosed (weakStateSet : Set (WeakDual ℂ A)) := by have h1 : IsClosed {φ : WeakDual ℂ A | ∀ a : A, (0 : ℂ) ≤ φ (star a * a)} := by rw [Set.ofPred_forall] @@ -80,8 +81,15 @@ theorem isClosed_weakStateSet : IsClosed (weakStateSet : Set (WeakDual ℂ A)) : isClosed_singleton.preimage (WeakDual.eval_continuous 1) exact h1.inter h2 -theorem isCompact_weakStateSet [Nontrivial A] : +/-- **Banach–Alaoglu for states.** The state set is weak-* compact. -/ +theorem isCompact_weakStateSet : IsCompact (weakStateSet : Set (WeakDual ℂ A)) := by + rcases subsingleton_or_nontrivial A with hA | hA + · convert isCompact_empty + refine Set.eq_empty_of_forall_notMem fun φ hφ => ?_ + have h := hφ.2 + rw [Subsingleton.elim (1 : A) 0, map_zero] at h + exact zero_ne_one h refine (WeakDual.isCompact_closedBall (0 : StrongDual ℂ A) 1).of_isClosed_subset isClosed_weakStateSet ?_ intro φ hφ diff --git a/Physicslib4/Spacetime/CausalComplement.lean b/Physicslib4/Spacetime/CausalComplement.lean index 84507bd..2814d8e 100644 --- a/Physicslib4/Spacetime/CausalComplement.lean +++ b/Physicslib4/Spacetime/CausalComplement.lean @@ -180,6 +180,8 @@ noncomputable def causalClosure : ClosureOperator (Set M.Carrier) := def IsCausallyComplete (B : Set M.Carrier) : Prop := M.spacelikeComplement (M.spacelikeComplement B) = B +/-- A region is causally complete iff it is a closed element of the `causalClosure` +closure operator. -/ theorem isCausallyComplete_iff_isClosed {B : Set M.Carrier} : M.IsCausallyComplete B ↔ M.causalClosure.IsClosed B := by rw [ClosureOperator.isClosed_iff] @@ -331,6 +333,8 @@ private theorem coe_iInf {ι : Sort*} (B : ι → M.CausallyCompleteRegion) : rw [Set.iInf_eq_iInter] at h' exact h' +/-- The causal complement is **antitone**: a larger causally complete region has a +smaller (or equal) causal complement. -/ theorem causalComplement_antitone : Antitone M.causalComplement := by intro B₁ B₂ h rw [← Subtype.coe_le_coe] diff --git a/Physicslib4/Spacetime/CrossMetricIsometry.lean b/Physicslib4/Spacetime/CrossMetricIsometry.lean index 6d8bcc7..95085b6 100644 --- a/Physicslib4/Spacetime/CrossMetricIsometry.lean +++ b/Physicslib4/Spacetime/CrossMetricIsometry.lean @@ -562,6 +562,9 @@ theorem pullback_alexandrovBasis_image (M : Spacetime) (ψ : Diffeo M M) (M.pullbackTimeOrientation ψ t) t (pullback_preservesFutureOrientationTwoSided M ψ t) hB +-- Needed for the same reason as the `respectTransparency` options in `Pullback.lean` (see the +-- comment on `bilinearPrecomp` there): `TangentSpace M.model x` is only semireducibly, not +-- reducibly, defeq to `SpacetimeModel`. set_option backward.isDefEq.respectTransparency false in /-- **`ψ⁻¹` carries Alexandrov basis sets to pullback Alexandrov basis sets** (`lmm:pullback-alexandrov-homeomorphism`, backward matching hypothesis). diff --git a/Physicslib4/Spacetime/IsometryCausality.lean b/Physicslib4/Spacetime/IsometryCausality.lean index 160cdc8..028f024 100644 --- a/Physicslib4/Spacetime/IsometryCausality.lean +++ b/Physicslib4/Spacetime/IsometryCausality.lean @@ -199,10 +199,10 @@ noncomputable def futureOrientationPreserving (M : Spacetime) (t : M.TimeOrienta Iff.rfl /-- The **oriented identity component**: the identity-component isometries that -also preserve the future orientation. This folds orientation-preservation into -the identity-component group of Axiom 5, sidestepping the statement that every -identity-component isometry preserves orientation, which the C⁰ topology used -here does not give. -/ +also preserve the future orientation. This is the identity-component group of +Axiom 5: every identity-component isometry preserves the future orientation, but +the proof needs a Myers–Steenrod-type rigidity theorem that Mathlib does not +provide, so orientation preservation is part of the definition instead. -/ noncomputable def orientedIdentityComponent (M : Spacetime) (t : M.TimeOrientation) : Subgroup (Isometry M) := Isometry.identityComponent M ⊓ futureOrientationPreserving M t diff --git a/Physicslib4/Spacetime/LorentzCauchySchwarz.lean b/Physicslib4/Spacetime/LorentzCauchySchwarz.lean index 5e72888..0e5f7d0 100644 --- a/Physicslib4/Spacetime/LorentzCauchySchwarz.lean +++ b/Physicslib4/Spacetime/LorentzCauchySchwarz.lean @@ -13,16 +13,19 @@ import Mathlib.LinearAlgebra.BilinearForm.Basic This file formalises the blueprint declaration `lmm:reverse-cauchy-schwarz` (Chapter 10, `sections/sec10/spacetime`). -For a symmetric Lorentzian bilinear form `g` (signature `diag(-1,1,1,1)`) and two -timelike vectors `v, w` (i.e. `g v v < 0`, `g w w < 0`) the *reverse* Cauchy-Schwarz -inequality holds: +For a symmetric Lorentzian bilinear form `g` (signature `diag(-1,1,1,1)`) and a +timelike vector `w` (i.e. `g w w < 0`) the *reverse* Cauchy-Schwarz inequality +holds for every `v`: ``` g v v * g w w ≤ (g v w) ^ 2. ``` -The abstract statement `reverse_cauchy_schwarz_of_lorentzianAt` depends only on the -pointwise `LorentzianAt` condition, so it transfers verbatim to the metric `g|_p` of -any spacetime; `Spacetime.reverse_cauchy_schwarz` is that specialisation. +The abstract statement `reverse_cauchy_schwarz_of_lorentzianAt` only needs `w` to +be timelike, which is stronger than the blueprint's `lmm:reverse-cauchy-schwarz` +(stated symmetrically, with both `v` and `w` timelike); the specialisation +`Spacetime.reverse_cauchy_schwarz` keeps the blueprint's symmetric hypotheses to +match its statement. It depends only on the pointwise `LorentzianAt` condition, +so it transfers verbatim to the metric `g|_p` of any spacetime. ## Proof strategy @@ -41,13 +44,16 @@ namespace Physicslib4 open scoped BigOperators -/-- **Reverse Cauchy-Schwarz inequality** for two timelike vectors of a symmetric -Lorentzian bilinear form on a real four-dimensional vector space. -/ +/-- **Reverse Cauchy-Schwarz inequality** for a timelike vector `w` of a symmetric +Lorentzian bilinear form on a real four-dimensional vector space, and an arbitrary +vector `v`. Only `w` needs to be timelike: this is stronger than the blueprint's +`lmm:reverse-cauchy-schwarz`, which states the symmetric hypothesis `v, w` both +timelike. -/ theorem reverse_cauchy_schwarz_of_lorentzianAt {V : Type*} [AddCommGroup V] [Module ℝ V] {B : LinearMap.BilinForm ℝ V} (_hsymm : ∀ v w, B v w = B w v) (hL : LorentzianAt (fun v w => B v w)) - {v w : V} (_hv : B v v < 0) (hw : B w w < 0) : + {v w : V} (hw : B w w < 0) : B v v * B w w ≤ (B v w) ^ 2 := by obtain ⟨b, hb⟩ := hL -- Beta-reduced form of the Gram-matrix hypothesis, usable directly by `rw`. @@ -95,10 +101,13 @@ theorem reverse_cauchy_schwarz_of_lorentzianAt namespace Spacetime /-- **Reverse Cauchy-Schwarz inequality** for the metric of a spacetime: two timelike -tangent vectors at a point satisfy `g v v * g w w ≤ (g v w) ^ 2`. -/ +tangent vectors at a point satisfy `g v v * g w w ≤ (g v w) ^ 2`. The hypothesis on `v` +is kept (unused in the proof) to match the blueprint's symmetric statement of +`lmm:reverse-cauchy-schwarz`; see `reverse_cauchy_schwarz_of_lorentzianAt` for the +sharper form that only needs `w` timelike. -/ theorem reverse_cauchy_schwarz (M : Spacetime) (x : M.Carrier) {v w : TangentSpace M.model x} - (hv : M.IsTimelike v) (hw : M.IsTimelike w) : + (_hv : M.IsTimelike v) (hw : M.IsTimelike w) : M.val x v v * M.val x w w ≤ (M.val x v w) ^ 2 := by let B : LinearMap.BilinForm ℝ (TangentSpace M.model x) := LinearMap.mk₂ ℝ (fun a c => M.val x a c) @@ -109,9 +118,8 @@ theorem reverse_cauchy_schwarz (M : Spacetime) (x : M.Carrier) intro a c; rw [hBapp, hBapp]; exact M.symm x a c have hL : LorentzianAt (fun a c => B a c) := by simpa only [hBapp] using M.lorentzian x - have hv' : B v v < 0 := by rw [hBapp]; exact hv have hw' : B w w < 0 := by rw [hBapp]; exact hw - have h := reverse_cauchy_schwarz_of_lorentzianAt (B := B) hsymm hL hv' hw' + have h := reverse_cauchy_schwarz_of_lorentzianAt (B := B) (v := v) hsymm hL hw' simpa only [hBapp] using h end Spacetime diff --git a/Physicslib4/Spacetime/LorentzCone.lean b/Physicslib4/Spacetime/LorentzCone.lean index f71d310..18629b1 100644 --- a/Physicslib4/Spacetime/LorentzCone.lean +++ b/Physicslib4/Spacetime/LorentzCone.lean @@ -70,7 +70,7 @@ theorem reverse_triangle_of_lorentzianAt have hc : (0 : ℝ) ≤ -(B v w) := by linarith -- Reverse Cauchy-Schwarz, rephrased for the nonnegative quantities. have hcs : B v v * B w w ≤ (B v w) ^ 2 := - reverse_cauchy_schwarz_of_lorentzianAt hsymm hL hv hw + reverse_cauchy_schwarz_of_lorentzianAt hsymm hL hw have hab : (-(B v v)) * (-(B w w)) ≤ (-(B v w)) ^ 2 := by nlinarith [hcs] -- Hence `√a · √b ≤ c`. have hprod : Real.sqrt (-(B v v)) * Real.sqrt (-(B w w)) ≤ -(B v w) := by diff --git a/Physicslib4/Spacetime/LorentzOrthogonal.lean b/Physicslib4/Spacetime/LorentzOrthogonal.lean index 7f8ba85..5059007 100644 --- a/Physicslib4/Spacetime/LorentzOrthogonal.lean +++ b/Physicslib4/Spacetime/LorentzOrthogonal.lean @@ -54,7 +54,7 @@ theorem nonneg_of_orthogonal_timelike {t u : V} (ht : B t t < 0) (h : B t u = 0) : 0 ≤ B u u := by by_contra hcon have hneg : B u u < 0 := not_le.mp hcon - have hcs := reverse_cauchy_schwarz_of_lorentzianAt hsymm hL ht hneg + have hcs := reverse_cauchy_schwarz_of_lorentzianAt (v := t) hsymm hL hneg rw [h] at hcs nlinarith [mul_pos_of_neg_of_neg ht hneg] @@ -216,8 +216,8 @@ theorem bilin_neg_of_inner_t_neg ring rw [evp, ewp, evw] at hcs exact sign_algebra ht hv hw htv htw - (reverse_cauchy_schwarz_of_lorentzianAt hsymm hL ht hv) - (reverse_cauchy_schwarz_of_lorentzianAt hsymm hL ht hw) hcs + (reverse_cauchy_schwarz_of_lorentzianAt (v := t) hsymm hL hv) + (reverse_cauchy_schwarz_of_lorentzianAt (v := t) hsymm hL hw) hcs namespace Spacetime diff --git a/Physicslib4/Spacetime/Minkowski.lean b/Physicslib4/Spacetime/Minkowski.lean index f18fac8..7e32709 100644 --- a/Physicslib4/Spacetime/Minkowski.lean +++ b/Physicslib4/Spacetime/Minkowski.lean @@ -93,6 +93,11 @@ noncomputable def minkowskiForm : -(v 0) * (w 0) + (v 1) * (w 1) + (v 2) * (w 2) + (v 3) * (w 3) := by rfl +-- Needed for the `val`/`symm`/`contMDiff` fields below: `TangentSpace 𝓘(ℝ, SpacetimeModel) x` +-- is only semireducibly, not reducibly, defeq to `SpacetimeModel`, and those fields apply +-- `minkowskiForm` directly to tangent vectors; at the default transparency the elaborator +-- cannot see through that unfolding (without the option the structure fields fail with +-- "Application type mismatch" between `TangentSpace 𝓘(ℝ, SpacetimeModel) x` and `SpacetimeModel`). set_option backward.isDefEq.respectTransparency false in /-- *Standard Minkowski spacetime* is the spacetime whose underlying real, @@ -589,22 +594,15 @@ theorem isOpen_minkowskiBackwardCone (q : SpacetimeModel) : /-! ### Minkowski spacetime -/ +-- Same reason as the option on `StandardMinkowskiSpacetime` above: the `nonvanishing` proof +-- below needs `TangentSpace 𝓘(ℝ, SpacetimeModel) x` to unify with `SpacetimeModel` at a looser +-- transparency than the default. set_option backward.isDefEq.respectTransparency false in /-- -*Minkowski spacetime* is the carrier of `StandardMinkowskiSpacetime` -viewed as a topological space under the Alexandrov topology. - -We expose `MinkowskiSpacetime` as a `def` returning a `Spacetime`, -following the convention used for `StandardMinkowskiSpacetime`. The -"underlying topology is the Alexandrov topology" character of Minkowski -spacetime is captured at the typeclass level by the alternative -`MinkowskiSpacetimeCarrier` synonym below, which carries the Alexandrov -topology directly. - -A time orientation `t` on standard Minkowski spacetime is required to -talk about the Alexandrov topology. It is the constant timelike vector -field `∂_0` pointing in the `0`-th coordinate direction, with its -non-vanishing discharged below. +The time orientation on standard Minkowski spacetime, required to talk +about the Alexandrov topology (see `MinkowskiSpacetime`). It is the +constant timelike vector field `∂_0` pointing in the `0`-th coordinate +direction, with its non-vanishing discharged below. -/ noncomputable def standardMinkowskiTimeOrientation : StandardMinkowskiSpacetime.TimeOrientation where @@ -1725,18 +1723,23 @@ lemma mfderiv_extChartAt_symm_minkowski (x₀ : MinkowskiSpacetimeCarrier) mfderivWithin_univ] at key exact key +-- Same reason as the option on `StandardMinkowskiSpacetime` above: the `val`/`symm` fields +-- below apply `minkowskiForm` to tangent vectors, which needs `TangentSpace _ x` to unify with +-- `SpacetimeModel` at a looser transparency than the default. set_option backward.isDefEq.respectTransparency false in /-- *Minkowski spacetime* is standard Minkowski spacetime re-topologised with the Alexandrov topology generated by intersections of chronological -futures and pasts. +futures and pasts. We expose it as a `def` returning a `Spacetime`, +following the convention used for `StandardMinkowskiSpacetime`. We return a `Spacetime` whose carrier is `MinkowskiSpacetimeCarrier`, whose smooth-manifold data is inherited from `StandardMinkowskiSpacetime`, and whose metric is `minkowskiForm`. The retopologisation of the carrier to the Alexandrov topology is handled by the `TopologicalSpace` instance -on `MinkowskiSpacetimeCarrier`; the smooth-manifold data is transported -along the homeomorphism `euclideanHomeoMinkowski` via the named instances +on `MinkowskiSpacetimeCarrier`, which carries the Alexandrov topology +directly; the smooth-manifold data is transported along the homeomorphism +`euclideanHomeoMinkowski` via the named instances `instChartedSpaceMinkowskiCarrier` and `instIsManifoldMinkowskiCarrier` (the content of section 2.4 of the blueprint). -/ diff --git a/Physicslib4/Spacetime/Pullback.lean b/Physicslib4/Spacetime/Pullback.lean index 0e56aa5..1f1d027 100644 --- a/Physicslib4/Spacetime/Pullback.lean +++ b/Physicslib4/Spacetime/Pullback.lean @@ -85,6 +85,9 @@ noncomputable def bilinearPrecomp (g : SpacetimeModel →L[ℝ] SpacetimeModel (A : SpacetimeModel →L[ℝ] SpacetimeModel) (v w : SpacetimeModel) : bilinearPrecomp g A v w = g (A v) (A w) := rfl +-- Needed because `TangentSpace M.model x` is only semireducibly, not reducibly, defeq to +-- `SpacetimeModel` (see the note on `bilinearPrecomp` above); the same reason applies at every +-- other `respectTransparency` option in this file. set_option backward.isDefEq.respectTransparency false in /-- **The pullback of a spacetime metric** (`def:pullback-metric`). @@ -102,6 +105,7 @@ noncomputable def pullbackVal (M : Spacetime) (ψ : Diffeo M M) (x : M.Carrier) TangentSpace M.model x →L[ℝ] TangentSpace M.model x →L[ℝ] ℝ := bilinearPrecomp (M.val (ψ x)) (mfderiv M.model M.model ψ x) +-- See the comment at `pullbackVal` above. set_option backward.isDefEq.respectTransparency false in @[simp] theorem pullbackVal_apply (M : Spacetime) (ψ : Diffeo M M) (x : M.Carrier) (v w : TangentSpace M.model x) : @@ -109,6 +113,7 @@ set_option backward.isDefEq.respectTransparency false in = M.val (ψ x) (mfderiv M.model M.model ψ x v) (mfderiv M.model M.model ψ x w) := rfl +-- See the comment at `pullbackVal` above. set_option backward.isDefEq.respectTransparency false in /-- **The pullback metric is symmetric** (`lmm:pullback-metric-symm`). -/ theorem pullbackVal_symm (M : Spacetime) (ψ : Diffeo M M) (x : M.Carrier) @@ -132,6 +137,7 @@ theorem pullbackVal_nondegenerate (M : Spacetime) (ψ : Diffeo M M) (x : M.Carri rw [← mfderivEquiv_coe ψ x] at hnd exact (ContinuousLinearEquiv.map_eq_zero_iff (e := mfderivEquiv ψ x)).mp hnd +-- See the comment at `pullbackVal` above. set_option backward.isDefEq.respectTransparency false in /-- **The pullback metric is Lorentzian** (`lmm:pullback-metric-lorentzian`): at every `x` there is a basis of `T_xM` whose Gram matrix under `(ψ^*g)_x` is @@ -150,6 +156,7 @@ theorem pullbackVal_lorentzian (M : Spacetime) (ψ : Diffeo M M) (x : M.Carrier) simpa [Module.Basis.map_apply, mfderiv_eq_mfderivEquiv, ContinuousLinearEquiv.apply_symm_apply] using hb i j +-- `respectTransparency`: see the comment at `pullbackVal` above. set_option backward.isDefEq.respectTransparency false in set_option maxHeartbeats 1000000 in -- The final `simp`/`simpa` calls in this proof unfold several layers of @@ -558,6 +565,7 @@ theorem isFuturePointing_pullback_iff_of_isNull (M : Spacetime) (ψ : Diffeo M M exact ((ContinuousLinearMap.continuous di).tendsto (mfderiv M.model M.model ψ x v)).comp hutends +-- See the comment at `pullbackVal` above. set_option backward.isDefEq.respectTransparency false in /-- **The pullback preserves the future orientation two-sidedly** diff --git a/Physicslib4/Spectral/BorelClasses.lean b/Physicslib4/Spectral/BorelClasses.lean index e1d5bfb..8e6ac4e 100644 --- a/Physicslib4/Spectral/BorelClasses.lean +++ b/Physicslib4/Spectral/BorelClasses.lean @@ -276,6 +276,9 @@ obligation. Blueprint reference: `def:L0-class`. -/ +-- `MeasurableSpace X` is unused in the body (only `TopologicalSpace X` is needed, via +-- `C(X, ℝ)`); kept for parity with the other definitions in this file (see the docstring above +-- for why the blueprint's measurability conjunct is dropped). @[nolint unusedArguments] def L0 (X : Type*) [TopologicalSpace X] [MeasurableSpace X] : Set (Set X) := {E | ∃ (f : ℕ → C(X, ℝ)) (C : ℝ), (∀ n x, ‖f n x‖ ≤ C) ∧ diff --git a/Physicslib4/Spectral/OperatorIntegral.lean b/Physicslib4/Spectral/OperatorIntegral.lean index 79e6aff..0236cec 100644 --- a/Physicslib4/Spectral/OperatorIntegral.lean +++ b/Physicslib4/Spectral/OperatorIntegral.lean @@ -47,6 +47,33 @@ open Filter Topology ContinuousLinearMap MeasureTheory variable {X : Type*} [MeasurableSpace X] variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] +/-- +The pointwise estimate behind every uniform-convergence-of-products argument below: +if `f` is bounded by `Mf`, `r` is bounded by `Mr`, and `s`/`r` approximate `f`/`g` to +within `ε / (2 * Mr)` and `ε / (2 * Mf)` respectively, then `s * r` approximates +`f * g` to within `ε`. Shared by `ProjectionValuedMeasure.integral_bddMeasurable_mul` +(diagonal case) and `tendstoUniformly_mul_of_bounded` (product-filter case). +-/ +private lemma norm_mul_sub_mul_lt {f g s r : ℂ} {Mf Mr ε : ℝ} + (hMf_pos : 0 < Mf) (hMr_pos : 0 < Mr) (hf : ‖f‖ ≤ Mf) (hr : ‖r‖ ≤ Mr) + (hs : ‖f - s‖ < ε / (2 * Mr)) (hg : ‖g - r‖ < ε / (2 * Mf)) : + ‖f * g - s * r‖ < ε := by + have hterm1 : ‖f‖ * ‖g - r‖ < ε / 2 := by + calc + ‖f‖ * ‖g - r‖ ≤ Mf * ‖g - r‖ := mul_le_mul_of_nonneg_right hf (norm_nonneg _) + _ < Mf * (ε / (2 * Mf)) := mul_lt_mul_of_pos_left hg hMf_pos + _ = ε / 2 := by field_simp + have hterm2 : ‖f - s‖ * ‖r‖ < ε / 2 := by + calc + ‖f - s‖ * ‖r‖ ≤ ‖f - s‖ * Mr := mul_le_mul_of_nonneg_left hr (norm_nonneg _) + _ < (ε / (2 * Mr)) * Mr := mul_lt_mul_of_pos_right hs hMr_pos + _ = ε / 2 := by field_simp + calc + ‖f * g - s * r‖ = ‖f * (g - r) + (f - s) * r‖ := by congr 1; ring + _ ≤ ‖f * (g - r)‖ + ‖(f - s) * r‖ := norm_add_le _ _ + _ ≤ ‖f‖ * ‖g - r‖ + ‖f - s‖ * ‖r‖ := add_le_add (norm_mul_le _ _) (norm_mul_le _ _) + _ < ε := by linarith + namespace ProjectionValuedMeasure /-! @@ -961,8 +988,7 @@ theorem norm_integral_le (μ : ProjectionValuedMeasure X H) {f : X → ℂ} rintro r ⟨χ, ψ, hχ, hψ, rfl⟩ exact hmat χ ψ hχ hψ) (by nlinarith [hset_nonneg g]) -- (II) the sharp bound for simple functions: ‖∫ s dμ‖ ≤ ‖s‖∞, cited as the - -- black-box declaration `norm_integral_simple_le` (proved by another prover in - -- the real file). + -- black-box declaration `norm_integral_simple_le`, proved above in this file. have hsimple : ∀ (s : SimpleFunc X ℂ), ‖μ.integral (fun x => s x)‖ ≤ snorm (fun x => s x) := by intro s exact le_trans (norm_integral_simple_le μ s) (by @@ -1411,8 +1437,10 @@ theorem integral_bddMeasurable_mul (μ : ProjectionValuedMeasure X H) {f g : X _ ≤ Cf * Cg := mul_le_mul (hCf x) (hCg x) (norm_nonneg _) (le_trans (norm_nonneg _) (hCf x)) -- `sᵢ rᵢ → f g` uniformly along the diagonal: the diagonal case of - -- `tendstoUniformly_mul_of_bounded` (which lives below this declaration and so - -- cannot be cited here), proved with the blueprint's ε/2-splitting. + -- `tendstoUniformly_mul_of_bounded`, proved from the shared estimate + -- `norm_mul_sub_mul_lt` (the two results cannot share a single declaration of + -- `TendstoUniformly` directly, since `tendstoUniformly_mul_of_bounded` is stated + -- for the product filter `atTop ×ˢ atTop`, not the diagonal). have hdiag : TendstoUniformly (fun n x => s n x * r n x) (f * g) atTop := by rw [Metric.tendstoUniformly_iff] intro ε hε @@ -1436,50 +1464,18 @@ theorem integral_bddMeasurable_mul (μ : ProjectionValuedMeasure X H) {f g : X hr' (ε / (2 * (Mf' + 1))) (by positivity) have hr1 : ∀ᶠ n in atTop, ∀ x, dist (g x) (r n x) < 1 := hr' 1 zero_lt_one filter_upwards [hsε, hrε, hr1] with n hsn hrn hr1n x - calc - dist ((f * g) x) (s n x * r n x) = ‖f x * g x - s n x * r n x‖ := by - simp [dist_eq_norm] - _ ≤ ‖(f x - s n x) * r n x‖ + ‖f x * (g x - r n x)‖ := by - rw [show f x * g x - s n x * r n x = (f x - s n x) * r n x + f x * (g x - r n x) by ring] - exact norm_add_le _ _ - _ ≤ ‖f x - s n x‖ * ‖r n x‖ + ‖f x‖ * ‖g x - r n x‖ := by - exact add_le_add (norm_mul_le _ _) (norm_mul_le _ _) - _ ≤ ‖f x - s n x‖ * (Mg' + 1) + Mf' * ‖g x - r n x‖ := by - have hrx : ‖r n x‖ ≤ Mg' + 1 := by - have hd : ‖g x - r n x‖ < 1 := by simpa [dist_eq_norm] using hr1n x - have hd' : ‖r n x - g x‖ ≤ 1 := by - simpa [norm_sub_rev] using le_of_lt hd - calc - ‖r n x‖ = ‖(r n x - g x) + g x‖ := by - congr 1 - ring - _ ≤ ‖r n x - g x‖ + ‖g x‖ := norm_add_le _ _ - _ ≤ Mg' + 1 := by - nlinarith [hd', hMg0 x] - exact add_le_add - (mul_le_mul_of_nonneg_left hrx (norm_nonneg _)) - (mul_le_mul_of_nonneg_right (hMf0 x) (norm_nonneg _)) - _ < ε := by - have h1 : ‖f x - s n x‖ * (Mg' + 1) < ε / 2 := by - have hd : ‖f x - s n x‖ < ε / (2 * (Mg' + 1)) := by - simpa [dist_eq_norm] using hsn x - have heq : (ε / (2 * (Mg' + 1))) * (Mg' + 1) = ε / 2 := by - field_simp [hMg'1.ne'] - exact lt_of_lt_of_eq (mul_lt_mul_of_pos_right hd hMg'1) heq - have h2 : Mf' * ‖g x - r n x‖ < ε / 2 := by - have hd : ‖g x - r n x‖ < ε / (2 * (Mf' + 1)) := by - simpa [dist_eq_norm] using hrn x - have hle : Mf' * ‖g x - r n x‖ ≤ Mf' * (ε / (2 * (Mf' + 1))) := - mul_le_mul_of_nonneg_left (le_of_lt hd) hMf' - have hlt : Mf' * (ε / (2 * (Mf' + 1))) < ε / 2 := by - calc - Mf' * (ε / (2 * (Mf' + 1))) < (Mf' + 1) * (ε / (2 * (Mf' + 1))) := - mul_lt_mul_of_pos_right (lt_add_of_pos_right Mf' one_pos) - (by positivity) - _ = ε / 2 := by - field_simp [hMf'1.ne'] - exact lt_of_le_of_lt hle hlt - nlinarith + have hrx : ‖r n x‖ ≤ Mg' + 1 := by + have hd : ‖g x - r n x‖ < 1 := by simpa [dist_eq_norm] using hr1n x + have hd' : ‖r n x - g x‖ ≤ 1 := by simpa [norm_sub_rev] using le_of_lt hd + calc + ‖r n x‖ = ‖(r n x - g x) + g x‖ := by congr 1; ring + _ ≤ ‖r n x - g x‖ + ‖g x‖ := norm_add_le _ _ + _ ≤ Mg' + 1 := by nlinarith [hd', hMg0 x] + have hlt : ‖f x * g x - s n x * r n x‖ < ε := + norm_mul_sub_mul_lt (Mf := Mf' + 1) (Mr := Mg' + 1) hMf'1 hMg'1 + (le_trans (hMf0 x) (by linarith)) hrx + (by simpa [dist_eq_norm] using hsn x) (by simpa [dist_eq_norm] using hrn x) + simpa [dist_eq_norm] using hlt -- `∫ sᵢrᵢ dμ → ∫ fg dμ` have hlim1 : Tendsto (fun n => μ.integral (fun x => s n x * r n x)) atTop (𝓝 (μ.integral (f * g))) := by @@ -1662,35 +1658,9 @@ theorem tendstoUniformly_mul_of_bounded {Y : Type*} {f g : Y → ℂ} have h1 : ‖f x - s i x‖ < ε / (2 * M_r) := hi x have h2 : ‖g x - r j x‖ < ε / (2 * M_f) := hj.1 x have h3 : ‖r j x‖ ≤ M_r := hj.2 x - have hMf_ne : M_f ≠ 0 := by linarith - have hMr_ne : M_r ≠ 0 := by linarith - have hterm1 : ‖f x‖ * ‖g x - r j x‖ < ε / 2 := by - calc - ‖f x‖ * ‖g x - r j x‖ ≤ M_f * ‖g x - r j x‖ := - mul_le_mul_of_nonneg_right (hMf x) (norm_nonneg _) - _ < M_f * (ε / (2 * M_f)) := mul_lt_mul_of_pos_left h2 hMf_pos - _ = ε / 2 := by - field_simp [hMf_ne] - have hterm2 : ‖f x - s i x‖ * ‖r j x‖ < ε / 2 := by - calc - ‖f x - s i x‖ * ‖r j x‖ ≤ ‖f x - s i x‖ * M_r := - mul_le_mul_of_nonneg_left h3 (norm_nonneg _) - _ < (ε / (2 * M_r)) * M_r := mul_lt_mul_of_pos_right h1 hMr_pos - _ = ε / 2 := by - field_simp [hMr_ne] - have hnorm : ‖(f * g) x - s i x * r j x‖ ≤ - ‖f x‖ * ‖g x - r j x‖ + ‖f x - s i x‖ * ‖r j x‖ := by - calc - ‖(f * g) x - s i x * r j x‖ = ‖f x * g x - s i x * r j x‖ := by rfl - _ = ‖f x * (g x - r j x) + (f x - s i x) * r j x‖ := by - congr 1 - ring - _ ≤ ‖f x * (g x - r j x)‖ + ‖(f x - s i x) * r j x‖ := norm_add_le _ _ - _ ≤ ‖f x‖ * ‖g x - r j x‖ + ‖f x - s i x‖ * ‖r j x‖ := by - exact add_le_add (norm_mul_le _ _) (norm_mul_le _ _) - have hsum : ‖f x‖ * ‖g x - r j x‖ + ‖f x - s i x‖ * ‖r j x‖ < ε := by - linarith - simpa [dist_eq_norm] using lt_of_le_of_lt hnorm hsum + have hlt : ‖f x * g x - s i x * r j x‖ < ε := + norm_mul_sub_mul_lt (Mf := M_f) (Mr := M_r) hMf_pos hMr_pos (hMf x) h3 h1 h2 + simpa [dist_eq_norm] using hlt end Spectral end Physicslib4 diff --git a/Physicslib4/Spectral/Unbounded/DirectSum.lean b/Physicslib4/Spectral/Unbounded/DirectSum.lean index bc3b1cb..51000d0 100644 --- a/Physicslib4/Spectral/Unbounded/DirectSum.lean +++ b/Physicslib4/Spectral/Unbounded/DirectSum.lean @@ -259,6 +259,9 @@ private theorem single_mem_range_subSmul [DecidableEq ι] (hA : ∀ i, IsSelfAdj obtain ⟨ξ, hξ⟩ := exists_sub_smul_eq_of_isSelfAdjoint (hA i) hc η refine mem_range.mpr ⟨⟨lp.single 2 i ξ, hT.memDomain _ (finite_support_single i ξ)⟩, ?_⟩ rw [subSmul_apply] + -- `rw` fails here: the domain membership proof `hT.memDomain _ _` in the LHS's subtype + -- element is only semireducibly, not reducibly, defeq to the membership proof + -- `(subSmul T c).domain` expects, so plain `rw` cannot match the pattern under the subtype. erw [apply_single_of_isDirectSumOperator hT i ξ] rw [← hξ, ← lp.single_smul, ← lp.single_sub] @@ -291,7 +294,7 @@ private theorem sum_single_mem_graph [DecidableEq ι] (hT : IsDirectSumOperator (∑ i ∈ s, lp.single 2 i (η i), ∑ i ∈ s, lp.single 2 i (A i (η i))) ∈ T.graph := by have h := T.graph.sum_mem (t := s) fun i _ => by have := T.mem_graph ⟨lp.single 2 i (η i), hT.memDomain _ (finite_support_single i (η i))⟩ - erw [apply_single_of_isDirectSumOperator hT i (η i)] at this + rw [apply_single_of_isDirectSumOperator hT i (η i)] at this exact this convert h using 1 ext <;> simp [Prod.fst_sum, Prod.snd_sum] @@ -464,6 +467,9 @@ private theorem summand_mem_range_subSmul (hA : ∀ n, IsSelfAdjoint (A n)) obtain ⟨ξ, hξ⟩ := exists_sub_smul_eq_of_isSelfAdjoint (hA m) hc η refine mem_range.mpr ⟨⟨ξ, mem_domain_of_mem_summand hT m ξ⟩, ?_⟩ rw [subSmul_apply] + -- `rw` fails here for the same reason as `single_mem_range_subSmul` above: the subtype's + -- domain-membership proof is only semireducibly defeq across `T.domain` and + -- `(subSmul T c).domain`. erw [apply_of_mem_summand hT m ξ] simp [← hξ] diff --git a/blueprint/src/sections/sec10/gns-construction-details.tex b/blueprint/src/sections/sec10/gns-construction-details.tex index 2ca3d55..a9028f4 100644 --- a/blueprint/src/sections/sec10/gns-construction-details.tex +++ b/blueprint/src/sections/sec10/gns-construction-details.tex @@ -321,6 +321,8 @@ \subsection{GNS Construction Theorem} \pi_\omega(a)' = U\pi_\omega(a)U^{-1}, \end{align} and two GNS triples associated to $(\mathfrak{U}, \omega)$ can differ at most by a unitary transformation, completing our proof of the GNS Construction Theorem. + +\noindent\textit{Formalization note.} The Lean proof of existence (\texttt{gns\_construction}) does not follow the construction above. It takes the Hilbert space to be Mathlib's GNS space of the positive linear functional $\omega$ (\texttt{PositiveLinearMap.GNS}, the completion of the quotient \texttt{PositiveLinearMap.PreGNS}), together with Mathlib's left-multiplication representation on it. Lemmas~\ref{lmm:cauchy-schwarz-inequality}, \ref{lmm:lmm1} and~\ref{lmm:lmm2} are formalized separately and are not used there. \end{proof} \subsection{Auxiliary Results Used in the Proof} diff --git a/blueprint/src/sections/sec10/haag-kastler-axioms-in-curved-spacetime.tex b/blueprint/src/sections/sec10/haag-kastler-axioms-in-curved-spacetime.tex index a5a10fc..83115bd 100644 --- a/blueprint/src/sections/sec10/haag-kastler-axioms-in-curved-spacetime.tex +++ b/blueprint/src/sections/sec10/haag-kastler-axioms-in-curved-spacetime.tex @@ -70,11 +70,11 @@ \section{Haag Kastler Axioms in Curved Spacetime}\label{sctn:haag-kastler-axioms \begin{definition}[Local Observable] \label{def:local-observable} - \lean{Physicslib4.AQFT.HaagKastlerCurved.IsLocalObservable} + \lean{Physicslib4.AQFT.HaagKastlerCurved.IsLocalObservable, Physicslib4.AQFT.HaagKastlerCurved.exists_isLocalObservable} \leanfile{Physicslib4/AQFT/HaagKastlerCurved/LocalAlgebra.lean} \leanok \uses{def:lorentzian-spacetime, thrm:gns-construction-theorem, def:local-algebras-in-curved-spacetime, def:state} - For Lorentzian spacetime $M$ the image $\pi_\omega(a)$ of a self-adjoint member $a$ of the local algebra $\mathfrak{U}(\mathbf{B})$ under the GNS *-homomorphism $\pi_\omega$ of a state $\omega$ on $\mathfrak{U}(\mathbf{B})$ is self-adjoint and thus corresponds to an ``observable''. Any ``observable'' corresponding to such a self-adjoint $\pi_\omega(a)$ is called a \textit{local observable}. + For Lorentzian spacetime $M$ the image $\pi_\omega(a)$ of a self-adjoint member $a$ of the local algebra $\mathfrak{U}(\mathbf{B})$ under the GNS *-homomorphism $\pi_\omega$ of a state $\omega$ on $\mathfrak{U}(\mathbf{B})$ is self-adjoint and thus corresponds to an ``observable''. Any ``observable'' corresponding to such a self-adjoint $\pi_\omega(a)$ is called a \textit{local observable}. The Lean records the existence content of this definition as \texttt{exists\_isLocalObservable}: for every state $\omega$ on $\mathfrak{U}(\mathbf{B})$ and every self-adjoint $a \in \mathfrak{U}(\mathbf{B})$, the GNS construction gives a $*$-representation $\pi$ in which $\pi(a)$ is a self-adjoint local observable. \end{definition} and can be stated as follows: @@ -123,6 +123,8 @@ \section{Haag Kastler Axioms in Curved Spacetime}\label{sctn:haag-kastler-axioms \end{center} commutes. + + \emph{Formalization note.} The isometry group carries the topology induced by $\varphi \mapsto (\varphi, \varphi^{-1})$ into $C(M,M) \times C(M,M)$ with the compact-open topology. The Lean takes the group of Axiom 5 to be the identity component of this group intersected with the isometries preserving the future orientation (\texttt{orientedIdentityComponent}). This is the group of isometries connected to the identity: an isometry connected to the identity preserves the future orientation, but the proof needs a Myers--Steenrod-type rigidity theorem, which Mathlib does not provide, so the Lean includes orientation preservation in the definition rather than proving it. \end{definition} \begin{definition}[Haag-Kastler Net in Curved Spacetime] @@ -134,6 +136,19 @@ \section{Haag Kastler Axioms in Curved Spacetime}\label{sctn:haag-kastler-axioms A \emph{Haag-Kastler net in curved spacetime} on a Lorentzian spacetime is the bundling of the data of \ref{def:local-algebras-in-curved-spacetime} together with the properties of \ref{def:isotony-in-curved-spacetime}, \ref{def:local-commutativity-in-curved-spacetime}, \ref{def:local-completeness-in-curved-spacetime}, and \ref{def:isometric-covariance-in-curved-spacetime}. In the Lean formalization this is a single structure whose fields are the assignment $\mathbf{B} \mapsto \mathfrak{U}(\mathbf{B})$ and proofs that this assignment satisfies the four remaining axioms. Theorems about AQFT in curved spacetime take an instance of this structure as a hypothesis and invoke each axiom as a projection. \end{definition} +\begin{theorem}[The Curved Haag-Kastler Axioms are Jointly Satisfiable] + \label{thrm:haag-kastler-net-in-curved-spacetime-nonempty} + \lean{Physicslib4.Spacetime.LorentzianSpacetime.nonempty_haagKastlerNet_identityComponent, Physicslib4.Spacetime.LorentzianSpacetime.isometricCovariance_trivial_identityComponent, Physicslib4.Spacetime.LorentzianSpacetime.identityComponent_faithfulSMul} + \uses{def:lorentzian-spacetime, def:haag-kastler-net-in-curved-spacetime, def:isometric-covariance-in-curved-spacetime} + \leanok + Let $M$ be a Lorentzian spacetime, with the group of Axiom 5 taken to be the isometries of $M$ connected to the identity (as in the formalization note of \ref{def:isometric-covariance-in-curved-spacetime}). The identity component of the isometry group acts faithfully on $M$: an isometry in it is determined by its action on points. So does its orientation-preserving subgroup, the group of Axiom 5. The trivial net, which assigns the C*-algebra $\mathbb{C}$ to every basis set and the identity map to every inclusion and every isometry, satisfies isometric covariance (\ref{def:isometric-covariance-in-curved-spacetime}). Moreover it satisfies all five axioms, so a Haag-Kastler net in curved spacetime (\ref{def:haag-kastler-net-in-curved-spacetime}) over $M$ exists. Existence holds for every abstract Lorentzian spacetime of the interface (\texttt{HaagKastlerCurved.nonempty\_haagKastlerNet}), and the concrete form specializes it. +\end{theorem} +\begin{proof} + \leanok + \uses{def:haag-kastler-net-in-curved-spacetime, def:isometric-covariance-in-curved-spacetime} + Faithfulness is inherited from the faithful action of the full isometry group on $M$, since faithfulness passes to subgroups: first to the identity component, then to its orientation-preserving subgroup. For the trivial net each local algebra is $\mathbb{C}$ and every map is the identity of $\mathbb{C}$, so the laws $\alpha_{\mathbf{1}} = \mathrm{id}$ and $\alpha_{\varphi'\circ\varphi} = \alpha_{\varphi'}\alpha_\varphi$ and the commuting square of Axiom 5 hold trivially; isotony holds because the identity is functorial, local commutativity because $\mathbb{C}$ is commutative, and local completeness because a self-adjoint $\pi(b)$ is also the image $\pi\big(\tfrac12(b + b^*)\big)$ of the self-adjoint part of $b$. The Lean proves this for an arbitrary abstract Lorentzian spacetime (\texttt{HaagKastlerCurved.nonempty\_haagKastlerNet}) and specializes it to the concrete one. +\end{proof} + Generally these axioms follow in a straightforward manner from those of AQFT in Minkowski spacetime. The only ``surprise'' in this presentation is the absence of a quasilocal algebra. However, as we found, its absence is simply a reflection of the observational constraints of Lorentzian spacetime which don't exist in Minkowski spacetime. \subsection{Einstein Causality in Curved Spacetime} @@ -154,6 +169,19 @@ \subsection{Einstein Causality in Curved Spacetime} This is the image under $\pi$ of the curved local commutativity (\ref{def:local-commutativity-in-curved-spacetime}). It is the curved-spacetime counterpart of \ref{thrm:einstein-causality}, with the containing local algebra playing the role of the quasilocal algebra. \end{proof} +\begin{lemma}[Local Commutativity for Subregions (Concrete Spacetime)] + \label{lmm:local-commutativity-mono-in-curved-spacetime} + \lean{Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.commute_of_spacelike_mono_geometric} + \uses{def:lorentzian-spacetime, def:haag-kastler-net-in-curved-spacetime, def:completely-spacelike, def:isotony-in-curved-spacetime} + \leanok + Let $L$ be a Lorentzian spacetime whose group of Axiom 5 is the isometries connected to the identity, and $\mathfrak{U}$ a Haag-Kastler net over it. Let $\mathbf{B}_1, \mathbf{B}_2 \subseteq \mathbf{B}$ be completely spacelike-separated regions, $\mathbf{B}$ a basis set, and $\mathbf{B}_1' \subseteq \mathbf{B}_1$, $\mathbf{B}_2' \subseteq \mathbf{B}_2$ basis sets. Then for all $a \in \mathfrak{U}(\mathbf{B}_1')$ and $b \in \mathfrak{U}(\mathbf{B}_2')$ the isotony images $i(a), i(b) \in \mathfrak{U}(\mathbf{B})$ commute. +\end{lemma} +\begin{proof} + \leanok + \uses{def:local-commutativity-in-curved-spacetime, def:completely-spacelike} + Complete spacelike separation is inherited by subsets: every point of $\mathbf{B}_1'$ is spacelike to every point of $\mathbf{B}_2'$ because the same holds for $\mathbf{B}_1$ and $\mathbf{B}_2$ (\ref{def:completely-spacelike}). So $\mathbf{B}_1'$ and $\mathbf{B}_2'$ are completely spacelike basis sets inside $\mathbf{B}$, and local commutativity (\ref{def:local-commutativity-in-curved-spacetime}) gives $i(a)\,i(b) = i(b)\,i(a)$. +\end{proof} + \subsection{Local von Neumann Algebras in Curved Spacetime} The von Neumann net carries over to curved spacetime, again relative to a containing basis algebra $\mathfrak{U}(\mathbf{B})$. Given a $*$-representation $\pi$ of $\mathfrak{U}(\mathbf{B})$ on a Hilbert space $H$, the local von Neumann algebra of a subregion $\mathbf{B}' \subseteq \mathbf{B}$ is the bicommutant $R(\mathbf{B}') = \pi(\mathfrak{U}(\mathbf{B}'))''$, where $\mathfrak{U}(\mathbf{B}')$ is embedded into $\mathfrak{U}(\mathbf{B})$ by the isotony witness of Axiom 2. @@ -272,13 +300,13 @@ \subsection{Local von Neumann Algebras in Curved Spacetime} \begin{theorem}[Geometric Covariance of the von Neumann Net (Curved Spacetime)] \label{thrm:von-neumann-geometric-covariance-in-curved-spacetime} - \lean{Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localVonNeumann, Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localOperators, Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAutHom_commIsotony} + \lean{Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localVonNeumann, Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localOperators, Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAutHom_commIsotony, Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localVonNeumannAlgebra} \leanfile{Physicslib4/AQFT/HaagKastlerCurved/GeometricCovariance.lean} - \uses{def:local-von-neumann-in-curved-spacetime, def:stabilizer-action-in-curved-spacetime, thrm:gns-unitary-stabilizer-in-curved-spacetime} + \uses{def:local-von-neumann-in-curved-spacetime, def:local-von-neumann-algebra-in-curved-spacetime, def:stabilizer-action-in-curved-spacetime, thrm:gns-unitary-stabilizer-in-curved-spacetime} \leanok The curved, stabilizer-subgroup analogue of geometric covariance. For $g \in \mathrm{Stab}(\mathbf{B})$, the implementing unitary $U(g)$ of the stabilizer GNS representation (\ref{thrm:gns-unitary-stabilizer-in-curved-spacetime}) conjugates the local von Neumann algebra of a subregion $\mathbf{B}_1 \subseteq \mathbf{B}$ onto that of $g \cdot \mathbf{B}_1$: \[ U(g)\, R(\mathbf{B}_1)\, U(g)^{-1} = R(g \cdot \mathbf{B}_1). \] - That $g \cdot \mathbf{B}_1$ is again a basis set is not a hypothesis: the abstract \texttt{LorentzianSpacetime} interface asserts that isometries preserve basis sets, which for a concrete Lorentzian spacetime with the isometries connected to the identity is \ref{lmm:axiom5-basis-preservation}. The inclusion $g \cdot \mathbf{B}_1 \subseteq \mathbf{B}$ is not a hypothesis: it follows from $\mathbf{B}_1 \subseteq \mathbf{B}$ and $g \cdot \mathbf{B} = \mathbf{B}$. The coherence relating the stabilizer action $\hat\alpha_g$ to the isotony embeddings of Axiom 2 (\ref{def:isotony-in-curved-spacetime}) is not a hypothesis: it is condition (3) of Axiom 5 (\ref{def:isometric-covariance-in-curved-spacetime}), transported along $g \cdot \mathbf{B} = \mathbf{B}$. + That $g \cdot \mathbf{B}_1$ is again a basis set is not a hypothesis: the abstract \texttt{LorentzianSpacetime} interface asserts that isometries preserve basis sets, which for a concrete Lorentzian spacetime with the isometries connected to the identity is \ref{lmm:axiom5-basis-preservation}. The inclusion $g \cdot \mathbf{B}_1 \subseteq \mathbf{B}$ is not a hypothesis: it follows from $\mathbf{B}_1 \subseteq \mathbf{B}$ and $g \cdot \mathbf{B} = \mathbf{B}$. The coherence relating the stabilizer action $\hat\alpha_g$ to the isotony embeddings of Axiom 2 (\ref{def:isotony-in-curved-spacetime}) is not a hypothesis: it is condition (3) of Axiom 5 (\ref{def:isometric-covariance-in-curved-spacetime}), transported along $g \cdot \mathbf{B} = \mathbf{B}$. The same identity holds for the bundled von Neumann algebras of \ref{def:local-von-neumann-algebra-in-curved-spacetime} (\texttt{lieConj\_image\_localVonNeumannAlgebra}), whose underlying sets are $R(\mathbf{B}_1)$ and $R(g \cdot \mathbf{B}_1)$. \end{theorem} \begin{proof} \leanok @@ -554,11 +582,13 @@ \subsection{The Stabilizer GNS Unitary in Curved Spacetime} \begin{theorem}[GNS Unitary Representation of the Stabilizer] \label{thrm:gns-unitary-stabilizer-in-curved-spacetime} - \lean{Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer} + \lean{Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer, Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitaryRep_stabilizer} \leanfile{Physicslib4/AQFT/HaagKastlerCurved/StabilizerAction.lean} \uses{def:stabilizer-action-in-curved-spacetime, def:state, thrm:gns-construction-theorem} \leanok Let $\omega$ be a state on $\mathfrak{U}(\mathbf{B})$ invariant under the stabilizer action, $\omega(\hat\alpha_\varphi a) = \omega(a)$ for all $\varphi \in \mathrm{Stab}(\mathbf{B})$. Then the action is implemented on the GNS Hilbert space of $\omega$ by a unitary representation $U$ of $\mathrm{Stab}(\mathbf{B})$: there is a GNS triple $(H, \pi, \Omega)$ and unitaries $U(\varphi)$ with $U(\varphi)\,\pi(a)\Omega = \pi(\hat\alpha_\varphi a)\,\Omega$, $U(\varphi)\Omega = \Omega$, the group laws, and $U(1) = \mathrm{id}$. + + The Lean form is stronger: $U$ also satisfies operator covariance $U(\varphi)\,\pi(a)\,U(\varphi)^{-1} = \pi(\hat\alpha_\varphi a)$, and $\Omega$ is cyclic for $\pi$. The bundled form \texttt{exists\_gns\_unitaryRep\_stabilizer} packages $U$ as a group homomorphism from $\mathrm{Stab}(\mathbf{B})$ into the unitary group of $H$, with the same reproduction, implementation, invariance, covariance and cyclicity clauses. \end{theorem} \begin{proof} \leanok @@ -568,16 +598,16 @@ \subsection{The Stabilizer GNS Unitary in Curved Spacetime} \begin{theorem}[Strongly Continuous Stabilizer GNS Unitary] \label{thrm:gns-unitary-stabilizer-strongly-continuous-in-curved-spacetime} - \lean{Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer_strongContinuous} + \lean{Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer_strongContinuous, Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitaryRep_stabilizer_strongContinuous} \leanfile{Physicslib4/AQFT/HaagKastlerCurved/StabilizerAction.lean} \uses{thrm:gns-unitary-stabilizer-in-curved-spacetime} \leanok - If moreover the isometry group carries a topology (the abstract interface supplies none, so it is an added hypothesis) and the matrix coefficients $\varphi \mapsto \omega\big(a^*\,\hat\alpha_\varphi b\big)$ are continuous on $\mathrm{Stab}(\mathbf{B})$, then the representation $U$ is strongly continuous: $\varphi \mapsto U(\varphi)\psi$ is continuous for every GNS vector $\psi$. + If moreover the isometry group carries a topology (the abstract interface supplies none, so it is an added hypothesis) and the matrix coefficients $\varphi \mapsto \omega\big(a^*\,\hat\alpha_\varphi b\big)$ are continuous on $\mathrm{Stab}(\mathbf{B})$, then the representation $U$ is strongly continuous: $\varphi \mapsto U(\varphi)\psi$ is continuous for every GNS vector $\psi$. The bundled form \texttt{exists\_gns\_unitaryRep\_stabilizer\_strongContinuous} returns the same strongly continuous $U$ as a group homomorphism from $\mathrm{Stab}(\mathbf{B})$ into the unitary group of $H$, with the same reproduction, implementation, invariance and operator covariance $U(\varphi)\,\pi(a)\,U(\varphi)^{-1} = \pi(\hat\alpha_\varphi a)$ clauses. \end{theorem} \begin{proof} \leanok - \uses{thrm:gns-unitary-stabilizer-in-curved-spacetime} - The stabilizer subgroup inherits its topology as a subspace of the isometry group. + \uses{thrm:gns-unitary-stabilizer-in-curved-spacetime, lmm:weak-continuity-implies-strong, def:cyclic-vector} + The stabilizer subgroup inherits its topology as a subspace of the isometry group. On the dense set of vectors $\pi(a)\Omega$ (dense since $\Omega$ is cyclic) the matrix coefficients $\langle \pi(a)\Omega, U(\varphi)\pi(b)\Omega\rangle = \omega(a^*\,\hat\alpha_\varphi b)$ are continuous, so the unitaries are weakly continuous there and hence strongly continuous by \ref{lmm:weak-continuity-implies-strong}. \end{proof} \begin{theorem}[Irreducible Covariant Representation of a Pure Invariant State (Curved Spacetime)] diff --git a/blueprint/src/sections/sec10/haag-kastler-axioms.tex b/blueprint/src/sections/sec10/haag-kastler-axioms.tex index e8b09bc..4fbda78 100644 --- a/blueprint/src/sections/sec10/haag-kastler-axioms.tex +++ b/blueprint/src/sections/sec10/haag-kastler-axioms.tex @@ -355,7 +355,7 @@ \subsection{The Quasilocal Colimit} The completeness hypothesis is hidden in the word ``C*-algebra'' and is essential: a $*$-algebra can carry more than one C*-norm, so what could be unique is only the \emph{complete} one. The counterexample is the group $*$-algebra $\mathbb{C}[F_2]$ of the free group on two generators, which carries the distinct full and reduced C*-norms, neither of them complete. -This is the Bratteli--Robinson 2.2.6 material that Chapter 6.2 discusses informally. Uniqueness is not a mere technicality there: it is what would make the quasilocal algebra well defined \emph{up to isomorphism} rather than merely existing, since without it the completion could depend on a choice of norm and different choices could give non-isomorphic C*-algebras. What \ref{lmm:quasilocal-completion-cstar} establishes is existence; uniqueness up to isomorphism is the unclaimed statement discussed above. +This is the Bratteli--Robinson 2.2.6 material that Section~\ref{sctn:completion-of-the-set-theoretic-union} discusses informally. Uniqueness is not a mere technicality there: it is what would make the quasilocal algebra well defined \emph{up to isomorphism} rather than merely existing, since without it the completion could depend on a choice of norm and different choices could give non-isomorphic C*-algebras. What \ref{lmm:quasilocal-completion-cstar} establishes is existence; uniqueness up to isomorphism is the unclaimed statement discussed above. \emph{Design note: the quasilocal algebra is constructed, by colimit then completion.} The project constructs the quasilocal algebra \emph{from the net alone} --- the directed colimit of the local C*-algebras along the isotony family, followed by completion --- which is exactly the chain running from \ref{def:completion-standing-hypotheses} through \ref{lmm:quasilocal-completion-cstar}. The reason is physical, as the next paragraphs explain. @@ -449,11 +449,13 @@ \subsection{The Quasilocal Colimit} \begin{definition}[Quasilocal Observable] \label{def:quasilocal-observable} - \lean{Physicslib4.AQFT.HaagKastler.IsQuasilocalObservable} + \lean{Physicslib4.AQFT.HaagKastler.IsQuasilocalObservable, Physicslib4.AQFT.HaagKastler.exists_isQuasilocalObservable} \leanfile{Physicslib4/AQFT/HaagKastler/QuasilocalObservable.lean} \leanok \uses{def:quasilocal-algebra, thrm:gns-construction-theorem, def:state} The image $\pi_\omega(a)$ of a self-adjoint member $a$ of the quasilocal algebra $\mathfrak{U}$ under a GNS *-homomorphism $\pi_\omega$ is self-adjoint and thus corresponds to an ``observable''. Any ``observable'' corresponding to such a self-adjoint $\pi_\omega(a)$ is called a \textit{quasilocal observable}. + + The existence content of this definition is \texttt{exists\_isQuasilocalObservable}: for every state $\omega$ on $\mathfrak{U}$ and every self-adjoint $a \in \mathfrak{U}$, the GNS construction (\ref{thrm:gns-construction-theorem}) supplies a Hilbert space $H$ and a $*$-representation $\pi$ of $\mathfrak{U}$ on $H$ in which $\pi(a)$ is a quasilocal observable and is self-adjoint. \end{definition} \begin{definition}[Axiom 4: Quasilocal Completeness] @@ -712,7 +714,7 @@ \subsection{The von Neumann Density Theorem} Unitality is the hypothesis that does the work and it is stated directly rather than derived from a stronger one. It is essential, not decorative: for the zero representation $\pi = 0$ on a nonzero $H$ the image $\pi(\mathfrak{U}) = \{0\}$ is already strongly closed while $\pi(\mathfrak{U})'' = \mathbb{C}\cdot 1$, so density fails outright. The case of interest here is the GNS representation $\pi_\omega$ attached to a state $\omega$ (\ref{def:state}, \ref{thrm:gns-construction-theorem}), which is unital because $\mathfrak{U}$ is a unital C*-algebra and $\omega$ a state; but the theorem holds for any unital $*$-representation, and stating it for the GNS one only would be assuming more than the density theorem needs. Below, $\pi_\omega$ is written wherever the GNS case is the one being discussed. - \emph{Its role: this is what makes the bridge principle tenable.} The observables of a region are read off from the local von Neumann algebra $R(\mathbf{B}) = \pi_\omega(\mathfrak{U}(\mathbf{B}))''$ of \ref{def:local-von-neumann}, and a bicommutant is in general strictly larger than the image it is formed from. The statement above is about the global bicommutant $\pi_\omega(\mathfrak{U})''$, and it covers the local ones because $\pi_\omega(\mathfrak{U}(\mathbf{B})) \subseteq \pi_\omega(\mathfrak{U})$ and taking commutants reverses inclusions twice over, so $R(\mathbf{B}) \subseteq \pi_\omega(\mathfrak{U})''$ for every $\mathbf{B}$. Taken naively, the strictness looks like a refutation of Axiom 4 (\ref{def:quasilocal-completeness}): the bicommutant would contain self-adjoint operators that are not quasilocal observables, hence physical observables outside the quasilocal ones. This node is what blocks that reading. Every element of the bicommutant is approximated in the strong topology by elements of $\pi_\omega(\mathfrak{U})$, and strong convergence \emph{implies} convergence of all the expectation values $\langle \psi, T \psi \rangle$ that a measurement can return --- the implication runs this way and not the other, since the strong topology is strictly finer than the weak one in which those expectation values converge --- so no measurement of finite precision can distinguish an element of the bicommutant from a quasilocal observable close to it. The extra elements are therefore not new measurable quantities, and their existence does not falsify the identification of physical observables with quasilocal ones; it refines it. This is the content that Chapter 6.3 argues informally, citing Murphy Lemma 4.1.4: a unital $*$-subalgebra of $\mathcal{B}(H)$ containing the identity is strongly dense in its bicommutant. + \emph{Its role: this is what makes the bridge principle tenable.} The observables of a region are read off from the local von Neumann algebra $R(\mathbf{B}) = \pi_\omega(\mathfrak{U}(\mathbf{B}))''$ of \ref{def:local-von-neumann}, and a bicommutant is in general strictly larger than the image it is formed from. The statement above is about the global bicommutant $\pi_\omega(\mathfrak{U})''$, and it covers the local ones because $\pi_\omega(\mathfrak{U}(\mathbf{B})) \subseteq \pi_\omega(\mathfrak{U})$ and taking commutants reverses inclusions twice over, so $R(\mathbf{B}) \subseteq \pi_\omega(\mathfrak{U})''$ for every $\mathbf{B}$. Taken naively, the strictness looks like a refutation of Axiom 4 (\ref{def:quasilocal-completeness}): the bicommutant would contain self-adjoint operators that are not quasilocal observables, hence physical observables outside the quasilocal ones. This node is what blocks that reading. Every element of the bicommutant is approximated in the strong topology by elements of $\pi_\omega(\mathfrak{U})$, and strong convergence \emph{implies} convergence of all the expectation values $\langle \psi, T \psi \rangle$ that a measurement can return --- the implication runs this way and not the other, since the strong topology is strictly finer than the weak one in which those expectation values converge --- so no measurement of finite precision can distinguish an element of the bicommutant from a quasilocal observable close to it. The extra elements are therefore not new measurable quantities, and their existence does not falsify the identification of physical observables with quasilocal ones; it refines it. This is the content that Section~\ref{sctn:all-observables-of-interest} argues informally, citing Murphy Lemma 4.1.4: a unital $*$-subalgebra of $\mathcal{B}(H)$ containing the identity is strongly dense in its bicommutant. One refinement of the argument is worth recording, since it is easy to overstate. The approximants furnished by the density theorem are elements of $\pi_\omega(\mathfrak{U})$, but they are not guaranteed to be \emph{self-adjoint}, so the reading ``every self-adjoint element of the bicommutant is a strong limit of self-adjoint quasilocal observables'' is a strictly stronger claim than the one asserted here. That stronger claim is the Kaplansky density theorem, which additionally preserves self-adjointness and norm bounds. Nothing in the tenability argument above needs it --- indistinguishability by expectation values only requires approximants at all --- but it is the result to invoke if the self-adjoint form is ever wanted. \end{theorem} @@ -1174,11 +1176,11 @@ \subsection{Irreducibility and Schur's Lemma} \begin{theorem}[An Irreducible Representation Generates a Factor] \label{thrm:irreducible-factor} - \lean{Physicslib4.GNS.center_gnsVonNeumann_eq_of_isIrreducible} + \lean{Physicslib4.GNS.center_gnsVonNeumann_eq_of_isIrreducible, Physicslib4.GNS.isFactor_gnsVonNeumann_of_isIrreducible} \leanfile{Physicslib4/GNS/Irreducibility.lean} \uses{def:irreducible-representation, def:local-von-neumann} \leanok - The von Neumann algebra $\pi(A)''$ generated by an irreducible representation has \emph{trivial center}: an operator lies in the center $\pi(A)'' \cap (\pi(A)'')'$ if and only if it is a scalar multiple of the identity. + The von Neumann algebra $\pi(A)''$ generated by an irreducible representation has \emph{trivial center}: an operator lies in the center $\pi(A)'' \cap (\pi(A)'')'$ if and only if it is a scalar multiple of the identity. In the factor vocabulary this says that $\pi(A)''$ is a \emph{factor}, a von Neumann algebra whose center is exactly the scalars; that restatement is \texttt{isFactor\_gnsVonNeumann\_of\_isIrreducible}. \end{theorem} \begin{proof} \leanok @@ -1882,16 +1884,48 @@ \subsection{Covariant States and the Covariance Action} \begin{theorem}[GNS Unitary Implementation of an Invariant State] \label{thrm:invariant-state-gns-unitary} - \lean{Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_unitary} + \lean{Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_unitary, Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_unitaryRep} \leanfile{Physicslib4/AQFT/HaagKastler/QuasilocalIntertwiner.lean} \uses{def:invariant-state, thrm:gns-construction-theorem} \leanok For an invariant state $\omega$, the covariance action is implemented on the GNS Hilbert space by a family of unitaries $U(L)$ satisfying $U(L)\,\pi(a)\Omega = \pi(\beta_L a)\,\Omega$ and $U(L)\Omega = \Omega$. + + The Lean form is stronger: $U$ is a unitary representation of the inhomogeneous Lorentz group, $U(L'L) = U(L')\,U(L)$ and $U(1) = \mathrm{id}$, it satisfies operator covariance $U(L)\,\pi(a)\,U(L)^{-1} = \pi(\beta_L a)$, and $\Omega$ is cyclic for $\pi$. The bundled form \texttt{exists\_gns\_unitaryRep} packages $U$ as a group homomorphism from the inhomogeneous Lorentz group into the unitary group of $H$, with the same reproduction, implementation, invariance, covariance and cyclicity clauses. \end{theorem} \begin{proof} \leanok \uses{def:invariant-state, lmm:quasilocal-action-coherence} - The unitaries are obtained by extending the densely-defined isometry $\pi(a)\Omega \mapsto \pi(\beta_L a)\Omega$ - isometric because $\omega$ preserves the GNS inner product $\langle \pi(a)\Omega, \pi(b)\Omega\rangle = \omega(a^* b)$ - to the whole GNS space. + The unitaries are obtained by extending the densely-defined isometry $\pi(a)\Omega \mapsto \pi(\beta_L a)\Omega$ - isometric because $\omega$ preserves the GNS inner product $\langle \pi(a)\Omega, \pi(b)\Omega\rangle = \omega(a^* b)$ - to the whole GNS space. The group laws and operator covariance follow from the coherence of the covariance action (\ref{lmm:quasilocal-action-coherence}). +\end{proof} + +\begin{lemma}[Weak Continuity Implies Strong Continuity] + \label{lmm:weak-continuity-implies-strong} + \lean{Physicslib4.strongContinuous_of_weak} + \uses{def:unitary-operator} + \leanok + Let $X$ be a topological space, $H$ a complex inner product space, and $x \mapsto U(x)$ a family of unitary operators on $H$ (\ref{def:unitary-operator}) indexed by $X$. Suppose $D \subseteq H$ is dense and the matrix coefficients $x \mapsto \langle \xi, U(x)\eta \rangle$ are continuous for all $\xi, \eta \in D$. Then $x \mapsto U(x)\psi$ is continuous for every $\psi \in H$. +\end{lemma} +\begin{proof} + \leanok + \uses{def:unitary-operator} + Since $\|U(x)\eta\| = \|\eta\|$, the bound $|\langle \xi - \xi', U(x)\eta\rangle| \le \|\xi - \xi'\|\,\|\eta\|$ is uniform in $x$, so approximating $\xi \in H$ by $\xi' \in D$ shows that $x \mapsto \langle \xi, U(x)\eta\rangle$ is continuous for every $\xi \in H$ and $\eta \in D$. For $\eta \in D$ and $x_0 \in X$, + \begin{align} + \|U(x)\eta - U(x_0)\eta\|^2 = 2\|\eta\|^2 - 2\,\mathrm{Re}\,\langle U(x_0)\eta, U(x)\eta\rangle, + \end{align} + which tends to $0$ as $x \to x_0$ by the previous step with $\xi = U(x_0)\eta$; so $x \mapsto U(x)\eta$ is continuous. Finally, for $\psi \in H$ and $\eta \in D$, $\|U(x)\psi - U(x)\eta\| = \|\psi - \eta\|$ for every $x$, so $x \mapsto U(x)\psi$ is a uniform limit of continuous maps, hence continuous. +\end{proof} + +\begin{theorem}[Strongly Continuous GNS Unitary of an Invariant State] + \label{thrm:invariant-state-gns-unitary-strongly-continuous} + \lean{Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_unitary_strongContinuous, Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_unitaryRep_strongContinuous} + \uses{thrm:invariant-state-gns-unitary, def:inhomogeneous-lorentz-group} + \leanok + Equip the inhomogeneous Lorentz group (\ref{def:inhomogeneous-lorentz-group}) with the topology induced by the embedding $(L, a) \mapsto (L, L^{-1}, a)$ into the product of the operator-norm topology on linear maps of $\mathbb{R}^4$ (twice) and $\mathbb{R}^4$. Let $\omega$ be an invariant state whose matrix coefficients $L \mapsto \omega(a^* \beta_L b)$ are continuous for all $a, b \in \mathfrak{U}$. Then the GNS unitary representation of \ref{thrm:invariant-state-gns-unitary} can be chosen \emph{strongly continuous}: there is a GNS triple $(H, \pi, \Omega)$ and unitaries $U(L)$ with $U(L)\,\pi(a)\Omega = \pi(\beta_L a)\Omega$, $U(L)\Omega = \Omega$, $U(L'L) = U(L')\,U(L)$, $U(1) = \mathrm{id}$, $U(L)\,\pi(a)\,U(L)^{-1} = \pi(\beta_L a)$, and $L \mapsto U(L)\psi$ continuous for every $\psi \in H$. The bundled form \texttt{exists\_gns\_unitaryRep\_strongContinuous} returns $U$ as a group homomorphism into the unitary group of $H$. +\end{theorem} +\begin{proof} + \leanok + \uses{thrm:invariant-state-gns-unitary, lmm:weak-continuity-implies-strong, def:invariant-state, def:cyclic-vector} + Take the GNS triple and the unitaries of \ref{thrm:invariant-state-gns-unitary}. The vectors $\pi(a)\Omega$ are dense because $\Omega$ is cyclic, and on them the matrix coefficients are $\langle \pi(a)\Omega, U(L)\pi(b)\Omega\rangle = \langle \pi(a)\Omega, \pi(\beta_L b)\Omega\rangle = \omega(a^* \beta_L b)$, continuous in $L$ by hypothesis. Lemma~\ref{lmm:weak-continuity-implies-strong} with $D = \{\pi(a)\Omega : a \in \mathfrak{U}\}$ gives strong continuity. \end{proof} Combining invariance with purity makes the GNS representation both covariant and irreducible. We emphasise that this is a \emph{precursor} to a vacuum representation, not a vacuum itself: a genuine vacuum additionally requires the spectrum condition (positivity of the energy-momentum spectrum), which is not imposed at this point. It is expressed below through the positive-energy condition \ref{def:positive-energy}, imposed on the translation subgroups in the definition of a vacuum state \ref{def:vacuum-state}. @@ -1933,7 +1967,7 @@ \subsection{Covariant States and the Covariance Action} \begin{lemma}[Uniqueness of the Generator] \label{lmm:exp-generator-unique} - \lean{Physicslib4.AQFT.generator_eq_of_eq_expUnitary, Physicslib4.AQFT.exp_generator_unique} + \lean{Physicslib4.AQFT.generator_eq_of_eq_expUnitary} \uses{def:hall-10.13, def:exp-unitary-group, def:hall-9.5} \leanok Let $V(t)$, $t \in \mathbb{R}$, be a family of unitary operators on $H$, and let $B$ be a self-adjoint operator with $V(t) = e^{itB}$ (\ref{def:exp-unitary-group}) for all $t$. Then $B$ is the infinitesimal generator of $V$, with equality of domains. In particular the positive operator witnessing positive energy is unique. diff --git a/blueprint/src/sections/sec10/spacetime.tex b/blueprint/src/sections/sec10/spacetime.tex index d99bfb3..111ad98 100644 --- a/blueprint/src/sections/sec10/spacetime.tex +++ b/blueprint/src/sections/sec10/spacetime.tex @@ -103,6 +103,54 @@ \section{Spacetime}\label{sctn:spacetime} Bilinearity and symmetry give $g(v+w,v+w) = g(v,v) + 2g(v,w) + g(w,w)$. Writing $a = -g(v,v) > 0$, $b = -g(w,w) > 0$ and $c = -g(v,w) \ge 0$, this equals $-(a + b + 2c) < 0$, so $v+w$ is timelike. The reverse Cauchy-Schwarz inequality (\ref{lmm:reverse-cauchy-schwarz}) gives $g(v,w)^2 \ge g(v,v)\,g(w,w)$, that is $c^2 \ge ab$, hence $c \ge \sqrt{ab} = \sqrt a \,\sqrt b$. Therefore $-g(v+w,v+w) = a + b + 2c \ge a + b + 2\sqrt a\,\sqrt b = (\sqrt a + \sqrt b)^2$, and taking square roots yields the reverse triangle inequality. \end{proof} +\begin{definition}[Lorentz, Proper and Orthochronous Transformations] + \label{def:lorentz-transformation} + \lean{Physicslib4.IsLorentz, Physicslib4.IsProper, Physicslib4.IsOrthochronous} + \uses{def:standard-minkowski-spacetime} + \leanok + Let $\eta(v,w) = -v^0 w^0 + v^1 w^1 + v^2 w^2 + v^3 w^3$ be the Minkowski form on $\mathbb{R}^4$ and $e_0 = (1,0,0,0)$. A linear automorphism $L$ of $\mathbb{R}^4$ is + \begin{itemize} + \item \textit{Lorentz} if $\eta(Lv, Lw) = \eta(v,w)$ for all $v, w$; + \item \textit{proper} if $\det L = 1$; + \item \textit{orthochronous} if the time component of $L e_0$ is positive, $(L e_0)^0 > 0$. + \end{itemize} + The proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$ consists of the linear automorphisms with all three properties. +\end{definition} + +\begin{lemma}[Orthochronous Lorentz Transformations Preserve the Future Cone] + \label{lmm:orthochronous-preserves-future-timelike} + \lean{Physicslib4.isOrthochronous_pos_time_of_timelike} + \uses{def:lorentz-transformation} + \leanok + Let $L$ be an orthochronous Lorentz transformation and let $v \in \mathbb{R}^4$ be timelike, $\eta(v,v) < 0$, with positive time component $v^0 > 0$. Then $(Lv)^0 > 0$. +\end{lemma} +\begin{proof} + \leanok + \uses{def:lorentz-transformation} + For timelike $x, y$ the spatial parts satisfy $\lvert \mathbf{x}\cdot\mathbf{y}\rvert \le \lVert\mathbf{x}\rVert\,\lVert\mathbf{y}\rVert < \lvert x^0\rvert\,\lvert y^0\rvert$ by the Cauchy-Schwarz inequality on $\mathbb{R}^3$, so $\eta(x,y) = -x^0 y^0 + \mathbf{x}\cdot\mathbf{y} < 0$ forces $x^0 y^0 > 0$. Since $L$ is Lorentz, $Lv$ and $Le_0$ are timelike and $\eta(Lv, Le_0) = \eta(v, e_0) = -v^0 < 0$. Hence $(Lv)^0$ has the sign of $(Le_0)^0$, which is positive because $L$ is orthochronous. +\end{proof} + +\begin{lemma}[Closure of the Lorentz Conditions] + \label{lmm:lorentz-closure} + \lean{Physicslib4.isLorentz_refl, Physicslib4.isLorentz_trans, Physicslib4.isLorentz_symm, Physicslib4.isProper_refl, Physicslib4.isProper_trans, Physicslib4.isProper_symm, Physicslib4.isOrthochronous_refl, Physicslib4.isOrthochronous_trans, Physicslib4.isOrthochronous_symm} + \uses{def:lorentz-transformation} + \leanok + The identity is Lorentz, proper and orthochronous. Each of the Lorentz and proper conditions is preserved by composition and inversion. If $L_1, L_2$ are orthochronous Lorentz transformations, then $L_2 \circ L_1$ and $L_1^{-1}$ are orthochronous. Hence $SO(1,3)^{\uparrow}$ is a group. +\end{lemma} +\begin{proof} + \leanok + \uses{def:lorentz-transformation, lmm:orthochronous-preserves-future-timelike} + The Lorentz condition composes by rewriting twice, and inverts by applying it to $L^{-1}v, L^{-1}w$. Properness follows from $\det(L_2 L_1) = \det L_2 \det L_1$ and $\det L^{-1} = (\det L)^{-1}$. For composition of orthochronous maps, $L_1 e_0$ is timelike with positive time component, so $(L_2 L_1 e_0)^0 > 0$ by \ref{lmm:orthochronous-preserves-future-timelike}. For the inverse, $u = L^{-1} e_0$ is timelike, so $u^0 \neq 0$; if $u^0 < 0$, then applying \ref{lmm:orthochronous-preserves-future-timelike} to $-u$ gives $(-e_0)^0 > 0$, which is false. Hence $u^0 > 0$. +\end{proof} + +\begin{definition}[Inhomogeneous Lorentz Group] + \label{def:inhomogeneous-lorentz-group} + \lean{Physicslib4.AQFT.HaagKastler.InhomogeneousLorentzGroup} + \uses{def:lorentz-transformation, lmm:lorentz-closure, def:standard-minkowski-spacetime} + \leanok + The \textit{inhomogeneous Lorentz group} (the identity component of the Poincar\'e group) is the set of pairs $(L, a)$ with $L \in SO(1,3)^{\uparrow}$ and $a \in \mathbb{R}^4$, with product $(L_1, a_1)(L_2, a_2) = (L_1 L_2,\, a_1 + L_1 a_2)$. It acts on standard Minkowski spacetime by the affine maps $(L, a) \cdot x = Lx + a$. +\end{definition} + \begin{definition}[Time Orientation] \label{def:time-orientable} \lean{Physicslib4.Spacetime.TimeOrientation, Physicslib4.Spacetime.IsTimeOrientable} @@ -322,7 +370,7 @@ \section{Spacetime}\label{sctn:spacetime} \end{proof} \medskip -\noindent\textbf{The Levi-Civita connection and geodesics.} The trip definitions below require their curves to be geodesics. The following declarations make ``geodesic'' precise. They are stated for a general pseudo-Riemannian manifold, of which every spacetime is an instance, and they use covariant derivatives in Mathlib's sense (\texttt{IsCovariantDerivativeOn}, \texttt{CovariantDerivative}). No chart appears in any definition; charts and local frames are used only inside proofs. +\noindent\textbf{The Levi-Civita connection and geodesics.} The trip definitions below require their curves to be geodesics. The following declarations make ``geodesic'' precise. The Levi-Civita connection is stated for a general pseudo-Riemannian manifold, of which every spacetime is an instance; geodesics are defined for spacetimes, whose smooth paths (\ref{def:paths}) they are built on. All of these declarations use covariant derivatives in Mathlib's sense (\texttt{IsCovariantDerivativeOn}, \texttt{CovariantDerivative}). No chart appears in any definition; charts and local frames are used only inside proofs. \begin{definition}[Pseudo-Riemannian Metric] \label{def:pseudo-riemannian-metric} @@ -473,12 +521,12 @@ \section{Spacetime}\label{sctn:spacetime} \lean{Physicslib4.Spacetime.SmoothPath.mfderiv_apply_tangent_eq_zero} \uses{def:paths} \leanok - Let $\mu : \Sigma \to M$ be a smooth path, $s \in \Sigma$, and $f : M \to \mathbb{R}$ differentiable at $\mu(s)$ with $f(\mu(t)) = 0$ for all $t \in \Sigma$ in a neighbourhood of $s$. Then $f(\mu(s)) = 0$ and $df_{\mu(s)}\big(\dot\mu(s)\big) = 0$, where $\dot\mu(s)$ is the tangent vector of $\mu$ at $s$ (the within-derivative on $\Sigma$). + Let $\mu : \Sigma \to M$ be a smooth path, $s \in \Sigma$, and $f : M \to \mathbb{R}$ differentiable at $\mu(s)$ with $f(\mu(t)) = 0$ for all $t \in \Sigma$ in a neighbourhood of $s$. Then $df_{\mu(s)}\big(\dot\mu(s)\big) = 0$, where $\dot\mu(s)$ is the tangent vector of $\mu$ at $s$ (the within-derivative on $\Sigma$). \end{lemma} \begin{proof} \leanok \uses{def:paths, lmm:path-parameter-unique-diff} - The first claim is the hypothesis at $t = s$. By the chain rule within $\Sigma$ (\texttt{HasMFDerivAt.comp\_hasMFDerivWithinAt}) the within-derivative of $f \circ \mu$ at $s$ is $df_{\mu(s)} \circ d\mu$, whose value on $1$ is $df_{\mu(s)}(\dot\mu(s))$. On the other hand $f \circ \mu$ is eventually equal to $0$ within $\Sigma$ at $s$, so it also has within-derivative $0$ (\texttt{HasMFDerivWithinAt.congr\_of\_eventuallyEq}). Within-derivatives on $\Sigma$ are unique (\ref{lmm:path-parameter-unique-diff}), so the two agree. + By the chain rule within $\Sigma$ (\texttt{HasMFDerivAt.comp\_hasMFDerivWithinAt}) the within-derivative of $f \circ \mu$ at $s$ is $df_{\mu(s)} \circ d\mu$, whose value on $1$ is $df_{\mu(s)}(\dot\mu(s))$. On the other hand $f \circ \mu$ is eventually equal to $0$ within $\Sigma$ at $s$, so it also has within-derivative $0$ (\texttt{HasMFDerivWithinAt.congr\_of\_eventuallyEq}). Within-derivatives on $\Sigma$ are unique (\ref{lmm:path-parameter-unique-diff}), so the two agree. \end{proof} \begin{lemma}[Covariant Derivative Along a Curve is Local] @@ -536,7 +584,7 @@ \section{Spacetime}\label{sctn:spacetime} \lean{Physicslib4.Spacetime.IsGeodesic} \uses{def:paths, thrm:levi-civita-exists-unique, lmm:covariant-derivative-along-curve-local, lmm:velocity-extends} \leanok - Let $(M,g)$ be a pseudo-Riemannian manifold (for instance a spacetime) with Levi-Civita connection $\nabla$ (\ref{thrm:levi-civita-exists-unique}). A smooth path $\mu : \Sigma \to M$ is a \textit{geodesic} if for every $s$ in the interior of $\Sigma$ and every smooth vector field $X$ on $M$ with $X(\mu(t)) = \dot\mu(t)$ for all $t$ in a neighbourhood of $s$, + Let $(M,g)$ be a spacetime with Levi-Civita connection $\nabla$ (\ref{thrm:levi-civita-exists-unique}). A smooth path $\mu : \Sigma \to M$ is a \textit{geodesic} if for every $s$ in the interior of $\Sigma$ and every smooth vector field $X$ on $M$ with $X(\mu(t)) = \dot\mu(t)$ for all $t$ in a neighbourhood of $s$, \begin{align} (\nabla_X X)(\mu(s)) = 0. \end{align} @@ -785,6 +833,21 @@ \subsection{Causal diamonds} \medskip +\begin{lemma}[Precedence-Preserving Maps Preserve Spacelike Separation] + \label{lmm:precedence-preserving-spacelike} + \lean{Physicslib4.Spacetime.isSpacelikeRelated_congr, Physicslib4.Spacetime.isCompletelySpacelike_image} + \uses{def:spacelike-related, def:completely-spacelike, def:causal-trip} + \leanok + Let $M$ be a spacetime with a time orientation and let $\varphi : M \to M$ be any map with $\varphi(p) \prec \varphi(q) \iff p \prec q$ for all $p, q$. Then $\varphi(p_1)$ and $\varphi(p_2)$ are spacelike related if and only if $p_1$ and $p_2$ are, and $\varphi(\mathbf{O}_1)$, $\varphi(\mathbf{O}_2)$ are completely spacelike if and only if $\mathbf{O}_1$, $\mathbf{O}_2$ are. +\end{lemma} +\begin{proof} + \leanok + \uses{def:spacelike-related, def:completely-spacelike, def:causal-future-and-causal-past} + Spacelike relatedness of $p_1, p_2$ says $\lnot(p_1 \prec p_2)$ and $\lnot(p_2 \prec p_1)$, and both conditions are unchanged when $p_i$ is replaced by $\varphi(p_i)$. For the images of regions, every point of $\varphi(\mathbf{O}_i)$ has the form $\varphi(p_i)$ with $p_i \in \mathbf{O}_i$, so the pointwise statement applies. +\end{proof} + +\medskip + \begin{definition}[Spacelike Complement of a Region] \label{def:spacelike-complement} \lean{Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement} @@ -1551,6 +1614,53 @@ \subsection{Isometries and basis-set preservation} Suppose $\mu$ is a geodesic, let $s$ be an interior point of $\Sigma$, and let $X$ be a smooth vector field with $X(\varphi(\mu(t))) = (\varphi\circ\mu)^{\boldsymbol\cdot}(t) = d\varphi\,\dot\mu(t)$ for $t$ near $s$ (\ref{lmm:pushforward-path}). Then $\varphi^* X$ is smooth, and since $d\varphi$ is invertible, $(\varphi^* X)(\mu(t)) = (d\varphi)^{-1} X(\varphi(\mu(t))) = \dot\mu(t)$ for $t$ near $s$; so $\varphi^* X$ extends $\dot\mu$ near $s$ and the geodesic condition for $\mu$ gives $(\nabla_{\varphi^* X}\varphi^* X)(\mu(s)) = 0$. By naturality (\ref{lmm:isometry-preserves-levi-civita}), $(\nabla_X X)(\varphi(\mu(s))) = d\varphi\big((\nabla_{\varphi^* X}\varphi^* X)(\mu(s))\big) = 0$. The converse is the same argument for the isometry $\varphi^{-1}$ and the path $\varphi \circ \mu$. \end{proof} +\begin{definition}[Lorentz Pushforward of a Smooth Path] + \label{def:lorentz-path} + \lean{Physicslib4.lorentzPath} + \uses{def:inhomogeneous-lorentz-group, def:paths, def:time-orientable, lmm:pushforward-path} + \leanok + For $g = (L, a)$ in the inhomogeneous Lorentz group and a smooth path $\mu : \Sigma \to \mathbb{R}^4$ in standard Minkowski spacetime, the \textit{Lorentz pushforward} of $\mu$ is the smooth path $g\mu : \Sigma \to \mathbb{R}^4$, $s \mapsto L\mu(s) + a$, on the same parameter space. It is the Minkowski instance of the pushforward of a path under an isometry (\ref{lmm:pushforward-path}). The \textit{standard time orientation} of standard Minkowski spacetime (\ref{def:time-orientable}) is the constant vector field $e_0 = \partial_0$. +\end{definition} + +\begin{lemma}[Lorentz Pushforward of Causal Paths] + \label{lmm:lorentz-path-causal} + \lean{Physicslib4.lorentzPath_mfderivWithin, Physicslib4.lorentzPath_isCausal, Physicslib4.lorentzPath_isFutureOriented, Physicslib4.lorentzPath_isPastEndpoint, Physicslib4.lorentzPath_isFutureEndpoint} + \uses{def:lorentz-path, def:timelike-and-causal-smooth-curves, def:future-and-past-oriented-smooth-curves, def:endpoints, def:time-orientable} + \leanok + Let $g = (L, a)$ be in the inhomogeneous Lorentz group and $\mu$ a smooth path in standard Minkowski spacetime. The tangent of $g\mu$ at $s$ is $L\dot\mu(s)$. If $\mu$ is causal, so is $g\mu$; if $\mu$ is future-oriented with respect to the standard time orientation, so is $g\mu$; and if $p$ (respectively $q$) is the past (respectively future) endpoint of $\mu$, then $g \cdot p$ (respectively $g \cdot q$) is the past (respectively future) endpoint of $g\mu$. +\end{lemma} +\begin{proof} + \leanok + \uses{def:lorentz-path, def:lorentz-transformation, lmm:orthochronous-preserves-future-timelike, lmm:path-parameter-unique-diff} + The translation $a$ is constant, so the chain rule for the continuous linear map $L$ (using unique differentials on $\Sigma$) gives the tangent $L\dot\mu(s)$. Since $L$ is Lorentz, $\eta(L\dot\mu, L\dot\mu) = \eta(\dot\mu,\dot\mu)$, so timelike and null tangents stay timelike and null. A future-pointing timelike tangent has positive time component, which $L$ preserves by \ref{lmm:orthochronous-preserves-future-timelike}; a future-pointing null tangent is a limit of future-pointing timelike vectors, and their images under the continuous map $L$ converge to its image. The endpoints are attained at the same parameters, where $g\mu$ takes the values $g \cdot p$ and $g \cdot q$. +\end{proof} + +\begin{theorem}[Lorentz Invariance of Causal Precedence] + \label{thrm:causal-precedence-lorentz-invariant} + \lean{Physicslib4.causalSegmentPrecedes_smul, Physicslib4.causallyPrecedes_smul, Physicslib4.causallyPrecedes_smul_iff} + \uses{def:inhomogeneous-lorentz-group, def:causal-trip, def:time-orientable} + \leanok + For every $g$ in the inhomogeneous Lorentz group and all points $p, q$ of standard Minkowski spacetime (with its standard time orientation), $g \cdot p \prec g \cdot q$ if and only if $p \prec q$. +\end{theorem} +\begin{proof} + \leanok + \uses{lmm:lorentz-path-causal, lmm:isometry-preserves-geodesics, def:causal-trip} + If $\mu$ is a future-oriented causal geodesic from $p$ to $q$, then $g\mu$ is a future-oriented causal path from $g \cdot p$ to $g \cdot q$ (\ref{lmm:lorentz-path-causal}), and it is a geodesic (\ref{lmm:isometry-preserves-geodesics}). So $g$ maps causal trip segments to causal trip segments, and hence, segment by segment, causal trips to causal trips: $p \prec q$ implies $g \cdot p \prec g \cdot q$. The converse is the same statement for $g^{-1}$. +\end{proof} + +\begin{corollary}[Lorentz Invariance of Spacelike Separation] + \label{crllr:spacelike-lorentz-invariant} + \lean{Physicslib4.isSpacelikeRelated_smul_iff, Physicslib4.isCompletelySpacelike_smul_iff} + \uses{def:inhomogeneous-lorentz-group, def:spacelike-related, def:completely-spacelike, def:time-orientable} + \leanok + For every $g$ in the inhomogeneous Lorentz group, points $p_1, p_2$ and regions $\mathbf{O}_1, \mathbf{O}_2$ of standard Minkowski spacetime (with its standard time orientation), $g \cdot p_1$ and $g \cdot p_2$ are spacelike related if and only if $p_1$ and $p_2$ are, and $g\mathbf{O}_1$, $g\mathbf{O}_2$ are completely spacelike if and only if $\mathbf{O}_1$, $\mathbf{O}_2$ are. +\end{corollary} +\begin{proof} + \leanok + \uses{lmm:precedence-preserving-spacelike, thrm:causal-precedence-lorentz-invariant} + Apply \ref{lmm:precedence-preserving-spacelike} to the map $x \mapsto g \cdot x$, which preserves causal precedence in both directions by \ref{thrm:causal-precedence-lorentz-invariant}. +\end{proof} + \begin{lemma}[Isometries Preserve Chronology] \label{lmm:isometry-preserves-chronology} \lean{Physicslib4.Spacetime.Isometry.PreservesFutureOrientation, Physicslib4.Spacetime.Isometry.preservesFutureOrientation_one, Physicslib4.Spacetime.Isometry.preservesFutureOrientation_mul, Physicslib4.Spacetime.Isometry.pushforwardPath_isFutureOriented, Physicslib4.Spacetime.Isometry.chronologicallyPrecedes_pushforward, Physicslib4.Spacetime.Isometry.chronologicalFuture_image_subset, Physicslib4.Spacetime.Isometry.chronologicalFuture_image, Physicslib4.Spacetime.Isometry.chronologicalPast_image_subset, Physicslib4.Spacetime.Isometry.chronologicalPast_image} diff --git a/blueprint/src/sections/sec10/spectral-theorems.tex b/blueprint/src/sections/sec10/spectral-theorems.tex index f93d396..8c564c8 100644 --- a/blueprint/src/sections/sec10/spectral-theorems.tex +++ b/blueprint/src/sections/sec10/spectral-theorems.tex @@ -239,7 +239,7 @@ \subsubsection{Elementary Properties of Bounded Operators} \end{align} In other words for $\|\psi\| \neq 0$ the sequence $\{A_i\psi\}_{i \in \mathbb{N}}$ is a Cauchy sequence in $\mathbf{H}$. -In the case $\|\psi\| = 0$ the sequence $\{A_i\psi\}_{i \in \mathbb{N}}$ consists only of zeros. Hence, it's trivially a Cauchy sequence. +In the case $\|\psi\| = 0$ the sequence $\{A_i\psi\}_{i \in \mathbb{N}}$ consists only of zeros. Hence it is trivially a Cauchy sequence. As $\mathbf{H}$ is complete, this Cauchy sequence $\{A_i\psi\}_{i \in \mathbb{N}}$ converges to an element of $\mathbf{H}$. With this knowledge, we can define a map $A : \mathbf{H} \rightarrow \mathbf{H}$ pointwise as follows \begin{align} @@ -1106,7 +1106,7 @@ \subsubsection{Spectral Theorem for Bounded Self-Adjoint Operators} \begin{proposition}[A Bounded Quadratic Form Determines a Unique Bounded Operator] \label{prpstn:hall-a.63} - \lean{Physicslib4.Spectral.existsUnique_operator_of_isBoundedQuadraticForm} + \lean{Physicslib4.Spectral.existsUnique_operator_of_isBoundedQuadraticForm, Physicslib4.Spectral.IsBoundedQuadraticForm.isSelfAdjoint_toOperator} \leanok If $Q$ is a bounded quadratic form on $\mathbf{H}$, there is a unique $A \in \mathcal{B}(\mathbf{H})$ such that $Q(\psi) = \left< \psi, A\psi \right>$ for all $\psi \in \mathbf{H}$. If $Q(\psi)$ belongs to $\mathbb{R}$ for all $\psi \in \mathbf{H}$, then the operator $A$ is self-adjoint. \end{proposition} @@ -1256,7 +1256,7 @@ \subsubsection{Spectral Theorem for Bounded Self-Adjoint Operators} \begin{proposition}[Integral of an Indicator Function] \label{prpstn:integral-of-indicator} - \lean{Physicslib4.Spectral.ProjectionValuedMeasure.integral_indicator} + \lean{Physicslib4.Spectral.ProjectionValuedMeasure.integral_indicator, Physicslib4.Spectral.ProjectionValuedMeasure.integral_one} \leanok Let $\Omega(X)$ be a $\sigma$-algebra on a set $X$, let $\mu : \Omega(X) \rightarrow \mathcal{B}(\mathbf{H})$ be a projection-valued measure, and let $f \mapsto \int_X f \, d\mu$ be the map of Theorem~\ref{thrm:operator-valued-integration}. Then for all $E \in \Omega(X)$, \begin{align} @@ -1303,6 +1303,8 @@ \subsubsection{Spectral Theorem for Bounded Self-Adjoint Operators} \begin{proof} \leanok \uses{def:adjoint-bounded, def:identity-operator, prpstn:hall-a.43, prpstn:integral-as-limit-of-simple, prpstn:integral-norm-bound-simple, prpstn:integral-of-indicator, thrm:complex-valued-simple-approximation-theorem, thrm:operator-valued-integration} +\noindent\textit{Formalization note.} The Lean proof (\texttt{norm\_integral\_le}) follows the argument below, and also proves a preliminary bound $\|\int_X f\,d\mu\| \le 6\sup|f|$ by Lemma~\ref{lmm:lemma-1}, which it uses to pass to the limit of simple functions. + The next result we must prove is that for all bounded, measurable, complex-valued functions $f$ on $X$, we have \begin{align} \left\| \, \int_X f \, d\mu \, \right\| \le \sup\limits_{\lambda \in X} \left\lvert f(\lambda) \right\rvert, @@ -1686,7 +1688,7 @@ \subsubsection{Spectral Theorem for Bounded Self-Adjoint Operators} \begin{align} \left< \psi, A_s\psi \right> = \int_X f \, d\mu_\psi. \end{align} -Now previously we proved that there is a unique linear map +We proved above that there is a unique linear map \begin{align} f \longrightarrow \int_X f \, d\mu \end{align} @@ -2990,7 +2992,7 @@ \subsubsection{Spectral Theorem for Bounded Self-Adjoint Operators} \label{thrm:boundedness-theorem} \lean{IsCompact.exists_bound_of_continuousOn} \leanok - A continuous real-valued function on a non-empty compact set $C$ is bounded on $C$. + A continuous real-valued function on a compact set $C$ is bounded on $C$. \end{theorem} \begin{proof} \leanok @@ -3252,7 +3254,7 @@ \subsubsection{Spectral Theorem for Bounded Self-Adjoint Operators} \end{proposition} \begin{proof} \leanok - \uses{def:adjoint-bounded, prpstn:cfc-multiplicative, prpstn:cfc-non-negative, prpstn:cfc-norm, prpstn:cfc-self-adjoint, prpstn:cfc-spectral-mapping, def:bounded-operator-notation, prpstn:hall-8.3, def:non-negative-operator, def:bounded-operator-resolvent-and-spectrum, prpstn:hall-7.5, thrm:boundedness-theorem, thrm:operator-valued-integration, lmm:spectral-mapping-theorem, thrm:composition-theorem, prpstn:hall-7.7, lmm:hall-prblm-7.4.8, lmm:spectrum-is-compact-metric-measurable, prpstn:continuity-of-the-adjoint} + \uses{prpstn:cfc-multiplicative, prpstn:cfc-self-adjoint, prpstn:cfc-non-negative, prpstn:cfc-norm, prpstn:cfc-spectral-mapping} Each of the four properties is established separately below: multiplicativity in Proposition~\ref{prpstn:cfc-multiplicative}, self-adjointness in Proposition~\ref{prpstn:cfc-self-adjoint}, non-negativity in Proposition~\ref{prpstn:cfc-non-negative}, and the two norm-and-spectrum identities in Proposition~\ref{prpstn:cfc-norm} and Proposition~\ref{prpstn:cfc-spectral-mapping} respectively. \end{proof} @@ -4833,7 +4835,7 @@ \subsubsection{Spectral Theorem for Bounded Self-Adjoint Operators} \end{theorem} \begin{proof} \leanok - \uses{def:adjoint-bounded, prpstn:Q-continuous-under-limits, def:separates-points, lmm:bounded-operators-form-a-banach-space, lmm:hall-prblm-8.3.3c, lmm:pointwise-limits-of-borel-measurable-functions, lmm:projection-norm-decreasing, prpstn:bounded-operators-are-continuous, prpstn:continuity-of-norm-and-inner-product, prpstn:integral-multiplicative, prpstn:integral-norm-bound, prpstn:mua-countably-additive, prpstn:mua-empty-and-whole, prpstn:mua-integrates-to-A, prpstn:mua-multiplicative, prpstn:mua-projection, prpstn:orthogonal-sum-converges, prpstn:orthogonal-sum-is-projection, prpstn:orthogonal-sum-range, prpstn:pvm-agree-on-continuous, prpstn:pvm-agree-on-measurable, prpstn:pvm-agree-on-polynomials, thrm:bounded-convergence-theorem, thrm:bounded-linear-transformation-theorem, thrm:hall-prblm-8.3.4, thrm:monotone-convergence-theorem, def:identity-operator, def:indicator-function, prpstn:basic-integral-properties, def:bounded-operator-notation, def:bounded-operator-resolvent-and-spectrum, def:hall-8.8, def:projection-valued-measure, thrm:operator-valued-integration, lmm:lemma-3, prpstn:hall-8.9, def:bounded-orthogonal-projection, lmm:lemma-4, prpstn:hall-8.3, thrm:projection-valued-measures-associated-measure, def:hall-8.6, prpstn:hall-a.63, thrm:riesz-representation, prpstn:hall-7.7, thrm:boundedness-theorem, lmm:spectral-mapping-theorem, lmm:spectrum-is-compact-metric-measurable} + \uses{def:projection-valued-measure, prpstn:mua-projection, prpstn:mua-multiplicative, prpstn:mua-empty-and-whole, prpstn:mua-countably-additive, prpstn:mua-integrates-to-A} This proof broadly consists of two parts \textbf{Part 1:} prove that $\mu^A$ is a projection valued measure on $\sigma(A)$ and \textbf{Part 2:} prove that $\mu^A$ satisfies \begin{align} \int_{\sigma(A)} \lambda \, d\mu^A(\lambda) = A. @@ -5806,7 +5808,7 @@ \subsubsection{Spectral Theorem for Bounded Self-Adjoint Operators} \end{theorem} \begin{proof} \leanok - \uses{def:adjoint-bounded, thrm:gelfand-formula, prpstn:resolvent-holomorphy-and-neumann-series, crllr:crllr-1, def:F-class, def:L0-class, def:algebra-of-sets, def:hall-8.6, def:hall-8.8, def:separates-points, def:spectral-radius, lmm:associated-measures-are-finite, lmm:bounded-operators-form-a-banach-space, lmm:hall-7.6, lmm:hall-7.8, lmm:hall-8.1, lmm:hall-ex-8.3.1, lmm:hall-prblm-7.4.8, lmm:hall-prblm-8.3.3a, lmm:hall-prblm-8.3.3b, lmm:hall-prblm-8.3.3c, lmm:lemma-1, lmm:lemma-2, lmm:pointwise-limits-of-borel-measurable-functions, lmm:spectral-mapping-theorem, lmm:spectrum-is-compact-metric-measurable, prpstn:F-bounded, prpstn:F-closed-under-limits, prpstn:F-contains-continuous, prpstn:F-homogeneous, prpstn:F-sesquilinear, prpstn:L0-complement, prpstn:L0-contains-closed, prpstn:L0-contains-empty, prpstn:L0-union, prpstn:bounded-operators-are-continuous, prpstn:cfc-multiplicative, prpstn:cfc-non-negative, prpstn:cfc-norm, prpstn:cfc-self-adjoint, prpstn:cfc-spectral-mapping, prpstn:continuity-of-the-adjoint, prpstn:hall-7.2, prpstn:hall-7.3, prpstn:hall-7.5, prpstn:hall-7.7, prpstn:hall-8.3, prpstn:hall-8.4, prpstn:hall-8.7, prpstn:hall-a.34, prpstn:hall-a.62, thrm:analytic-equivalence-theorem, thrm:archimedean-property, thrm:bounded-convergence-theorem, thrm:bounded-linear-transformation-theorem, thrm:boundedness-theorem, thrm:complex-valued-simple-approximation-theorem, thrm:composition-theorem, thrm:existence-of-bump-functions, thrm:fundamental-theorem-of-algebra, thrm:maximum-modulus-principle, thrm:monotone-class-theorem, thrm:monotone-convergence-theorem-nonincreasing, thrm:riesz-representation, def:identity-operator, def:indicator-function, def:orthogonal-complement, prpstn:basic-integral-properties, def:bounded-operator-notation, def:projection-valued-measure, def:bounded-operator-resolvent-and-spectrum, def:bounded-orthogonal-projection, thrm:operator-valued-integration, thrm:hall-8.10, thrm:hall-prblm-8.3.4} + \uses{thrm:hall-8.10, thrm:hall-prblm-8.3.4} \textit{Existence.} Theorem~\ref{thrm:hall-8.10} constructs a projection-valued measure $\mu^A$ on the Borel $\sigma$-algebra of $\sigma(A)$, with values in orthogonal projections on $\mathbf{H}$, and establishes \begin{align} \int_{\sigma(A)} \lambda \, d\mu^A(\lambda) = A. diff --git a/blueprint/src/sections/sec10/unbounded-spectral-theorems.tex b/blueprint/src/sections/sec10/unbounded-spectral-theorems.tex index 3edf28a..acc6bea 100644 --- a/blueprint/src/sections/sec10/unbounded-spectral-theorems.tex +++ b/blueprint/src/sections/sec10/unbounded-spectral-theorems.tex @@ -1814,6 +1814,8 @@ \subsubsection{Integration Against a Projection-Valued Measure} the second equality using that $\psi \in \mathbf{H}$ already forces $\sum_n \left\| \psi_n \right\|_n^2 < \infty$ (Parseval for the Hilbert space direct sum). This is exactly the domain $V$ in Proposition~\ref{prpstn:hall-9.26}. We have shown: $\{\mathbf{H}_n\}$ is an internal orthogonal decomposition of $\mathbf{H}$ into separable closed subspaces, each $A_n$ is a bounded self-adjoint operator on $\mathbf{H}_n$, and $A_f$ is a symmetric operator (we check this directly: for $\phi, \psi \in W_f$, $\left< \phi, A_f\psi \right> = L_f(\phi,\psi)$ and $\left< A_f\phi, \psi \right> = \overline{\left< \psi, A_f\phi \right>} = \overline{L_f(\psi,\phi)}$; Part 2 of Proposition~\ref{prpstn:quadratic-forms-on-a-subspace-properties}, applicable since $Q_f$ is real-valued, gives $L_f(\phi,\psi) = \overline{L_f(\psi,\phi)}$, so $\left< \phi, A_f\psi \right> = \left< A_f\phi, \psi \right>$) whose domain contains the algebraic span $W_0$ of the $\mathbf{H}_n$'s (finite sums), and which acts as $A_n$ on each $\mathbf{H}_n$. This is exactly the hypothesis of Proposition~\ref{prpstn:hall-9.26-internal}, which concludes that $A_f$ is essentially self-adjoint with $\text{Dom}(A_f^{\text{cl}}) = \text{Dom}(A_f^*)$ equal to the space $V$ identified there — which we showed above coincides with $W_f = \text{Dom}(A_f)$. Since $A_f$ is symmetric, Proposition~\ref{prpstn:hall-9.4} shows $A_f^*$ is an extension of $A_f$, and having just shown $\text{Dom}(A_f^*) = \text{Dom}(A_f)$, this extension is trivial: $A_f^* = A_f$, i.e. $A_f$ is self-adjoint. + +\noindent\textit{Formalization note.} The Lean proof of this proposition (\texttt{isSelfAdjoint\_pmapIntegral\_of\_real}) does not use the direct-sum decomposition above. It shows that $A_f$ is symmetric with dense domain, and that $A_f - c\mathbf{1}$ maps onto $\mathbf{H}$ for every non-real $c$, with the bounded right inverse $\int_X (f-c)^{-1}\,d\mu$ (\texttt{range\_subSmul\_pmapIntegral\_eq\_top}); a symmetric operator with this property is self-adjoint (\texttt{adjoint\_eq\_of\_range\_eq\_top}). Proposition~\ref{prpstn:hall-9.26-internal} and the results on internal orthogonal decompositions are not used there. \end{proof} \subsubsection{The Spectral Theorem for Bounded Normal Operators} @@ -1845,7 +1847,7 @@ \subsubsection{The Spectral Theorem for Bounded Normal Operators} Hall's proof that the bounded self-adjoint spectral theorem extends to normal operators proceeds in two stages, mirroring the two-stage proof of the self-adjoint case itself. The first stage builds a continuous functional calculus for $A$; the second turns that functional calculus into a projection-valued measure. Hall's own observation is that this second stage, once a continuous functional calculus is in hand, uses nothing about the operator beyond the functional calculus itself — not self-adjointness, not realness of the spectrum. Rather than treat this as license to say the self-adjoint case's construction "carries over unchanged" — citing one proof to justify another, exactly the pattern Lemma~\ref{lmm:b-squared-inequality-symmetric} and Proposition~\ref{prpstn:quadratic-forms-on-a-subspace-properties} were introduced earlier to avoid — when we reach that stage we will extract the construction as its own proposition, parameterized by an abstract continuous functional calculus on a compact metric space, so that the self-adjoint and normal cases each cite that one statement rather than one citing the other's proof. For now, our task is the first stage: building the continuous functional calculus for a normal operator. -For a self-adjoint operator, this calculus was built by approximating continuous \textit{real-valued} functions on the (real) spectrum by real polynomials in $\lambda$, using the real Stone–Weierstrass theorem. For a normal operator, $\sigma(A)$ is a general compact subset of $\mathbb{C}$, and we need to approximate \textit{complex-valued} functions; the complex-valued Stone–Weierstrass theorem requires an algebra of functions closed under complex conjugation, so plain polynomials in $\lambda$ no longer suffice — we need polynomials in $\lambda$ \textit{and} $\overline{\lambda}$. On the operator side, the counterpart to conjugation is the adjoint, so a polynomial $p(\lambda,\overline\lambda)$ should correspond to the operator $p(A,A^*)$ obtained by substituting $A$ for $\lambda$ and $A^*$ for $\overline\lambda$. This substitution is where normality is essential: the polynomial ring $\mathbb{C}[\lambda,\overline\lambda]$ is commutative, so a given function $p(\lambda,\overline\lambda)$ has a unique representation as such a polynomial — but substituting $\lambda \mapsto A$, $\overline\lambda \mapsto A^*$ turns this into an expression in the generally \textit{non}commutative algebra $\mathcal{B}(\mathbf{H})$, and the substitution is well defined as an algebra homomorphism only if it respects every relation that holds in the source ring — in particular $\lambda\overline\lambda = \overline\lambda\lambda$, which forces $AA^* = A^*A$. Without normality, there would be nothing to substitute \textit{into}: no consistent way to assign a single operator to $p(\lambda,\overline\lambda)$ independent of how it is written. +For a self-adjoint operator, this calculus was built by approximating continuous \textit{real-valued} functions on the (real) spectrum by real polynomials in $\lambda$, using the real Stone–Weierstrass theorem. For a normal operator, $\sigma(A)$ is a general compact subset of $\mathbb{C}$, and we need to approximate \textit{complex-valued} functions; the complex-valued Stone–Weierstrass theorem requires an algebra of functions closed under complex conjugation, so plain polynomials in $\lambda$ do not suffice — we need polynomials in $\lambda$ \textit{and} $\overline{\lambda}$. On the operator side, the counterpart to conjugation is the adjoint, so a polynomial $p(\lambda,\overline\lambda)$ should correspond to the operator $p(A,A^*)$ obtained by substituting $A$ for $\lambda$ and $A^*$ for $\overline\lambda$. This substitution is where normality is essential: the polynomial ring $\mathbb{C}[\lambda,\overline\lambda]$ is commutative, so a given function $p(\lambda,\overline\lambda)$ has a unique representation as such a polynomial — but substituting $\lambda \mapsto A$, $\overline\lambda \mapsto A^*$ turns this into an expression in the generally \textit{non}commutative algebra $\mathcal{B}(\mathbf{H})$, and the substitution is well defined as an algebra homomorphism only if it respects every relation that holds in the source ring — in particular $\lambda\overline\lambda = \overline\lambda\lambda$, which forces $AA^* = A^*A$. Without normality, there would be nothing to substitute \textit{into}: no consistent way to assign a single operator to $p(\lambda,\overline\lambda)$ independent of how it is written. The key technical result we are aiming for is a version of the spectral mapping theorem for this two-variable substitution: $\sigma\big(p(A,A^*)\big) = \big\{ p(\lambda,\overline\lambda) \mid \lambda \in \sigma(A) \big\}$. Unlike the ordinary (one-variable) spectral mapping theorem, this is genuinely harder to prove, and the route we follow — matching Hall's — uses the bounded self-adjoint spectral theorem itself, applied to an auxiliary self-adjoint operator, together with the notion of an \textit{almost eigenvector}. diff --git a/blueprint/src/sections/sec4/isotony.tex b/blueprint/src/sections/sec4/isotony.tex index b51b344..95dcf28 100644 --- a/blueprint/src/sections/sec4/isotony.tex +++ b/blueprint/src/sections/sec4/isotony.tex @@ -34,7 +34,7 @@ \section{Isotony}\label{sctn:isotony} It turns out that for C*-algebras we have the following theorem (Proposition 2.2.7 of \href{https://doi.org/10.1007/978-3-662-02520-8}{Bratteli and Robinson}) \begin{theorem}[C*-Spectrum is Invariant Under Inclusion] - Let $\mathfrak{U}_1$ be a C*-subalgebra of the C*-algebra $\mathfrak{U}_2$. If $a$ is an element of $\mathfrak{U}_1$, then the spectrum $\sigma_{\mathfrak{U}_1}(a)$ of $a$ when viewed as an element of $\mathfrak{U}_1$ is the same as the spectrum $\sigma_{\mathfrak{U}_2}(a)$ of $a$ when viewed as an element of $\mathfrak{U}_2$* + Let $\mathfrak{U}_1$ be a C*-subalgebra of the C*-algebra $\mathfrak{U}_2$. If $a$ is an element of $\mathfrak{U}_1$, then the spectrum $\sigma_{\mathfrak{U}_1}(a)$ of $a$ when viewed as an element of $\mathfrak{U}_1$ is the same as the spectrum $\sigma_{\mathfrak{U}_2}(a)$ of $a$ when viewed as an element of $\mathfrak{U}_2$, \begin{align} \sigma_{\mathfrak{U}_1}(a) = \sigma_{\mathfrak{U}_2}(a). \end{align} diff --git a/blueprint/src/sections/sec5/quasilocal-algebra.tex b/blueprint/src/sections/sec5/quasilocal-algebra.tex index cdea4d5..b922a67 100644 --- a/blueprint/src/sections/sec5/quasilocal-algebra.tex +++ b/blueprint/src/sections/sec5/quasilocal-algebra.tex @@ -31,5 +31,5 @@ \section{Quasilocal Algebra}\label{sctn:quasilocal-algebra} $\mathfrak{U}(\mathbf{B})$, so the statement can equally be read as commutation in $\mathfrak{U}$; the two readings are equivalent. The formulation through a containing $\mathfrak{U}(\mathbf{B})$ is preferred because it uses only the net and isotony, and because it -is the formulation that survives in curved spacetime (Chapter 9), where a containing basis element +is the formulation that survives in curved spacetime (Chapter~\ref{chptr:haag-kastler-axioms-in-curved-spacetime}), where a containing basis element need not exist and the quasilocal algebra need not exist either. diff --git a/home_page/axioms.md b/home_page/axioms.md index 6a29e74..aa53a02 100644 --- a/home_page/axioms.md +++ b/home_page/axioms.md @@ -10,30 +10,30 @@ usemathjax: true Axiom 6 (Primitivity) from the original 1964 Haag–Kastler paper is not carried into the sharpened axiom set — see Chapter 8 for the discussion of why it is dropped. The five sharpened axioms in each setting, bundled together as a `HaagKastlerNet`, are what is actually formalised. -**Minkowski spacetime** — bundled as `Physicslib4.AQFT.HaagKastler.HaagKastlerNet` (Definition 508): +**Minkowski spacetime** — bundled as `Physicslib4.AQFT.HaagKastler.HaagKastlerNet` (Definition 517): | Axiom | Node | Lean | |---|---|---| -| 1. Local Algebras | Definition 467 | `Physicslib4.AQFT.HaagKastler.LocalNet` | -| 2. Isotony | Definition 468 | `Physicslib4.AQFT.HaagKastler.Isotony` | -| 3. Local Commutativity | Definition 485 | `Physicslib4.AQFT.HaagKastler.LocalCommutativity` | -| 4. Quasilocal Completeness | Definition 489 | `Physicslib4.AQFT.HaagKastler.ObservableCorrespondence` | -| 5. Lorentz Covariance | Definition 507 | `Physicslib4.AQFT.HaagKastler.LorentzCovariance` | +| 1. Local Algebras | Definition 476 | `Physicslib4.AQFT.HaagKastler.LocalNet` | +| 2. Isotony | Definition 477 | `Physicslib4.AQFT.HaagKastler.Isotony` | +| 3. Local Commutativity | Definition 494 | `Physicslib4.AQFT.HaagKastler.LocalCommutativity` | +| 4. Quasilocal Completeness | Definition 498 | `Physicslib4.AQFT.HaagKastler.ObservableCorrespondence` | +| 5. Lorentz Covariance | Definition 516 | `Physicslib4.AQFT.HaagKastler.LorentzCovariance` | -**Curved spacetime** — bundled as `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet` (Definition 633): +**Curved spacetime** — bundled as `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet` (Definition 644): | Axiom | Node | Lean | |---|---|---| -| 1. Local Algebras | Definition 627 | `Physicslib4.AQFT.HaagKastlerCurved.LocalNet` | -| 2. Isotony | Definition 628 | `Physicslib4.AQFT.HaagKastlerCurved.Isotony` | -| 3. Local Commutativity | Definition 629 | `Physicslib4.AQFT.HaagKastlerCurved.LocalCommutativity` | -| 4. Local Completeness | Definition 631 | `Physicslib4.AQFT.HaagKastlerCurved.LocalAlgebra` | -| 5. Isometric Covariance | Definition 632 | `Physicslib4.AQFT.HaagKastlerCurved.IsometricCovariance` | +| 1. Local Algebras | Definition 638 | `Physicslib4.AQFT.HaagKastlerCurved.LocalNet` | +| 2. Isotony | Definition 639 | `Physicslib4.AQFT.HaagKastlerCurved.Isotony` | +| 3. Local Commutativity | Definition 640 | `Physicslib4.AQFT.HaagKastlerCurved.LocalCommutativity` | +| 4. Local Completeness | Definition 642 | `Physicslib4.AQFT.HaagKastlerCurved.LocalAlgebra` | +| 5. Isometric Covariance | Definition 643 | `Physicslib4.AQFT.HaagKastlerCurved.IsometricCovariance` | -**General Covariance (Definition 677, `Physicslib4.AQFT.HaagKastlerCurved.IsGenerallyCovariant`)** is deliberately *not* a sixth axiom. Axioms 1–5 each constrain a single net over a single fixed spacetime, whereas general covariance relates two nets over two spacetimes; it is therefore a property of the section $$L \mapsto \mathfrak{U}_L$$ assigning a net to every Lorentzian spacetime, not an extra field of the net structure. +**General Covariance (Definition 690, `Physicslib4.AQFT.HaagKastlerCurved.IsGenerallyCovariant`)** is deliberately *not* a sixth axiom. Axioms 1–5 each constrain a single net over a single fixed spacetime, whereas general covariance relates two nets over two spacetimes; it is therefore a property of the section $$L \mapsto \mathfrak{U}_L$$ assigning a net to every Lorentzian spacetime, not an extra field of the net structure. ## Design notes -- **Isotony supplies its embeddings as data.** Axiom 2 (Definitions 468 and 628) fixes the family $$i_{\mathbf{B}_1 \mathbf{B}_2}$$ together with identity and composition laws, making $$\mathbf{B} \mapsto \mathfrak{U}(\mathbf{B})$$ a functor on the inclusion order. Axiom 3 uses that family rather than choosing witnesses of its own, and Axiom 5 (Definitions 507 and 632) states its coherence condition for the same family, so that covariance-compatibility of a quasilocal algebra is a lemma (Lemma 588) rather than a hypothesis. This is what makes the quasilocal colimit well posed in Minkowski spacetime, and it means that curved-spacetime statements about nested regions need no coherence hypotheses. -- **Local commutativity is stated in a common containing diamond.** Definition 485 requires completely spacelike local algebras to commute in every local algebra $$\mathfrak{U}(\mathbf{B})$$ containing both, through the Axiom 2 isotony maps, exactly as Definition 629 does in curved spacetime; the two differ only in that a containing diamond always exists in Minkowski spacetime. Commutation in the quasilocal algebra is the equivalent form of Lemma 487 (`Physicslib4.AQFT.HaagKastler.localCommutativity_iff_exists_commute_ι`). -- **Axiom 4 is a bridge principle; the quasilocal algebra is a theorem.** That a quasilocal algebra exists is Theorem 490 (`Physicslib4.AQFT.HaagKastler.exists_quasilocalAlgebra`), proved from the colimit-and-completion construction. Axiom 4 (Definition 489) is the bridge principle relating physical observables to quasilocal ones, which has, by design, no mathematical consumers. +- **Isotony supplies its embeddings as data.** Axiom 2 (Definitions 477 and 639) fixes the family $$i_{\mathbf{B}_1 \mathbf{B}_2}$$ together with identity and composition laws, making $$\mathbf{B} \mapsto \mathfrak{U}(\mathbf{B})$$ a functor on the inclusion order. Axiom 3 uses that family rather than choosing witnesses of its own, and Axiom 5 (Definitions 516 and 643) states its coherence condition for the same family, so that covariance-compatibility of a quasilocal algebra is a lemma (Lemma 597) rather than a hypothesis. This is what makes the quasilocal colimit well posed in Minkowski spacetime, and it means that curved-spacetime statements about nested regions need no coherence hypotheses. +- **Local commutativity is stated in a common containing diamond.** Definition 494 requires completely spacelike local algebras to commute in every local algebra $$\mathfrak{U}(\mathbf{B})$$ containing both, through the Axiom 2 isotony maps, exactly as Definition 640 does in curved spacetime; the two differ only in that a containing diamond always exists in Minkowski spacetime. Commutation in the quasilocal algebra is the equivalent form of Lemma 496 (`Physicslib4.AQFT.HaagKastler.localCommutativity_iff_exists_commute_ι`). +- **Axiom 4 is a bridge principle; the quasilocal algebra is a theorem.** That a quasilocal algebra exists is Theorem 499 (`Physicslib4.AQFT.HaagKastler.exists_quasilocalAlgebra`), proved from the colimit-and-completion construction. Axiom 4 (Definition 498) is the bridge principle relating physical observables to quasilocal ones, which has, by design, no mathematical consumers. diff --git a/home_page/guide.md b/home_page/guide.md index 02a3178..ab019b0 100644 --- a/home_page/guide.md +++ b/home_page/guide.md @@ -9,11 +9,11 @@ usemathjax: true ## How the blueprint is organised -The blueprint is 341 pages long and splits cleanly in two. +The blueprint is 345 pages long and splits cleanly in two. **Chapters 1–9 are mathematical background and are not formalised in Lean.** They motivate and analyse each of the original Haag–Kastler axioms in turn, and then generalise them to curved spacetime. Along the way they cite twelve supporting results, numbered 1 through 12 — Gelfand–Naimark, the Bounded Linear Transformation Theorem, the existence of a Lorentz metric, and so on. These are quoted from the literature where needed; none of them carries a Lean declaration. -**Chapter 10 collects the formalisation-ready content, and it is the content of Chapter 10 that is formalised in Lean.** Its items are numbered consecutively, running from Definition 13 through Definition 677, and comprise **662 declarations in total: 150 definitions, 156 theorems, 253 lemmas, 99 propositions, and 4 corollaries**, mapped onto **1,146 distinct Lean declarations** (where Mathlib already supplies a result, the blueprint names the Mathlib declaration directly). Three further numbered items — Conventions 22, 213 and 260, standing hypotheses of the two spectral-theory sections — share the numbering but are conventions rather than declarations, and are not counted. Chapter 10 is divided into eight top-level sections, §10.1 through §10.8. +**Chapter 10 collects the formalisation-ready content, and it is the content of Chapter 10 that is formalised in Lean.** Its items are numbered consecutively, running from Definition 13 through Definition 690, and comprise **675 declarations in total: 153 definitions, 159 theorems, 259 lemmas, 99 propositions, and 5 corollaries**, mapped onto **1,188 distinct Lean declarations** (where Mathlib already supplies a result, the blueprint names the Mathlib declaration directly). Three further numbered items — Conventions 22, 213 and 260, standing hypotheses of the two spectral-theory sections — share the numbering but are conventions rather than declarations, and are not counted. Chapter 10 is divided into eight top-level sections, §10.1 through §10.8. ## Chapters 1–9: unpacking the original axioms @@ -33,16 +33,16 @@ The lower numbers, 1 through 12, label supporting theorems and definitions intro ## The section pages -Only Chapter 10 is formalised in Lean. Its items are numbered consecutively from Definition 13 through Definition 677 — 662 declarations, plus the three standing conventions of §10.2 and §10.3. Each section page below summarises its section and lists **every** item, in numerical order, under the blueprint subsection in which it appears. Each entry links to the node in the web blueprint and names the principal Lean declaration that realises it; where a node maps onto several declarations, the count of the remainder is shown. The lists are derived mechanically from the blueprint's own `\lean` and `\leanok` annotations. +Only Chapter 10 is formalised in Lean. Its items are numbered consecutively from Definition 13 through Definition 690 — 675 declarations, plus the three standing conventions of §10.2 and §10.3. Each section page below summarises its section and lists **every** item, in numerical order, under the blueprint subsection in which it appears. Each entry links to the node in the web blueprint and names the principal Lean declaration that realises it; where a node maps onto several declarations, the count of the remainder is shown. The lists are derived mechanically from the blueprint's own `\lean` and `\leanok` annotations. - [§10.1 GNS Construction (pp. 29–38)](sec10-1-gns.html) -- [§10.2 Spectral Theorems (pp. 38–155)](sec10-2-spectral.html) -- [§10.3 Unbounded Spectral Theorems (pp. 155–235)](sec10-3-unbounded.html) -- [§10.4 Stone's Theorem (pp. 235–249)](sec10-4-stone.html) -- [§10.5 Spacetime and causal structure (pp. 249–287)](sec10-5-spacetime.html) -- [§10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 287–328)](sec10-6-haag-kastler.html) -- [§10.7 Haag–Kastler Axioms in curved spacetime (pp. 328–338)](sec10-7-curved.html) -- [§10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 338–339)](sec10-8-general-covariance.html) +- [§10.2 Spectral Theorems (pp. 39–156)](sec10-2-spectral.html) +- [§10.3 Unbounded Spectral Theorems (pp. 156–236)](sec10-3-unbounded.html) +- [§10.4 Stone's Theorem (pp. 236–250)](sec10-4-stone.html) +- [§10.5 Spacetime and causal structure (pp. 250–289)](sec10-5-spacetime.html) +- [§10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 289–330)](sec10-6-haag-kastler.html) +- [§10.7 Haag–Kastler Axioms in curved spacetime (pp. 330–340)](sec10-7-curved.html) +- [§10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 340–343)](sec10-8-general-covariance.html) ## Where the Lean lives diff --git a/home_page/index.md b/home_page/index.md index 2d0d458..127df30 100644 --- a/home_page/index.md +++ b/home_page/index.md @@ -25,27 +25,27 @@ In 1964, Rudolf Haag and Daniel Kastler introduced a set of axioms for Algebraic ## Overview -The blueprint is 341 pages long and splits cleanly in two. +The blueprint is 345 pages long and splits cleanly in two. **Chapters 1–9 are mathematical background and are not formalised in Lean.** They motivate and analyse each of the original Haag–Kastler axioms in turn, and then generalise them to curved spacetime. Along the way they cite twelve supporting results, numbered 1 through 12 — Gelfand–Naimark, the Bounded Linear Transformation Theorem, the existence of a Lorentz metric, and so on. These are quoted from the literature where needed; none of them carries a Lean declaration. -**Chapter 10 collects the formalisation-ready content, and it is the content of Chapter 10 that is formalised in Lean.** Its items are numbered consecutively, running from Definition 13 through Definition 677, and comprise **662 declarations in total: 150 definitions, 156 theorems, 253 lemmas, 99 propositions, and 4 corollaries**, mapped onto **1,146 distinct Lean declarations** (where Mathlib already supplies a result, the blueprint names the Mathlib declaration directly). Three further numbered items — Conventions 22, 213 and 260, standing hypotheses of the two spectral-theory sections — share the numbering but are conventions rather than declarations, and are not counted. Chapter 10 is divided into eight top-level sections, §10.1 through §10.8. +**Chapter 10 collects the formalisation-ready content, and it is the content of Chapter 10 that is formalised in Lean.** Its items are numbered consecutively, running from Definition 13 through Definition 690, and comprise **675 declarations in total: 153 definitions, 159 theorems, 259 lemmas, 99 propositions, and 5 corollaries**, mapped onto **1,188 distinct Lean declarations** (where Mathlib already supplies a result, the blueprint names the Mathlib declaration directly). Three further numbered items — Conventions 22, 213 and 260, standing hypotheses of the two spectral-theory sections — share the numbering but are conventions rather than declarations, and are not counted. Chapter 10 is divided into eight top-level sections, §10.1 through §10.8. -**All 662 are formalised, statements and proofs alike, and the Lean contains no `sorry`.** Of the 512 theorems, lemmas, propositions and corollaries in Chapter 10, 472 carry a written proof in the blueprint; the other 40, in §10.2, §10.3 and §10.4, are standard results quoted from the literature (Heine–Borel, Stone–Weierstrass, dominated convergence, the Bochner integral and the like) that the blueprint states in full but deliberately does not prove. All 512 proofs are formalised in Lean, including the 40 quoted results, which are linked to the Mathlib declarations that supply them. +**All 675 are formalised, statements and proofs alike, and the Lean contains no `sorry`.** Of the 522 theorems, lemmas, propositions and corollaries in Chapter 10, 482 carry a written proof in the blueprint; the other 40, in §10.2, §10.3 and §10.4, are standard results quoted from the literature (Heine–Borel, Stone–Weierstrass, dominated convergence, the Bochner integral and the like) that the blueprint states in full but deliberately does not prove. All 522 proofs are formalised in Lean, including the 40 quoted results, which are linked to the Mathlib declarations that supply them. -At a glance, the 662 declarations break down by top-level section as follows (click a section to see its items): +At a glance, the 675 declarations break down by top-level section as follows (click a section to see its items): | Section | Topic | Pages | Definitions | Theorems | Lemmas | Propositions | Corollaries | Total | Formalised | |---|---|---|---|---|---|---|---|---|---| | [§10.1](sec10-1-gns.html) | GNS Construction | 29–38 | 2 | 1 | 3 | 0 | 0 | 6 | 6 | -| [§10.2](sec10-2-spectral.html) | Spectral Theorems (bounded) | 38–155 | 25 | 24 | 27 | 67 | 1 | 144 | 144 | -| [§10.3](sec10-3-unbounded.html) | Unbounded Spectral Theorems | 155–235 | 27 | 13 | 50 | 30 | 3 | 123 | 123 | -| [§10.4](sec10-4-stone.html) | Stone's Theorem | 235–249 | 3 | 2 | 34 | 2 | 0 | 41 | 41 | -| [§10.5](sec10-5-spacetime.html) | Spacetime and causal structure | 249–287 | 34 | 17 | 86 | 0 | 0 | 137 | 137 | -| [§10.6](sec10-6-haag-kastler.html) | Haag–Kastler Axioms (Minkowski) | 287–328 | 40 | 70 | 50 | 0 | 0 | 160 | 160 | -| [§10.7](sec10-7-curved.html) | Haag–Kastler Axioms (curved spacetime) | 328–338 | 17 | 29 | 3 | 0 | 0 | 49 | 49 | -| [§10.8](sec10-8-general-covariance.html) | General Covariance | 338–339 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -| **Total** | | | **150** | **156** | **253** | **99** | **4** | **662** | **662** | +| [§10.2](sec10-2-spectral.html) | Spectral Theorems (bounded) | 39–156 | 25 | 24 | 27 | 67 | 1 | 144 | 144 | +| [§10.3](sec10-3-unbounded.html) | Unbounded Spectral Theorems | 156–236 | 27 | 13 | 50 | 30 | 3 | 123 | 123 | +| [§10.4](sec10-4-stone.html) | Stone's Theorem | 236–250 | 3 | 2 | 34 | 2 | 0 | 41 | 41 | +| [§10.5](sec10-5-spacetime.html) | Spacetime and causal structure | 250–289 | 37 | 18 | 90 | 0 | 1 | 146 | 146 | +| [§10.6](sec10-6-haag-kastler.html) | Haag–Kastler Axioms (Minkowski) | 289–330 | 40 | 71 | 51 | 0 | 0 | 162 | 162 | +| [§10.7](sec10-7-curved.html) | Haag–Kastler Axioms (curved spacetime) | 330–340 | 17 | 30 | 4 | 0 | 0 | 51 | 51 | +| [§10.8](sec10-8-general-covariance.html) | General Covariance | 340–343 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | +| **Total** | | | **153** | **159** | **259** | **99** | **5** | **675** | **675** | ## Explore @@ -53,18 +53,18 @@ At a glance, the 662 declarations break down by top-level section as follows (cl - **[The axioms at a glance](axioms.html)** — the Minkowski and curved-spacetime axioms with their blueprint nodes and Lean declarations. - **Chapter 10, section by section:** - [§10.1 GNS Construction (pp. 29–38)](sec10-1-gns.html) - - [§10.2 Spectral Theorems (pp. 38–155)](sec10-2-spectral.html) - - [§10.3 Unbounded Spectral Theorems (pp. 155–235)](sec10-3-unbounded.html) - - [§10.4 Stone's Theorem (pp. 235–249)](sec10-4-stone.html) - - [§10.5 Spacetime and causal structure (pp. 249–287)](sec10-5-spacetime.html) - - [§10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 287–328)](sec10-6-haag-kastler.html) - - [§10.7 Haag–Kastler Axioms in curved spacetime (pp. 328–338)](sec10-7-curved.html) - - [§10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 338–339)](sec10-8-general-covariance.html) + - [§10.2 Spectral Theorems (pp. 39–156)](sec10-2-spectral.html) + - [§10.3 Unbounded Spectral Theorems (pp. 156–236)](sec10-3-unbounded.html) + - [§10.4 Stone's Theorem (pp. 236–250)](sec10-4-stone.html) + - [§10.5 Spacetime and causal structure (pp. 250–289)](sec10-5-spacetime.html) + - [§10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 289–330)](sec10-6-haag-kastler.html) + - [§10.7 Haag–Kastler Axioms in curved spacetime (pp. 330–340)](sec10-7-curved.html) + - [§10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 340–343)](sec10-8-general-covariance.html) ## Highlights -- **The Haag–Kastler axioms in Minkowski spacetime** ([§10.6](sec10-6-haag-kastler.html), [the axioms at a glance](axioms.html)). Five axioms, Definitions 467, 468, 485, 489 and 507: local C\*-algebras on the Alexandrov diamonds; isotony as a family of embeddings with identity and composition laws, so that the net is a functor on the inclusion order; local commutativity of completely spacelike local algebras in every diamond containing both; Axiom 4 as the bridge principle relating physical observables to quasilocal ones; and Lorentz covariance. They are bundled as a Haag–Kastler net (Definition 508, `Physicslib4.AQFT.HaagKastler.HaagKastlerNet`), and the quasilocal algebra of a net is constructed from the net alone, as a directed colimit followed by completion (Theorem 490, `Physicslib4.AQFT.HaagKastler.exists_quasilocalAlgebra`). -- **The Haag–Kastler axioms in Lorentzian spacetime** ([§10.7](sec10-7-curved.html)). The same five axioms on a Lorentzian spacetime, Definitions 627–632, with isometric covariance under the identity component of the isometry group in place of Lorentz covariance, and with commutation and completeness stated relative to a common containing region, since a quasilocal algebra need not exist. They are bundled as `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet` (Definition 633), and general covariance (Definition 677, [§10.8](sec10-8-general-covariance.html)) relates the nets over diffeomorphism-related spacetimes. +- **The Haag–Kastler axioms in Minkowski spacetime** ([§10.6](sec10-6-haag-kastler.html), [the axioms at a glance](axioms.html)). Five axioms, Definitions 476, 477, 494, 498 and 516: local C\*-algebras on the Alexandrov diamonds; isotony as a family of embeddings with identity and composition laws, so that the net is a functor on the inclusion order; local commutativity of completely spacelike local algebras in every diamond containing both; Axiom 4 as the bridge principle relating physical observables to quasilocal ones; and Lorentz covariance. They are bundled as a Haag–Kastler net (Definition 517, `Physicslib4.AQFT.HaagKastler.HaagKastlerNet`), and the quasilocal algebra of a net is constructed from the net alone, as a directed colimit followed by completion (Theorem 499, `Physicslib4.AQFT.HaagKastler.exists_quasilocalAlgebra`). +- **The Haag–Kastler axioms in Lorentzian spacetime** ([§10.7](sec10-7-curved.html)). The same five axioms on a Lorentzian spacetime, Definitions 638–643, with isometric covariance under the identity component of the isometry group in place of Lorentz covariance, and with commutation and completeness stated relative to a common containing region, since a quasilocal algebra need not exist. They are bundled as `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet` (Definition 644), and general covariance (Definition 690, [§10.8](sec10-8-general-covariance.html)) relates the nets over diffeomorphism-related spacetimes. - **The spectral theorem for bounded self-adjoint operators** ([§10.2](sec10-2-spectral.html)). Every bounded self-adjoint operator $$A$$ on a Hilbert space is $$\int_{\sigma(A)} \lambda \, d\mu^A(\lambda)$$ for a unique projection-valued measure $$\mu^A$$ on its spectrum (Theorem 162, `Physicslib4.Spectral.existsUnique_spectralMeasure`), obtained through the continuous functional calculus, an operator-valued Riesz representation theorem and the bounded Borel functional calculus. - **The spectral theorem for unbounded self-adjoint operators** ([§10.3](sec10-3-unbounded.html)). Every self-adjoint operator $$A$$, bounded or not, is $$\int_{\mathbb{R}} \lambda \, d\mu^A(\lambda)$$ for a unique projection-valued measure $$\mu^A$$ on $$\mathbb{R}$$, concentrated on the spectrum (Theorem 285, `Physicslib4.Spectral.Unbounded.existsUnique_spectralMeasure_unbounded`). The proof passes through the Cayley transform and the spectral theorem for bounded normal operators; the resulting functional calculus (Definition 287) underlies Stone's theorem in [§10.4](sec10-4-stone.html). @@ -74,7 +74,7 @@ The blueprint annotates every node with the Lean declarations that realise it, s **Formalised, but the Lean is shaped differently from the prose (one place).** The node below carries Lean declarations and a formalised proof; the caveat is one of presentation, not of coverage or content. -- **Axiom 5 in curved spacetime (Definition 632).** "Isometries connected to the identity" and "identity-component isometries preserving the future orientation" describe the same group, but the inclusion of the former in the latter rests on a Myers–Steenrod-type rigidity result that Mathlib does not provide. The Lean therefore intersects the identity component with the explicitly orientation-preserving subgroup. This is an implementation choice and does not alter the mathematical content of the axiom. +- **Axiom 5 in curved spacetime (Definition 643).** "Isometries connected to the identity" and "identity-component isometries preserving the future orientation" describe the same group, but the inclusion of the former in the latter rests on a Myers–Steenrod-type rigidity result that Mathlib does not provide. The Lean therefore intersects the identity component with the explicitly orientation-preserving subgroup. This is an implementation choice and does not alter the mathematical content of the axiom. ## Useful links diff --git a/home_page/sec10-2-spectral.md b/home_page/sec10-2-spectral.md index e0f39b5..f2a6ca2 100644 --- a/home_page/sec10-2-spectral.md +++ b/home_page/sec10-2-spectral.md @@ -1,5 +1,5 @@ --- -title: "§10.2 Spectral Theorems (pp. 38–155)" +title: "§10.2 Spectral Theorems (pp. 39–156)" usemathjax: true --- @@ -7,16 +7,16 @@ usemathjax: true ← [§10.1 GNS](sec10-1-gns.html) · [§10.3 Unbounded](sec10-3-unbounded.html) → -# §10.2 Spectral Theorems (pp. 38–155) +# §10.2 Spectral Theorems (pp. 39–156) -The largest section by declaration count (144 declarations, items 19–163, together with Convention 22), stating and proving the Spectral Theorem for Bounded, Self-Adjoint Operators (Theorem 162), a result the blueprint describes as sitting at the core of much of AQFT. It follows the presentation of Hall's *Quantum Theory for Mathematicians*. Standard results whose proofs lie outside the development — Heine–Borel, Riesz representation, Stone–Weierstrass, bounded convergence and the like — are stated in full but not proved in the prose, and are formalised all the same, usually directly by the corresponding Mathlib declaration. The whole section is a single subsection, §10.2.1 Spectral Theorem: Bounded Self-Adjoint Operators (p. 38), which proceeds in six layers. +The largest section by proposition count (144 declarations, items 19–163, together with Convention 22), stating and proving the Spectral Theorem for Bounded, Self-Adjoint Operators (Theorem 162), a result the blueprint describes as sitting at the core of much of AQFT. It follows the presentation of Hall's *Quantum Theory for Mathematicians*. Standard results whose proofs lie outside the development — Heine–Borel, Riesz representation, Stone–Weierstrass, bounded convergence and the like — are stated in full but not proved in the prose, and are formalised all the same, usually directly by the corresponding Mathlib declaration. The whole section is a single subsection, §10.2.1 Spectral Theorem: Bounded Self-Adjoint Operators (p. 39), which proceeds in six layers. - *Elementary properties of bounded operators (pp. 39–45, items 19–37).* Fixes the inner product, the induced norm and the bounded-operator notation (Definitions 19–21), the standing convention that the Hilbert space is non-zero (Convention 22), the identity operator, indicator functions and orthogonal complements (Definitions 23–26), and basic facts about integrals, Cauchy–Schwarz, the reverse triangle inequality and continuity of the norm and inner product (Propositions 27–30). $$\mathcal{B}(\mathbf{H})$$ is a Banach space (Lemma 31); bounded inverses, the resolvent and the spectrum are defined (Definitions 32–33); and the Riesz theorem (Theorem 34) yields the adjoint of a bounded operator (Definition 35), which is bounded (Proposition 36) and has the expected algebraic properties (Lemma 37). -- *Projection-valued measures (p. 45, items 38–67).* Orthogonal projections (Definition 38, Lemma 39) and projection-valued measures (Definition 40), with their associated scalar measures (Theorem 41) and operator-valued integration of bounded measurable functions (Theorem 42). The integral is built from bounded sesquilinear and quadratic forms (Definitions 43–44) and the simple-function approximation theorem (Theorem 45): the quadratic form of a bounded measurable function is bounded (Lemma 46, Propositions 47–49), and a bounded quadratic form determines a unique bounded operator (Propositions 50 and 55, Lemmas 51–54). The section then proves the norm bounds (Propositions 57 and 60, Lemmas 58–59), approximation by integrals of simple functions (Proposition 61), multiplicativity (Propositions 62–66), and compatibility with conjugation and the adjoint (Proposition 67). -- *Stage 1: the continuous functional calculus (p. 80, items 68–104).* The Neumann series (Lemma 70) shows the spectrum is closed, bounded and non-empty (Proposition 73), with holomorphy of the resolvent (Proposition 74). The spectrum of a self-adjoint operator is real (Proposition 78) and compact (Lemma 80). The spectral radius (Definition 81) is at most the operator norm (Corollary 82), is given by Gelfand's formula (Theorem 85), and equals the norm for a self-adjoint operator (Lemma 86). Polynomial spectral mapping (Lemma 89), Stone–Weierstrass (Theorem 92) and the Bounded Linear Transformation Theorem (Theorem 94) then give the continuous functional calculus (Proposition 95), which is multiplicative, produces self-adjoint operators, preserves non-negativity, and satisfies the norm and spectral-mapping identities (Propositions 99–104). -- *Stage 2: an operator-valued Riesz representation theorem (p. 107, items 105–133).* Riesz representation (Theorem 105) attaches measures to a self-adjoint operator (Proposition 106), and hence a bounded quadratic form to every bounded measurable function (Definition 107, Proposition 109). The class of functions with bounded quadratic form (Definition 110) contains the continuous functions (Proposition 115) and is closed under bounded pointwise limits (Proposition 117). A monotone-class argument through the algebra of sets $$\mathcal{L}_0$$ and bump functions (Definitions 119, 122 and 127, Theorems 128–129, Lemmas 121, 130 and 132–133) shows that the class contains every bounded Borel function. -- *The bounded Borel functional calculus and the spectral measure (pp. 127–146, items 134–155).* The bounded Borel functional calculus (Definition 134) sends real functions to self-adjoint operators (Lemma 135) and is multiplicative (Proposition 137, via the classes $$\mathcal{F}_1$$ and $$\mathcal{F}_2$$ of Definition 138 and Propositions 139–142). The spectral projections of Borel sets form the spectral measure of the operator (Theorem 143): each is an orthogonal projection, they multiply to the intersection, and they are countably additive (Propositions 144–147, using the convergence of sums of pairwise orthogonal projections, Lemma 149 and Propositions 150–152). The two bounded functional calculi agree, and the spectral measure integrates to the operator itself (Propositions 153–155). -- *Uniqueness and the spectral theorem (p. 147, items 156–163).* The spectral measure is unique (Theorem 156): two spectral measures of the same operator agree on polynomials, on continuous functions (via complex Stone–Weierstrass, Theorem 159, and Lemma 160) and on bounded measurable functions (Propositions 157, 158 and 161). This gives the Spectral Theorem for Bounded, Self-Adjoint Operators (Theorem 162) and the functional calculus it induces (Definition 163). +- *Projection-valued measures (p. 46, items 38–67).* Orthogonal projections (Definition 38, Lemma 39) and projection-valued measures (Definition 40), with their associated scalar measures (Theorem 41) and operator-valued integration of bounded measurable functions (Theorem 42). The integral is built from bounded sesquilinear and quadratic forms (Definitions 43–44) and the simple-function approximation theorem (Theorem 45): the quadratic form of a bounded measurable function is bounded (Lemma 46, Propositions 47–49), and a bounded quadratic form determines a unique bounded operator (Propositions 50 and 55, Lemmas 51–54). The section then proves the norm bounds (Propositions 57 and 60, Lemmas 58–59), approximation by integrals of simple functions (Proposition 61), multiplicativity (Propositions 62–66), and compatibility with conjugation and the adjoint (Proposition 67). +- *Stage 1: the continuous functional calculus (p. 81, items 68–104).* The Neumann series (Lemma 70) shows the spectrum is closed, bounded and non-empty (Proposition 73), with holomorphy of the resolvent (Proposition 74). The spectrum of a self-adjoint operator is real (Proposition 78) and compact (Lemma 80). The spectral radius (Definition 81) is at most the operator norm (Corollary 82), is given by Gelfand's formula (Theorem 85), and equals the norm for a self-adjoint operator (Lemma 86). Polynomial spectral mapping (Lemma 89), Stone–Weierstrass (Theorem 92) and the Bounded Linear Transformation Theorem (Theorem 94) then give the continuous functional calculus (Proposition 95), which is multiplicative, produces self-adjoint operators, preserves non-negativity, and satisfies the norm and spectral-mapping identities (Propositions 99–104). +- *Stage 2: an operator-valued Riesz representation theorem (p. 108, items 105–133).* Riesz representation (Theorem 105) attaches measures to a self-adjoint operator (Proposition 106), and hence a bounded quadratic form to every bounded measurable function (Definition 107, Proposition 109). The class of functions with bounded quadratic form (Definition 110) contains the continuous functions (Proposition 115) and is closed under bounded pointwise limits (Proposition 117). A monotone-class argument through the algebra of sets $$\mathcal{L}_0$$ and bump functions (Definitions 119, 122 and 127, Theorems 128–129, Lemmas 121, 130 and 132–133) shows that the class contains every bounded Borel function. +- *The bounded Borel functional calculus and the spectral measure (pp. 128–147, items 134–155).* The bounded Borel functional calculus (Definition 134) sends real functions to self-adjoint operators (Lemma 135) and is multiplicative (Proposition 137, via the classes $$\mathcal{F}_1$$ and $$\mathcal{F}_2$$ of Definition 138 and Propositions 139–142). The spectral projections of Borel sets form the spectral measure of the operator (Theorem 143): each is an orthogonal projection, they multiply to the intersection, and they are countably additive (Propositions 144–147, using the convergence of sums of pairwise orthogonal projections, Lemma 149 and Propositions 150–152). The two bounded functional calculi agree, and the spectral measure integrates to the operator itself (Propositions 153–155). +- *Uniqueness and the spectral theorem (p. 148, items 156–163).* The spectral measure is unique (Theorem 156): two spectral measures of the same operator agree on polynomials, on continuous functions (via complex Stone–Weierstrass, Theorem 159, and Lemma 160) and on bounded measurable functions (Propositions 157, 158 and 161). This gives the Spectral Theorem for Bounded, Self-Adjoint Operators (Theorem 162) and the functional calculus it induces (Definition 163). **Where the Lean lives:** `Physicslib4/Spectral/` (`Basic`, `Spectrum`, `Forms`, `ProjectionValuedMeasure`, `OperatorIntegral`, `ContinuousCalculus`, `BorelClasses`, `BorelCalculus`, `SpectralTheorem`) @@ -56,7 +56,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.2.1 · Spectral Theorem for Bounded Self-Adjoint Operators › Projection-Valued Measures (p. 45) — 30 items +§10.2.1 · Spectral Theorem for Bounded Self-Adjoint Operators › Projection-Valued Measures (p. 46) — 30 items - [**Definition 38**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:bounded-orthogonal-projection) — Orthogonal Projection · `IsStarProjection` (+2 more) - [**Lemma 39**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:projection-norm-decreasing) — Orthogonal Projections are Norm-Decreasing · `Physicslib4.Spectral.norm_apply_le_of_isStarProjection` @@ -75,8 +75,8 @@ Each blueprint subsection below is collapsed; click a heading to see its items. - [**Lemma 52**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:sesquilinear-pointwise-limit) — Pointwise Limits of Sesquilinear Forms are Sesquilinear · `Physicslib4.Spectral.exists_sesquilinearForm_of_tendsto` - [**Lemma 53**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bqf-linear-combination) — Linear Combinations of Bounded Quadratic Forms are Bounded Quadratic Forms · `Physicslib4.Spectral.IsBoundedQuadraticForm.sum` - [**Lemma 54**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bqf-pointwise-limit) — Uniformly Bounded Pointwise Limits of Bounded Quadratic Forms · `Physicslib4.Spectral.isBoundedQuadraticForm_of_tendsto` -- [**Proposition 55**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#prpstn:hall-a.63) — A Bounded Quadratic Form Determines a Unique Bounded Operator · `Physicslib4.Spectral.existsUnique_operator_of_isBoundedQuadraticForm` -- [**Proposition 56**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#prpstn:integral-of-indicator) — Integral of an Indicator Function · `Physicslib4.Spectral.ProjectionValuedMeasure.integral_indicator` +- [**Proposition 55**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#prpstn:hall-a.63) — A Bounded Quadratic Form Determines a Unique Bounded Operator · `Physicslib4.Spectral.existsUnique_operator_of_isBoundedQuadraticForm` (+1 more) +- [**Proposition 56**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#prpstn:integral-of-indicator) — Integral of an Indicator Function · `Physicslib4.Spectral.ProjectionValuedMeasure.integral_indicator` (+1 more) - [**Proposition 57**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#prpstn:integral-norm-bound) — Norm Bound for the Operator-Valued Integral · `Physicslib4.Spectral.ProjectionValuedMeasure.norm_integral_le` - [**Lemma 58**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:lemma2-of-operator-valued-integration) — Orthogonality of Spectral Projections on a Disjoint Cover · `Physicslib4.Spectral.ProjectionValuedMeasure.inner_eq_zero_of_disjoint` (+1 more) - [**Lemma 59**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:lemma-1) — Operator Norm via the Inner Product · `Physicslib4.Spectral.opNorm_eq_sSup_inner` @@ -92,7 +92,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.2.1 · Spectral Theorem for Bounded Self-Adjoint Operators › The Spectral Theorem › Stage 1: The Continuous Functional Calculus (p. 80) — 37 items +§10.2.1 · Spectral Theorem for Bounded Self-Adjoint Operators › The Spectral Theorem › Stage 1: The Continuous Functional Calculus (p. 81) — 37 items - [**Lemma 68**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:lemma-2) — Bounded Operator Product is Submultiplicative · `norm_mul_le` - [**Proposition 69**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#prpstn:hall-a.34) — Absolute Convergence Implies Convergence in a Banach Space · `Summable.of_norm` @@ -135,7 +135,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.2.1 · Spectral Theorem for Bounded Self-Adjoint Operators › The Spectral Theorem › Stage 2: An Operator-Valued Riesz Representation Theorem (p. 107) — 29 items +§10.2.1 · Spectral Theorem for Bounded Self-Adjoint Operators › The Spectral Theorem › Stage 2: An Operator-Valued Riesz Representation Theorem (p. 108) — 29 items - [**Theorem 105**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:riesz-representation) — Riesz Representation · `Physicslib4.Spectral.existsUnique_measure_of_positive_linear` - [**Proposition 106**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#prpstn:associated-measures-self-adjoint) — The Measures Associated to a Self-Adjoint Operator · `Physicslib4.Spectral.existsUnique_assocMeasure` @@ -170,7 +170,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.2.1 · Spectral Theorem for Bounded Self-Adjoint Operators › The Spectral Theorem › The Bounded Borel Functional Calculus (p. 127) — 9 items +§10.2.1 · Spectral Theorem for Bounded Self-Adjoint Operators › The Spectral Theorem › The Bounded Borel Functional Calculus (p. 128) — 9 items - [**Definition 134**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:hall-8.8) — The Bounded Borel Functional Calculus · `Physicslib4.Spectral.existsUnique_borelCalculus` (+1 more) - [**Lemma 135**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:lemma-3) — The Operator of a Real-Valued Function is Self-Adjoint · `Physicslib4.Spectral.isSelfAdjoint_borelCalculus` @@ -185,7 +185,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.2.1 · Spectral Theorem for Bounded Self-Adjoint Operators › The Spectral Theorem › The Spectral Measure (p. 133) — 13 items +§10.2.1 · Spectral Theorem for Bounded Self-Adjoint Operators › The Spectral Theorem › The Spectral Measure (p. 134) — 13 items - [**Theorem 143**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:hall-8.10) — The Spectral Measure of a Self-Adjoint Operator · `Physicslib4.Spectral.exists_spectralMeasure` - [**Proposition 144**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#prpstn:mua-projection) — Each Spectral Projection is an Orthogonal Projection · `Physicslib4.Spectral.isStarProjection_borelCalculus_indicator` @@ -204,7 +204,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.2.1 · Spectral Theorem for Bounded Self-Adjoint Operators › The Spectral Theorem › Uniqueness of the Spectral Measure (p. 147) — 8 items +§10.2.1 · Spectral Theorem for Bounded Self-Adjoint Operators › The Spectral Theorem › Uniqueness of the Spectral Measure (p. 148) — 8 items - [**Theorem 156**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:hall-prblm-8.3.4) — Uniqueness of the Spectral Measure · `Physicslib4.Spectral.pvm_eq_of_integral_id_eq` - [**Proposition 157**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#prpstn:pvm-agree-on-polynomials) — Two Spectral Measures Agree on Polynomials · `Physicslib4.Spectral.pvm_integral_polynomial_eq` diff --git a/home_page/sec10-3-unbounded.md b/home_page/sec10-3-unbounded.md index 85719f7..2c6fee8 100644 --- a/home_page/sec10-3-unbounded.md +++ b/home_page/sec10-3-unbounded.md @@ -1,5 +1,5 @@ --- -title: "§10.3 Unbounded Spectral Theorems (pp. 155–235)" +title: "§10.3 Unbounded Spectral Theorems (pp. 156–236)" usemathjax: true --- @@ -7,25 +7,25 @@ usemathjax: true ← [§10.2 Spectral](sec10-2-spectral.html) · [§10.4 Stone](sec10-4-stone.html) → -# §10.3 Unbounded Spectral Theorems (pp. 155–235) +# §10.3 Unbounded Spectral Theorems (pp. 156–236) 123 declarations (items 164–288, together with Conventions 213 and 260), extending §10.2 to the Spectral Theorem for Unbounded, Self-Adjoint Operators (Theorem 285) and the functional calculus it supplies. The blueprint motivates this with the few genuinely unbounded self-adjoint operators that arise in AQFT, the momentum operator being the standard example. The section reuses §10.2 without restating it, follows the same Hall presentation, and builds up unbounded operators from scratch. Its single subsection, §10.3.1 Spectral Theorem: Unbounded Self-Adjoint Operators (p. 156), proceeds in six stages. -- *Unbounded operators (pp. 156–181, items 164–212).* Unbounded operators (Definition 164), their adjoints (Definition 168, Propositions 169–170, Lemma 171), symmetric operators, extensions and self-adjointness (Definitions 172–173 and 175, Proposition 174), positive operators, with positivity agreeing with the bounded notion (Definition 176, Lemma 177), and closed and closable operators via the graph in $$\mathbf{H} \times \mathbf{H}$$ (Definitions 178 and 181, Proposition 182), with the closure of a symmetric operator symmetric (Lemma 183) and essential self-adjointness defined (Definition 184). Elementary properties follow: symmetric operators are closable (Proposition 186), the adjoint of a closure (Proposition 187), uniqueness of the self-adjoint extension of an essentially self-adjoint operator (Proposition 190), and kernel and range, with the orthogonal complement of the range equal to the kernel of the adjoint (Definitions 191–192, Proposition 195). Next come the resolvent set and spectrum of an unbounded operator (Definition 199), which agree with the bounded notions (Lemma 201) and are real for self-adjoint operators (Theorem 203). The layer closes with criteria for essential self-adjointness: dense range (Theorem 204), and, through Hilbert direct sums and internal orthogonal decompositions (Definitions 205, 207 and 208, Lemmas 206 and 210), direct sums of bounded self-adjoint operators (Propositions 211–212). -- *Integration against a projection-valued measure (p. 181, items 213–240).* Under standing hypotheses (Convention 213), and with sesquilinear and quadratic forms on subspaces (Definitions 216–217) and the needed measure theory (Theorems 220–221, Propositions 222–226, Definition 224), the integral of an unbounded measurable function against a projection-valued measure is constructed on its natural domain (Propositions 232 and 234). It coincides with the bounded integral of §10.2 (Proposition 235), ignores null sets (Lemma 236), is the limit of its truncations (Lemma 237), and is self-adjoint for real-valued functions (Proposition 240). -- *Bounded normal operators (p. 199, items 241–259).* For normal operators (Definition 241) the norm equals the spectral radius (Proposition 246). Spectral subspaces (Definition 247, Propositions 248–249) and almost eigenvectors (Definition 251, Lemmas 250 and 252–256) lead to the two-variable spectral mapping theorem for polynomials in $$A$$ and $$A^*$$ (Theorem 257, Corollary 258), and hence to the continuous functional calculus for a normal operator (Theorem 259). Lemma 253 is stated with a constant uniform over all eigenvalues $$\lvert \lambda \rvert \le R$$, which is what the proof of Theorem 257 consumes; the Lean proof of Theorem 257 itself takes a shorter route through Mathlib's continuous functional calculus (`cfc_map_spectrum`), as a formalization note in the blueprint records, while the almost-eigenvector lemmas are formalised in their own right. -- *From a continuous functional calculus to a projection-valued measure (p. 213, items 260–274).* Under standing hypotheses (Convention 260), an abstract continuous functional calculus (Definition 261) is extended, through its associated measures (Definition 263), to bounded measurable functions (Definition 266). The extension is linear, multiplicative and compatible with conjugation (Lemmas 267 and 271, Proposition 270), and it yields a projection-valued measure (Theorem 272). Combined with the previous stage, this gives the Spectral Theorem for Bounded Normal Operators (Theorem 274). -- *The Cayley transform and the unbounded spectral theorem (pp. 222–231, items 275–285).* Unitary operators are normal, with spectrum on the unit circle (Lemmas 275–276). The Cayley map (Lemma 277) and the Cayley transform of a self-adjoint operator (Theorem 278), with its spectral mapping (Lemma 279), reduce the unbounded self-adjoint case to the bounded normal one: a projection-valued measure is transported along a Borel bijection (Lemma 281), the Cayley transform omits the point $$1$$ (Lemma 282), and the spectral measure is transported through the Cayley transform (Theorem 284). The result is the Spectral Theorem for Unbounded, Self-Adjoint Operators (Theorem 285). -- *The functional calculus of an unbounded self-adjoint operator (p. 234, items 286–288).* These items supply what §10.4 needs. The spectral measure of a self-adjoint operator is named (Definition 286), the functional calculus $$f(A)$$ is the integral of $$f$$ against it (Definition 287), and for bounded $$f$$ it agrees with the bounded calculus of §10.2 (Lemma 288). Earlier in the section, a self-adjoint operator is shown to have no proper symmetric extension (Lemma 189), which §10.4 uses to identify the generator of $$e^{itA}$$ with $$A$$. +- *Unbounded operators (pp. 157–181, items 164–212).* Unbounded operators (Definition 164), their adjoints (Definition 168, Propositions 169–170, Lemma 171), symmetric operators, extensions and self-adjointness (Definitions 172–173 and 175, Proposition 174), positive operators, with positivity agreeing with the bounded notion (Definition 176, Lemma 177), and closed and closable operators via the graph in $$\mathbf{H} \times \mathbf{H}$$ (Definitions 178 and 181, Proposition 182), with the closure of a symmetric operator symmetric (Lemma 183) and essential self-adjointness defined (Definition 184). Elementary properties follow: symmetric operators are closable (Proposition 186), the adjoint of a closure (Proposition 187), uniqueness of the self-adjoint extension of an essentially self-adjoint operator (Proposition 190), and kernel and range, with the orthogonal complement of the range equal to the kernel of the adjoint (Definitions 191–192, Proposition 195). Next come the resolvent set and spectrum of an unbounded operator (Definition 199), which agree with the bounded notions (Lemma 201) and are real for self-adjoint operators (Theorem 203). The layer closes with criteria for essential self-adjointness: dense range (Theorem 204), and, through Hilbert direct sums and internal orthogonal decompositions (Definitions 205, 207 and 208, Lemmas 206 and 210), direct sums of bounded self-adjoint operators (Propositions 211–212). +- *Integration against a projection-valued measure (p. 182, items 213–240).* Under standing hypotheses (Convention 213), and with sesquilinear and quadratic forms on subspaces (Definitions 216–217) and the needed measure theory (Theorems 220–221, Propositions 222–226, Definition 224), the integral of an unbounded measurable function against a projection-valued measure is constructed on its natural domain (Propositions 232 and 234). It coincides with the bounded integral of §10.2 (Proposition 235), ignores null sets (Lemma 236), is the limit of its truncations (Lemma 237), and is self-adjoint for real-valued functions (Proposition 240). +- *Bounded normal operators (p. 201, items 241–259).* For normal operators (Definition 241) the norm equals the spectral radius (Proposition 246). Spectral subspaces (Definition 247, Propositions 248–249) and almost eigenvectors (Definition 251, Lemmas 250 and 252–256) lead to the two-variable spectral mapping theorem for polynomials in $$A$$ and $$A^*$$ (Theorem 257, Corollary 258), and hence to the continuous functional calculus for a normal operator (Theorem 259). Lemma 253 is stated with a constant uniform over all eigenvalues $$\lvert \lambda \rvert \le R$$, which is what the proof of Theorem 257 consumes; the Lean proof of Theorem 257 itself takes a shorter route through Mathlib's continuous functional calculus (`cfc_map_spectrum`), as a formalization note in the blueprint records, while the almost-eigenvector lemmas are formalised in their own right. +- *From a continuous functional calculus to a projection-valued measure (p. 215, items 260–274).* Under standing hypotheses (Convention 260), an abstract continuous functional calculus (Definition 261) is extended, through its associated measures (Definition 263), to bounded measurable functions (Definition 266). The extension is linear, multiplicative and compatible with conjugation (Lemmas 267 and 271, Proposition 270), and it yields a projection-valued measure (Theorem 272). Combined with the previous stage, this gives the Spectral Theorem for Bounded Normal Operators (Theorem 274). +- *The Cayley transform and the unbounded spectral theorem (pp. 223–232, items 275–285).* Unitary operators are normal, with spectrum on the unit circle (Lemmas 275–276). The Cayley map (Lemma 277) and the Cayley transform of a self-adjoint operator (Theorem 278), with its spectral mapping (Lemma 279), reduce the unbounded self-adjoint case to the bounded normal one: a projection-valued measure is transported along a Borel bijection (Lemma 281), the Cayley transform omits the point $$1$$ (Lemma 282), and the spectral measure is transported through the Cayley transform (Theorem 284). The result is the Spectral Theorem for Unbounded, Self-Adjoint Operators (Theorem 285). +- *The functional calculus of an unbounded self-adjoint operator (p. 235, items 286–288).* These items supply what §10.4 needs. The spectral measure of a self-adjoint operator is named (Definition 286), the functional calculus $$f(A)$$ is the integral of $$f$$ against it (Definition 287), and for bounded $$f$$ it agrees with the bounded calculus of §10.2 (Lemma 288). Earlier in the section, a self-adjoint operator is shown to have no proper symmetric extension (Lemma 189), which §10.4 uses to identify the generator of $$e^{itA}$$ with $$A$$. -**Where the Lean lives:** `Physicslib4/Spectral/Unbounded/` (`Basic`, `Spectrum`, `DirectSum`, `Integral`, `Normal`, `AbstractCalculus`, `Cayley`) +**Where the Lean lives:** `Physicslib4/Spectral/Unbounded/` (`Basic`, `Spectrum`, `DirectSum`, `Integral`, `Normal`, `AbstractCalculus`, `Cayley`, `FunctionalCalculus`) ## Items Each blueprint subsection below is collapsed; click a heading to see its items. Every entry links to its node in the web blueprint and names the principal Lean declaration.
-§10.3.1 · Unbounded Operators › Adjoint and Closure of an Unbounded Operator (p. 156) — 21 items +§10.3.1 · Unbounded Operators › Adjoint and Closure of an Unbounded Operator (p. 157) — 21 items - [**Definition 164**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:hall-3.1) — Unbounded Operator · `LinearPMap` (+1 more) - [**Proposition 165**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#prpstn:convergence-facts) — Convergence Facts for Sequences and Series · `tendsto_atTop_ciSup` (+6 more) @@ -52,7 +52,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.3.1 · Unbounded Operators › Elementary Properties of Adjoints and Closed Operators (p. 164) — 14 items +§10.3.1 · Unbounded Operators › Elementary Properties of Adjoints and Closed Operators (p. 165) — 14 items - [**Definition 185**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:closed-linear-map-on-a-subspace) — Closed Linear Map on a Subspace · `LinearPMap.IsClosed` (+1 more) - [**Proposition 186**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#prpstn:hall-9.8) — Closed Graph of the Adjoint; Closability of Symmetric Operators · `Physicslib4.Spectral.Unbounded.isClosable_of_isSymmetric` (+1 more) @@ -72,7 +72,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.3.1 · Unbounded Operators › The Spectrum of an Unbounded Operator (p. 170) — 5 items +§10.3.1 · Unbounded Operators › The Spectrum of an Unbounded Operator (p. 171) — 5 items - [**Definition 199**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:hall-9.16) — Resolvent Set and Spectrum of an Unbounded Operator · `Physicslib4.Spectral.Unbounded.pmapResolventSet` (+2 more) - [**Lemma 200**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:uniqueness-of-resolvent) — Uniqueness of the Resolvent · `Physicslib4.Spectral.Unbounded.existsUnique_isResolvent` @@ -83,7 +83,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.3.1 · Unbounded Operators › Conditions for Self-Adjointness and Essential Self-Adjointness (p. 173) — 9 items +§10.3.1 · Unbounded Operators › Conditions for Self-Adjointness and Essential Self-Adjointness (p. 174) — 9 items - [**Theorem 204**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:hall-9.21) — Essential Self-Adjointness via Dense Range · `Physicslib4.Spectral.Unbounded.isEssentiallySelfAdjoint_iff_dense_range` - [**Definition 205**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:hall-a.45) — Hilbert Space Direct Sum · `lp` (+1 more) @@ -98,7 +98,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.3.1 · Integration Against a Projection-Valued Measure (p. 181) — 28 items +§10.3.1 · Integration Against a Projection-Valued Measure (p. 182) — 28 items - [**Convention 213**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#conv:section-integration) — Standing Hypotheses: Integration against a Projection-Valued Measure *(a standing convention, not a declaration; it carries no Lean annotation)* - [**Lemma 214**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:associated-measure-total-mass) — The Associated Measure has Total Mass $$\left\| \psi \right\|^2$$ · `Physicslib4.Spectral.Unbounded.assoc_univ_eq` (+1 more) @@ -132,7 +132,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.3.1 · The Spectral Theorem for Bounded Normal Operators (p. 199) — 6 items +§10.3.1 · The Spectral Theorem for Bounded Normal Operators (p. 201) — 6 items - [**Definition 241**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:hall-10.19) — Normal Operator · `IsStarNormal` - [**Lemma 242**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:self-adjoint-is-normal) — Bounded Self-Adjoint Operators are Normal · `IsSelfAdjoint.isStarNormal` @@ -144,7 +144,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.3.1 · The Spectral Theorem for Bounded Normal Operators › Spectral Subspaces (p. 203) — 3 items +§10.3.1 · The Spectral Theorem for Bounded Normal Operators › Spectral Subspaces (p. 204) — 3 items - [**Definition 247**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:hall-7.14) — Spectral Subspaces · `Physicslib4.Spectral.Unbounded.spectralSubspace` - [**Proposition 248**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#prpstn:hall-7.15) — Properties of Spectral Subspaces · `Physicslib4.Spectral.Unbounded.pvmOperator` (+3 more) @@ -153,7 +153,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.3.1 · The Spectral Theorem for Bounded Normal Operators › Almost Eigenvectors (p. 206) — 7 items +§10.3.1 · The Spectral Theorem for Bounded Normal Operators › Almost Eigenvectors (p. 208) — 7 items - [**Lemma 250**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:normality-balances-norms) — Normality Balances the Two Norms · `Physicslib4.Spectral.Unbounded.norm_adjoint_sub_smul_apply` - [**Definition 251**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:hall-10.24) — $$\varepsilon$$-Almost Eigenvector · `Physicslib4.Spectral.Unbounded.IsAlmostEigenvector` @@ -166,7 +166,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.3.1 · The Spectral Theorem for Bounded Normal Operators › The Two-Variable Spectral Mapping Theorem (p. 211) — 2 items +§10.3.1 · The Spectral Theorem for Bounded Normal Operators › The Two-Variable Spectral Mapping Theorem (p. 212) — 2 items - [**Theorem 257**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:hall-10.23) — Spectral Mapping for Polynomials in $$A$$ and $$A^*$$ · `Physicslib4.Spectral.Unbounded.spectrum_mvApply` - [**Corollary 258**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#crllr:norm-of-polynomial-in-a-astar) — Norm of a Polynomial in $$A$$ and $$A^*$$ · `Physicslib4.Spectral.Unbounded.norm_mvApply` @@ -174,14 +174,14 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.3.1 · The Spectral Theorem for Bounded Normal Operators › The Continuous Functional Calculus for a Normal Operator (p. 212) — 1 item +§10.3.1 · The Spectral Theorem for Bounded Normal Operators › The Continuous Functional Calculus for a Normal Operator (p. 213) — 1 item - [**Theorem 259**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:continuous-functional-calculus-normal) — Continuous Functional Calculus for a Normal Operator · `Physicslib4.Spectral.Unbounded.mvPolyOn` (+9 more)
-§10.3.1 · From a Continuous Functional Calculus to a Projection-Valued Measure (p. 213) — 14 items +§10.3.1 · From a Continuous Functional Calculus to a Projection-Valued Measure (p. 215) — 14 items - [**Convention 260**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#conv:section-abstract) — Standing Hypotheses: The Abstract Functional Calculus *(a standing convention, not a declaration; it carries no Lean annotation)* - [**Definition 261**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:abstract-continuous-functional-calculus) — Abstract Continuous Functional Calculus · `Physicslib4.Spectral.Unbounded.IsAbstractCalculus` @@ -201,14 +201,14 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.3.1 · From a Continuous Functional Calculus to a Projection-Valued Measure › The Spectral Theorem for Bounded Normal Operators (p. 220) — 1 item +§10.3.1 · From a Continuous Functional Calculus to a Projection-Valued Measure › The Spectral Theorem for Bounded Normal Operators (p. 222) — 1 item - [**Theorem 274**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:hall-10.20) — Spectral Theorem for Bounded Normal Operators · `Physicslib4.Spectral.Unbounded.existsUnique_spectralMeasure_normal` (+1 more)
-§10.3.1 · The Cayley Transform (p. 222) — 5 items +§10.3.1 · The Cayley Transform (p. 223) — 5 items - [**Lemma 275**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:unitary-is-normal) — Unitary Operators are Normal · `Physicslib4.Spectral.Unbounded.unitary_mul_adjoint` - [**Lemma 276**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:unitary-spectrum-circle) — The Spectrum of a Unitary Operator Lies on the Unit Circle · `spectrum.subset_circle_of_unitary` (+1 more) @@ -219,7 +219,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.3.1 · Proof of the Spectral Theorem for Unbounded Self-Adjoint Operators (p. 226) — 9 items +§10.3.1 · Proof of the Spectral Theorem for Unbounded Self-Adjoint Operators (p. 227) — 9 items - [**Theorem 280**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:change-of-variables) — Change of Variables for a Pushforward Measure · `MeasureTheory.lintegral_map_equiv` (+1 more) - [**Lemma 281**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:borel-bijection-transports-pvm) — A Borel Bijection Transports a Projection-Valued Measure · `Physicslib4.Spectral.Unbounded.restrictPVM` (+2 more) diff --git a/home_page/sec10-4-stone.md b/home_page/sec10-4-stone.md index 6dd69cf..bba527f 100644 --- a/home_page/sec10-4-stone.md +++ b/home_page/sec10-4-stone.md @@ -1,5 +1,5 @@ --- -title: "§10.4 Stone's Theorem (pp. 235–249)" +title: "§10.4 Stone's Theorem (pp. 236–250)" usemathjax: true --- @@ -7,15 +7,15 @@ usemathjax: true ← [§10.3 Unbounded](sec10-3-unbounded.html) · [§10.5 Spacetime](sec10-5-spacetime.html) → -# §10.4 Stone's Theorem (pp. 235–249) +# §10.4 Stone's Theorem (pp. 236–250) 41 declarations (items 289–329), all formalised and proved. The section proves **Stone's Theorem** (Theorem 329): every strongly continuous one-parameter unitary group $$U$$ on a Hilbert space $$\mathbf{H}$$ is $$U(t) = e^{itA}$$ for a unique self-adjoint operator $$A$$, its infinitesimal generator. Together with the converse direction (Proposition 313) this is a one-to-one correspondence between strongly continuous one-parameter unitary groups and self-adjoint operators. In AQFT it is what turns the translation symmetry of a net into an energy–momentum operator, and it is what lets positive energy (`IsPositiveEnergy`, §10.6) be stated with a genuine unbounded generator. The section builds on the Spectral Theorem for Unbounded, Self-Adjoint Operators (Theorem 285) and the functional calculus it supplies (Definitions 286–287, Lemma 288), and proceeds in five subsections. -- *One-parameter unitary groups (p. 236, items 289–300).* A one-parameter unitary group (Definition 289), strong continuity and its reduction to continuity at $$0$$ (Lemma 290), the adjoint $$U(t)^* = U(-t)$$ (Lemma 291), and the infinitesimal generator (Definition 292), with its characterisation (Lemma 293) and the symmetry identity it satisfies (Lemma 294). Three generators are computed directly: the trivial group has generator $$0$$ (Lemma 295), conjugating a group by a unitary $$W$$ conjugates its generator (Lemmas 296–297), and for bounded self-adjoint $$P$$ the group $$\exp(itP)$$ has generator $$P$$ (Lemmas 298–300); these feed the positive-energy lemmas of §10.6. -- *Imported results (p. 239, items 301–305).* Standard analysis stated without proof and linked to Mathlib: the Bochner integral (Theorem 301), smooth approximate identities (Lemma 302), facts about differentiation (Proposition 303) and about convolution (Lemma 305). The scalar equation $$y' = cy$$ (Lemma 304) is proved from them. -- *From a self-adjoint operator to a unitary group (p. 241, items 306–313).* For self-adjoint $$A$$ the bounded functions $$e^{it\lambda}$$ give unitaries (Lemma 306) forming the group $$e^{itA}$$ (Definition 307), which satisfies the group law (Lemma 308) and is strongly continuous by dominated convergence (Lemma 309). A difference-quotient norm identity (Lemmas 310–311) gives the derivative at $$0$$ on the domain of $$A$$ (Lemma 312), and maximal symmetry of a self-adjoint operator identifies the generator with $$A$$ (Proposition 313). -- *Two intermediate results (p. 244, items 314–320).* Orbits $$t \mapsto U(t)\psi$$ are differentiable on the generator's domain (Lemma 314), the group commutes with its generator and preserves its domain (Lemmas 315–316), and averaging an orbit against a smooth bump, written as a convolution (Lemmas 317–319), shows that the generator is densely defined (Lemma 320). -- *Stone's Theorem (p. 246, items 321–329).* An ordinary differential equation along orbits (Lemma 321) and the vanishing of bounded solutions of $$y' = \pm y$$ (Lemma 322) make the deficiency subspaces trivial (Lemma 323), so the generator is essentially self-adjoint (Lemma 324). A norm-preservation argument (Lemmas 325–326) shows that the orbits of $$U$$ and of $$e^{itA}$$ agree on the generator's domain (Lemma 327), hence everywhere by density (Lemma 328), which proves Stone's Theorem (Theorem 329). +- *One-parameter unitary groups (p. 237, items 289–300).* A one-parameter unitary group (Definition 289), strong continuity and its reduction to continuity at $$0$$ (Lemma 290), the adjoint $$U(t)^* = U(-t)$$ (Lemma 291), and the infinitesimal generator (Definition 292), with its characterisation (Lemma 293) and the symmetry identity it satisfies (Lemma 294). Three generators are computed directly: the trivial group has generator $$0$$ (Lemma 295), conjugating a group by a unitary $$W$$ conjugates its generator (Lemmas 296–297), and for bounded self-adjoint $$P$$ the group $$\exp(itP)$$ has generator $$P$$ (Lemmas 298–300); these feed the positive-energy lemmas of §10.6. +- *Imported results (p. 240, items 301–305).* Standard analysis stated without proof and linked to Mathlib: the Bochner integral (Theorem 301), smooth approximate identities (Lemma 302), facts about differentiation (Proposition 303) and about convolution (Lemma 305). The scalar equation $$y' = cy$$ (Lemma 304) is proved from them. +- *From a self-adjoint operator to a unitary group (p. 242, items 306–313).* For self-adjoint $$A$$ the bounded functions $$e^{it\lambda}$$ give unitaries (Lemma 306) forming the group $$e^{itA}$$ (Definition 307), which satisfies the group law (Lemma 308) and is strongly continuous by dominated convergence (Lemma 309). A difference-quotient norm identity (Lemmas 310–311) gives the derivative at $$0$$ on the domain of $$A$$ (Lemma 312), and maximal symmetry of a self-adjoint operator identifies the generator with $$A$$ (Proposition 313). +- *Two intermediate results (p. 245, items 314–320).* Orbits $$t \mapsto U(t)\psi$$ are differentiable on the generator's domain (Lemma 314), the group commutes with its generator and preserves its domain (Lemmas 315–316), and averaging an orbit against a smooth bump, written as a convolution (Lemmas 317–319), shows that the generator is densely defined (Lemma 320). +- *Stone's Theorem (p. 247, items 321–329).* An ordinary differential equation along orbits (Lemma 321) and the vanishing of bounded solutions of $$y' = \pm y$$ (Lemma 322) make the deficiency subspaces trivial (Lemma 323), so the generator is essentially self-adjoint (Lemma 324). A norm-preservation argument (Lemmas 325–326) shows that the orbits of $$U$$ and of $$e^{itA}$$ agree on the generator's domain (Lemma 327), hence everywhere by density (Lemma 328), which proves Stone's Theorem (Theorem 329). The closing Summary of the blueprint section, the bijection between groups and generators, is recorded in prose and is not a separate declaration. @@ -26,7 +26,7 @@ The closing Summary of the blueprint section, the bijection between groups and g Each blueprint subsection below is collapsed; click a heading to see its items. Every entry links to its node in the web blueprint and names the principal Lean declaration; the four imported results name the Mathlib declarations that supply them.
-§10.4.1 One-Parameter Unitary Groups (p. 236) — 12 items +§10.4.1 One-Parameter Unitary Groups (p. 237) — 12 items - [**Definition 289**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:hall-10.11) — One-Parameter Unitary Group · `Physicslib4.Spectral.Stone.IsOneParameterUnitaryGroup` (+1 more) - [**Lemma 290**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:strongly-continuous-iff-at-zero) — Strong Continuity is Strong Continuity at $$0$$ · `Physicslib4.Spectral.Stone.isStronglyContinuous_iff_tendsto_zero` @@ -44,7 +44,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.4.2 Imported Results (p. 239) — 5 items +§10.4.2 Imported Results (p. 240) — 5 items - [**Theorem 301**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:bochner-integral) — Bochner Integral · `ContinuousLinearMap.integral_comp_comm` (+1 more) - [**Lemma 302**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:smooth-approximate-identity) — Smooth Approximate Identity · `ContDiffBump.normed` (+4 more) @@ -55,7 +55,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.4.3 From a Self-Adjoint Operator to a Unitary Group (p. 241) — 8 items +§10.4.3 From a Self-Adjoint Operator to a Unitary Group (p. 242) — 8 items - [**Lemma 306**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:exp-unitary) — The Bounded Integral of $$e^{it\lambda }$$ is Unitary · `Physicslib4.Spectral.Stone.integral_expFun_mem_unitary` - [**Definition 307**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:exp-unitary-group) — The Unitary Group $$e^{itA}$$ · `Physicslib4.Spectral.Stone.expFun` (+2 more) @@ -69,7 +69,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.4.4 Two Intermediate Results (p. 244) — 7 items +§10.4.4 Two Intermediate Results (p. 245) — 7 items - [**Lemma 314**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:orbit-hasDerivAt) — Differentiating an Orbit · `Physicslib4.Spectral.Stone.IsOneParameterUnitaryGroup.hasDerivAt_orbit` - [**Lemma 315**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:generator-commutes) — The Group Commutes with its Generator · `Physicslib4.Spectral.Stone.IsOneParameterUnitaryGroup.generator_apply_comm` @@ -82,7 +82,7 @@ Each blueprint subsection below is collapsed; click a heading to see its items.
-§10.4.5 Stone's Theorem (p. 246) — 9 items +§10.4.5 Stone's Theorem (p. 247) — 9 items - [**Lemma 321**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:orbit-inner-ode) — An Ordinary Differential Equation along Orbits · `Physicslib4.Spectral.Stone.hasDerivAt_inner_orbit` - [**Lemma 322**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bounded-exp-solution-zero) — Bounded Solutions of $$y' = \pm y$$ Vanish · `Physicslib4.Spectral.Stone.eq_zero_of_hasDerivAt_of_bounded` diff --git a/home_page/sec10-5-spacetime.md b/home_page/sec10-5-spacetime.md index 5d3672f..7a8545c 100644 --- a/home_page/sec10-5-spacetime.md +++ b/home_page/sec10-5-spacetime.md @@ -1,5 +1,5 @@ --- -title: "§10.5 Spacetime and causal structure (pp. 249–287)" +title: "§10.5 Spacetime and causal structure (pp. 250–289)" usemathjax: true --- @@ -7,19 +7,19 @@ usemathjax: true ← [§10.4 Stone](sec10-4-stone.html) · [§10.6 Minkowski](sec10-6-haag-kastler.html) → -# §10.5 Spacetime and causal structure (pp. 249–287) +# §10.5 Spacetime and causal structure (pp. 250–289) -137 items (Definitions 330–466), building the entire causal and topological apparatus the axioms are indexed on. The trips underlying the chronological and causal relations $$\ll$$ and $$\prec$$ are genuine geodesics of the Levi-Civita connection of the metric, which the section constructs for an arbitrary pseudo-Riemannian metric on top of Mathlib's covariant derivatives. It proceeds in nine layers. +146 items (Definitions 330–475), building the entire causal and topological apparatus the axioms are indexed on. The trips underlying the chronological and causal relations $$\ll$$ and $$\prec$$ are genuine geodesics of the Levi-Civita connection of the metric, which the section constructs for an arbitrary pseudo-Riemannian metric on top of Mathlib's covariant derivatives. It proceeds in nine layers. -- *Spacetime, tangent-vector causality, and curves (pp. 249–254, items 330–353).* Gives precise definitions of spacetime (Definition 330) and standard Minkowski spacetime (Definition 331); classifies tangent vectors as timelike, spacelike, or null (Definition 332) and proves the trichotomy (Lemma 333), the reverse Cauchy–Schwarz and reverse triangle inequalities for timelike vectors (Lemmas 334–335), and the cone geometry—orientation of pointing vectors, the sign lemma, definiteness of the spacelike complement of a timelike vector, and convexity of the cones (Lemmas 338–341)—alongside time orientations (Definition 336) and future- and past-pointing vectors (Definition 337). Then paths, curves, and oriented curves are defined as equivalence classes of paths up to reparametrisation (Definitions 342–347), with causal type and future/past orientation each shown well-defined on the quotient (Theorems 346 and 348) and a forgetful projection from oriented to unoriented curves (Theorem 349). Endpoints (Definition 350), shown to be well defined on curves and smooth curves (Lemma 351), come with two point-set lemmas—an extremal parameter lies in the frontier, and two endpoints force a compact parameter interval (Lemmas 352–353). -- *The Levi-Civita connection and geodesics (pp. 255–258, items 354–369).* Pseudo-Riemannian metrics (Definition 354) are smooth, symmetric, non-degenerate families of bilinear forms, with no signature condition, so everything here applies to every spacetime. Covariant derivatives are Mathlib's `IsCovariantDerivativeOn`; metric compatibility and the Levi-Civita condition (compatible and torsion-free) are defined on top of them (Definitions 355–356). The musical isomorphism $$v \mapsto g_x(v, \cdot)$$ is a linear isomorphism at each point (Lemma 357), and its inverse is a smooth section of $$\mathrm{Hom}(T^*M, TM)$$ (Lemma 358). The Koszul formula (Lemma 359), the tensoriality of the Koszul expression (Lemma 360), and the verification that the Koszul connection is a covariant derivative (Lemma 361) that is Levi-Civita (Lemma 362) give existence and uniqueness of the Levi-Civita connection (Theorem 363). Five lemmas about smooth paths—a function vanishing along a path has zero derivative along it, the covariant derivative along a curve is local, local vector fields globalise, a smooth path has a local left inverse, and its velocity extends to a smooth vector field (Lemmas 364–368)—make the geodesic condition chart-free, well defined and non-vacuous. A geodesic (Definition 369) is then an affinely parametrised smooth path $$\mu$$ with $$(\nabla_X X)(\mu(s)) = 0$$ at every interior parameter, for every smooth vector field $$X$$ extending the velocity near $$s$$. -- *Trips, futures and pasts (pp. 259–260, items 370–378).* Trips and causal trips (Definitions 370–371) are finite chains of future-oriented timelike (respectively causal) geodesic segments, geodesic in the sense of Definition 369, with matching past and future endpoints. Chronological and causal precedence are transitive (Theorem 372); the causality condition (Definition 373) makes them a strict partial order (Theorem 374); and chronological and causal futures and pasts (Definitions 375–376) come with their basic inclusion and monotonicity properties (Lemmas 377–378). -- *§10.5.1 Causal diamonds (p. 260, items 379–393).* Introduces the causal diamond $$J^+(p) \cap J^-(q)$$ and the chronological (Alexandrov) diamond $$I^+(p) \cap I^-(q)$$ (Definition 379), with the structural properties of the causal diamond—monotonicity under endpoint spread, causal convexity, and that nonemptiness forces $$p \prec q$$ (Lemma 380)—and the containment of chronological diamonds in causal ones, together with the identification of the Alexandrov basis as exactly the chronological diamonds (Theorem 381). It then develops spacelike separation of points and of regions (Definitions 382–383, Lemmas 384–385), the spacelike complement $$\mathbf{B}^\perp$$ (Definition 386) and its order structure—antitone, extensive on the double complement, with the triple complement collapsing, making complementation the Galois connection attached to the spacelike-separation relation (Lemma 387). The double complement is packaged as the causal closure operator (Definition 388, Lemma 389), whose fixed points are the causally complete regions (Definition 390). These form a complete lattice—meets are intersections, joins are causal closures of unions—on which the causal complement is an order-reversing involution (Theorem 391); the set-level De Morgan laws (Lemma 392) then lift to full binary and infinitary De Morgan laws on the lattice (Theorem 393). The blueprint is careful to record what does *not* hold: the full orthocomplement law $$\mathbf{B} \wedge \mathbf{B}^\perp = \bot$$ fails at this generality, because the trip-based causal relation is irreflexive and so a point is spacelike-separated from itself. -- *§10.5.2 Causal convexity (p. 265, items 394–396).* Defines a causally convex region as one containing every point causally between two of its own points (Definition 394), shows causal diamonds are causally convex (Lemma 395), and shows every spacelike complement—hence every causally complete region—is causally convex (Theorem 396). -- *§10.5.3 Causal convexity: closure structure, and the Alexandrov basis theorems (p. 265, items 397–420).* The causally convex regions are shown to form a closure system (Lemma 397), giving the causal-convex hull (Definition 398) as a genuine closure operator whose closed sets are exactly the causally convex regions (Lemmas 399, Theorem 400). The subsection then turns to the Alexandrov topology (Definition 401): every diamond is open (Lemma 402); chronological futures and pasts are open under an explicit "no endpoints" hypothesis (Lemma 403) and unconditionally on standard Minkowski spacetime, where the coordinate-cone description discharges the hypothesis (Lemma 404). On standard Minkowski spacetime the componentwise directional derivative is a Levi-Civita connection (Lemma 405), hence the Levi-Civita connection is flat (Lemma 406) and affinely parametrised straight lines are geodesics (Lemma 407), so trips there are the familiar straight-line segments. Minkowski spacetime and Lorentzian spacetime are then defined (Definitions 408–409, the latter as a spacetime whose Alexandrov topology is Hausdorff), with a bundled form of spacelike separation and basis openness (Lemma 410). A point lying in no diamond is pathological: its only Alexandrov neighbourhood is the whole space (Lemma 411), and the Hausdorff assumption rules this out, forcing the diamonds to cover the space (Lemma 412). Given downward-directedness the diamonds form a genuine topological basis (Theorem 413); past and future interpolation on standard Minkowski (Lemmas 414–415) supply downward-directedness there (Lemma 416), and common chronological predecessors and successors supply upward-directedness (Lemmas 417–419), so on standard Minkowski the diamonds are an unconditional basis (Theorem 420). Upward-directedness is what the quasilocal colimit of §10.6.1 later consumes. -- *§10.5.4 Dilations are causal automorphisms but not isometries (p. 272, items 421–423).* The dilations $$x \mapsto \lambda x$$ preserve the Minkowski cones (Lemma 421) and are therefore causal automorphisms (Theorem 422), yet scale the metric by $$\lambda^2$$ and so are not isometries for $$\lambda \neq 1$$ (Theorem 423). This is the counterexample that forces the general-covariance morphism of §10.8 to be specified geometrically rather than causally. -- *§10.5.5 Isometries and basis-set preservation (p. 272, items 424–434).* Single-metric transport: isometries preserve the causal classification of tangent vectors (Lemma 424), paths have unique differentials and well-defined pushforwards (Lemmas 425–426), and isometries respect the Levi-Civita structure: pullbacks of vector fields and of the metric pairing, their derivatives and Lie brackets transform naturally (Lemmas 427–429), so an isometry preserves the Levi-Civita connection (Lemma 430) and maps geodesics to geodesics and back (Lemma 431). Future-orientation-preserving isometries preserve chronology (Lemma 432) and map Alexandrov-basis diamonds to diamonds (Lemma 433), which is exactly the well-definedness condition for the Axiom 5 action (Lemma 434). -- *§10.5.6 Pullback metrics and cross-metric isometries (p. 275, items 435–466).* The single-metric lemmas above compare a spacetime with itself; general covariance instead compares two different metrics on one carrier, so the transport statements are redone cross-metric. This subsection defines the pullback $$\psi^* g$$ of a spacetime metric as a bundled family of continuous bilinear forms (Definition 435) and verifies every obligation in turn: the differential of a diffeomorphism is a linear equivalence (Lemma 436), with the two round-trip cancellation identities for a global `Diffeomorph` (Lemmas 437–439); the pullback metric is symmetric, non-degenerate, Lorentzian, and a smooth section of the bilinear-form bundle (Lemmas 440–443), so the pullback of a spacetime is a spacetime (Theorem 444). Two-sided preservation of future orientation is then defined (Definition 445) and the pullback time orientation is shown to be bundle-smooth, nowhere vanishing, and everywhere timelike (Lemmas 446–449), with transport of future-pointing timelike and null vectors (Lemmas 450–451) giving two-sided orientation preservation (Lemma 452). Cross-metric isometries are defined (Definition 453) and shown closed under inverses (Lemma 454), to preserve causal classification (Lemma 455), to push paths forward preserving the timelike/causal conditions and endpoints (Lemmas 456–459), and to transport chronological precedence and the chronological future and past (Lemmas 460–462), hence to preserve Alexandrov-basis sets (Lemma 463). A general topological lemma—a bijection matching generating families is a homeomorphism (Lemma 464)—then identifies the pullback Alexandrov topology (Lemma 465) and yields that the pullback of a Lorentzian spacetime is again a Lorentzian spacetime (Theorem 466). Without Theorem 466 the phrase "the net over $$\psi^*(M,g)$$" in §10.8 would have no referent. +- *Spacetime, tangent-vector causality, and curves (pp. 250–256, items 330–357).* Gives precise definitions of spacetime (Definition 330) and standard Minkowski spacetime (Definition 331); classifies tangent vectors as timelike, spacelike, or null (Definition 332) and proves the trichotomy (Lemma 333), the reverse Cauchy–Schwarz and reverse triangle inequalities for timelike vectors (Lemmas 334–335), and the cone geometry—orientation of pointing vectors, the sign lemma, definiteness of the spacelike complement of a timelike vector, and convexity of the cones (Lemmas 342–345)—alongside time orientations (Definition 340) and future- and past-pointing vectors (Definition 341). The Lorentz, proper and orthochronous transformations of standard Minkowski spacetime are defined (Definition 336); orthochronous Lorentz transformations preserve the future cone (Lemma 337), the identity is Lorentz, proper and orthochronous and each condition is closed under composition and inversion, so that $$SO(1,3)^{\uparrow}$$ is a group (Lemma 338), and the inhomogeneous Lorentz group is built on top of it (Definition 339). Then paths, curves, and oriented curves are defined as equivalence classes of paths up to reparametrisation (Definitions 346–351), with causal type and future/past orientation each shown well-defined on the quotient (Theorems 350 and 352) and a forgetful projection from oriented to unoriented curves (Theorem 353). Endpoints (Definition 354), shown to be well defined on curves and smooth curves (Lemma 355), come with two point-set lemmas—an extremal parameter lies in the frontier, and two endpoints force a compact parameter interval (Lemmas 356–357). +- *The Levi-Civita connection and geodesics (pp. 256–260, items 358–373).* Pseudo-Riemannian metrics (Definition 358) are smooth, symmetric, non-degenerate families of bilinear forms, with no signature condition, so everything here applies to every spacetime. Covariant derivatives are Mathlib's `IsCovariantDerivativeOn`; metric compatibility and the Levi-Civita condition (compatible and torsion-free) are defined on top of them (Definitions 359–360). The musical isomorphism $$v \mapsto g_x(v, \cdot)$$ is a linear isomorphism at each point (Lemma 361), and its inverse is a smooth section of $$\mathrm{Hom}(T^*M, TM)$$ (Lemma 362). The Koszul formula (Lemma 363), the tensoriality of the Koszul expression (Lemma 364), and the verification that the Koszul connection is a covariant derivative (Lemma 365) that is Levi-Civita (Lemma 366) give existence and uniqueness of the Levi-Civita connection (Theorem 367). Five lemmas about smooth paths—a function vanishing along a path has zero derivative along it, the covariant derivative along a curve is local, local vector fields globalise, a smooth path has a local left inverse, and its velocity extends to a smooth vector field (Lemmas 368–372)—make the geodesic condition chart-free, well defined and non-vacuous. A geodesic (Definition 373) is then an affinely parametrised smooth path $$\mu$$ in a spacetime with $$(\nabla_X X)(\mu(s)) = 0$$ at every interior parameter, for every smooth vector field $$X$$ extending the velocity near $$s$$. +- *Trips, futures and pasts (pp. 260–262, items 374–382).* Trips and causal trips (Definitions 374–375) are finite chains of future-oriented timelike (respectively causal) geodesic segments, geodesic in the sense of Definition 373, with matching past and future endpoints. Chronological and causal precedence are transitive (Theorem 376); the causality condition (Definition 377) makes them a strict partial order (Theorem 378); and chronological and causal futures and pasts (Definitions 379–380) come with their basic inclusion and monotonicity properties (Lemmas 381–382). +- *§10.5.1 Causal diamonds (p. 262, items 383–398).* Introduces the causal diamond $$J^+(p) \cap J^-(q)$$ and the chronological (Alexandrov) diamond $$I^+(p) \cap I^-(q)$$ (Definition 383), with the structural properties of the causal diamond—monotonicity under endpoint spread, causal convexity, and that nonemptiness forces $$p \prec q$$ (Lemma 384)—and the containment of chronological diamonds in causal ones, together with the identification of the Alexandrov basis as exactly the chronological diamonds (Theorem 385). It then develops spacelike separation of points and of regions (Definitions 386–387, Lemmas 388–389), shows that any map preserving causal precedence in both directions preserves spacelike separation of points and complete spacelike separation of regions (Lemma 390), and introduces the spacelike complement $$\mathbf{B}^\perp$$ (Definition 391) and its order structure—antitone, extensive on the double complement, with the triple complement collapsing, making complementation the Galois connection attached to the spacelike-separation relation (Lemma 392). The double complement is packaged as the causal closure operator (Definition 393, Lemma 394), whose fixed points are the causally complete regions (Definition 395). These form a complete lattice—meets are intersections, joins are causal closures of unions—on which the causal complement is an order-reversing involution (Theorem 396); the set-level De Morgan laws (Lemma 397) then lift to full binary and infinitary De Morgan laws on the lattice (Theorem 398). The blueprint is careful to record what does *not* hold: the full orthocomplement law $$\mathbf{B} \wedge \mathbf{B}^\perp = \bot$$ fails at this generality, because the trip-based causal relation is irreflexive and so a point is spacelike-separated from itself. +- *§10.5.2 Causal convexity (p. 267, items 399–401).* Defines a causally convex region as one containing every point causally between two of its own points (Definition 399), shows causal diamonds are causally convex (Lemma 400), and shows every spacelike complement—hence every causally complete region—is causally convex (Theorem 401). +- *§10.5.3 Causal convexity: closure structure, and the Alexandrov basis theorems (p. 267, items 402–425).* The causally convex regions are shown to form a closure system (Lemma 402), giving the causal-convex hull (Definition 403) as a genuine closure operator whose closed sets are exactly the causally convex regions (Lemmas 404, Theorem 405). The subsection then turns to the Alexandrov topology (Definition 406): every diamond is open (Lemma 407); chronological futures and pasts are open under an explicit "no endpoints" hypothesis (Lemma 408) and unconditionally on standard Minkowski spacetime, where the coordinate-cone description discharges the hypothesis (Lemma 409). On standard Minkowski spacetime the componentwise directional derivative is a Levi-Civita connection (Lemma 410), hence the Levi-Civita connection is flat (Lemma 411) and affinely parametrised straight lines are geodesics (Lemma 412), so trips there are the familiar straight-line segments. Minkowski spacetime and Lorentzian spacetime are then defined (Definitions 413–414, the latter as a spacetime whose Alexandrov topology is Hausdorff), with a bundled form of spacelike separation and basis openness (Lemma 415). A point lying in no diamond is pathological: its only Alexandrov neighbourhood is the whole space (Lemma 416), and the Hausdorff assumption rules this out, forcing the diamonds to cover the space (Lemma 417). Given downward-directedness the diamonds form a genuine topological basis (Theorem 418); past and future interpolation on standard Minkowski (Lemmas 419–420) supply downward-directedness there (Lemma 421), and common chronological predecessors and successors supply upward-directedness (Lemmas 422–424), so on standard Minkowski the diamonds are an unconditional basis (Theorem 425). Upward-directedness is what the quasilocal colimit of §10.6.1 later consumes. +- *§10.5.4 Dilations are causal automorphisms but not isometries (p. 274, items 426–428).* The dilations $$x \mapsto \lambda x$$ preserve the Minkowski cones (Lemma 426) and are therefore causal automorphisms (Theorem 427), yet scale the metric by $$\lambda^2$$ and so are not isometries for $$\lambda \neq 1$$ (Theorem 428). This is the counterexample that forces the general-covariance morphism of §10.8 to be specified geometrically rather than causally. +- *§10.5.5 Isometries and basis-set preservation (p. 274, items 429–443).* Single-metric transport: isometries preserve the causal classification of tangent vectors (Lemma 429), paths have unique differentials and well-defined pushforwards (Lemmas 430–431), and isometries respect the Levi-Civita structure: pullbacks of vector fields and of the metric pairing, their derivatives and Lie brackets transform naturally (Lemmas 432–434), so an isometry preserves the Levi-Civita connection (Lemma 435) and maps geodesics to geodesics and back (Lemma 436). The inhomogeneous Lorentz group acts on smooth paths by pushforward (Definition 437), carrying future-oriented causal paths to future-oriented causal paths (Lemma 438), so causal precedence (Theorem 439) and spacelike separation (Corollary 440) are Lorentz invariant. Future-orientation-preserving isometries preserve chronology (Lemma 441) and map Alexandrov-basis diamonds to diamonds (Lemma 442), which is exactly the well-definedness condition for the Axiom 5 action (Lemma 443). +- *§10.5.6 Pullback metrics and cross-metric isometries (p. 277, items 444–475).* The single-metric lemmas above compare a spacetime with itself; general covariance instead compares two different metrics on one carrier, so the transport statements are redone cross-metric. This subsection defines the pullback $$\psi^* g$$ of a spacetime metric as a bundled family of continuous bilinear forms (Definition 444) and verifies every obligation in turn: the differential of a diffeomorphism is a linear equivalence (Lemma 445), with the two round-trip cancellation identities for a global `Diffeomorph` (Lemmas 446–448); the pullback metric is symmetric, non-degenerate, Lorentzian, and a smooth section of the bilinear-form bundle (Lemmas 449–452), so the pullback of a spacetime is a spacetime (Theorem 453). Two-sided preservation of future orientation is then defined (Definition 454) and the pullback time orientation is shown to be bundle-smooth, nowhere vanishing, and everywhere timelike (Lemmas 455–458), with transport of future-pointing timelike and null vectors (Lemmas 459–460) giving two-sided orientation preservation (Lemma 461). Cross-metric isometries are defined (Definition 462) and shown closed under inverses (Lemma 463), to preserve causal classification (Lemma 464), to push paths forward preserving the timelike/causal conditions and endpoints (Lemmas 465–468), and to transport chronological precedence and the chronological future and past (Lemmas 469–471), hence to preserve Alexandrov-basis sets (Lemma 472). A general topological lemma—a bijection matching generating families is a homeomorphism (Lemma 473)—then identifies the pullback Alexandrov topology (Lemma 474) and yields that the pullback of a Lorentzian spacetime is again a Lorentzian spacetime (Theorem 475). Without Theorem 475 the phrase "the net over $$\psi^*(M,g)$$" in §10.8 would have no referent. **Where the Lean lives:** `Physicslib4/Geometry/PseudoRiemannian/` (`Basic`, `LeviCivita`, `Flat`), `Physicslib4/Spacetime/` (`Causality`, `Curves`, `AlongPath`, `CausalComplement`, `CausalStructure`, `Minkowski`, `MinkowskiDirected`, `LorentzianSpacetime`, `IsometryCausality`, …) @@ -28,7 +28,7 @@ usemathjax: true Each blueprint subsection below is collapsed; click a heading to see its items. Every entry links to its node in the web blueprint and names the principal Lean declaration.
-§10.5 opening run — spacetime, tangent-vector causality, curves, the Levi-Civita connection and geodesics, trips, futures and pasts (p. 249) — 49 items +§10.5 opening run — spacetime, tangent-vector causality, curves, the Levi-Civita connection and geodesics, trips, futures and pasts (p. 250) — 53 items - [**Definition 330**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:spacetime) — Spacetime · `Physicslib4.Spacetime` - [**Definition 331**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:standard-minkowski-spacetime) — Standard Minkowski Spacetime · `Physicslib4.StandardMinkowskiSpacetime` @@ -36,173 +36,182 @@ Each blueprint subsection below is collapsed; click a heading to see its items. - [**Lemma 333**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:causal-classification) — Causal Classification · `Physicslib4.Spacetime.isTimelike_or_isNull_or_isSpacelike` (+5 more) - [**Lemma 334**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:reverse-cauchy-schwarz) — Reverse Cauchy–Schwarz for Timelike Vectors · `Physicslib4.reverse_cauchy_schwarz_of_lorentzianAt` (+1 more) - [**Lemma 335**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:timelike-cone-convexity) — Timelike Cone Convexity and Reverse Triangle Inequality · `Physicslib4.add_isTimelike_of_lorentzianAt` (+3 more) -- [**Definition 336**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:time-orientable) — Time Orientation · `Physicslib4.Spacetime.TimeOrientation` (+1 more) -- [**Definition 337**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:future-and-past-pointing-vectors) — Future and Past Pointing Vectors · `Physicslib4.Spacetime.IsFuturePointing` (+1 more) -- [**Lemma 338**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pointing-orientation) — Orientation of Pointing Vectors · `Physicslib4.Spacetime.isTimelike_or_isNull_of_isFuturePointing` (+2 more) -- [**Lemma 339**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cone-sign-lemma) — Sign Lemma for the Future Cone · `Physicslib4.nonneg_of_orthogonal_timelike` (+4 more) -- [**Lemma 340**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:spacelike-complement-definite) — Definiteness of the Spacelike Complement · `Physicslib4.eq_zero_of_forall_bilin_eq_zero` (+2 more) -- [**Lemma 341**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:future-cone-convexity) — Convexity of the Future Cone · `Physicslib4.Spacetime.isFuturePointing_add` (+10 more) -- [**Definition 342**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:paths) — Paths · `Physicslib4.Spacetime.Path` (+1 more) -- [**Definition 343**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:curves) — Curves · `Physicslib4.Spacetime.Curve` (+1 more) -- [**Definition 344**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:timelike-and-causal-smooth-curves) — Timelike and Causal Smooth Curves · `Physicslib4.Spacetime.IsTimelikeSmoothCurve` (+1 more) -- [**Definition 345**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:future-and-past-oriented-smooth-curves) — Future and Past Oriented Smooth Curves · `Physicslib4.Spacetime.IsFutureOrientedSmoothCurve` (+1 more) -- [**Theorem 346**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:smooth-curve-causal-well-defined) — Reparametrisation-Invariance of the Causal Type · `Physicslib4.Spacetime.isTimelikeSmoothCurve_ofPath_iff` (+1 more) -- [**Definition 347**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:oriented-smooth-curve) — Oriented Smooth Curve · `Physicslib4.Spacetime.OrientedSmoothCurve` -- [**Theorem 348**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:oriented-curve-well-defined) — Reparametrisation-Invariance of Orientation · `Physicslib4.Spacetime.isFutureOrientedCurve_ofPath_iff` (+1 more) -- [**Theorem 349**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:oriented-curve-projection) — The Forgetful Projection of Oriented Curves · `Physicslib4.Spacetime.OrientedSmoothCurve.toSmoothCurve` (+1 more) -- [**Definition 350**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:endpoints) — Endpoints · `Physicslib4.Spacetime.IsEndpoint` (+2 more) -- [**Lemma 351**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:curve-endpoint-well-defined) — Endpoints of curves are well defined · `Physicslib4.Spacetime.IsCurveEndpoint` (+5 more) -- [**Lemma 352**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:extremal-parameter-mem-frontier) — An extremal parameter lies in the frontier · `Physicslib4.Spacetime.mem_frontier_of_isMin` (+3 more) -- [**Lemma 353**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:endpoint-parameter-space-eq-Icc) — Two endpoints force a compact parameter interval · `Physicslib4.Spacetime.parameterSpace_eq_Icc_of_endpoints` -- [**Definition 354**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:pseudo-riemannian-metric) — Pseudo-Riemannian Metric · `Physicslib4.Geometry.PseudoRiemannianMetric` (+1 more) -- [**Definition 355**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:metric-compatible-connection) — Covariant Derivative; Metric Compatibility · `Physicslib4.Geometry.CovariantDerivative.IsMetricCompatibleWith` -- [**Definition 356**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:levi-civita-connection) — Levi-Civita Connection · `Physicslib4.Geometry.CovariantDerivative.IsLeviCivitaFor` -- [**Lemma 357**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:musical-isomorphism) — Musical Isomorphism · `Physicslib4.Geometry.PseudoRiemannianMetric.bijective_val` -- [**Lemma 358**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:musical-inverse-smooth) — Smoothness of the Inverse Musical Isomorphism · `Physicslib4.Geometry.PseudoRiemannianMetric.contMDiff_flatEquiv_symm` (+1 more) -- [**Lemma 359**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:koszul-formula) — Koszul Formula · `Physicslib4.Geometry.CovariantDerivative.IsLeviCivitaFor.koszul` -- [**Lemma 360**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:koszul-expression-tensorial) — Tensoriality of the Koszul Expression · `Physicslib4.Geometry.PseudoRiemannianMetric.tensorialAt_koszulAux₁` (+2 more) -- [**Lemma 361**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:koszul-connection-is-covariant-derivative) — The Koszul Connection is a Covariant Derivative · `Physicslib4.Geometry.PseudoRiemannianMetric.isCovariantDerivativeOn_leviCivitaAux` -- [**Lemma 362**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:koszul-connection-is-levi-civita) — The Koszul Connection is Levi-Civita · `Physicslib4.Geometry.PseudoRiemannianMetric.isMetricCompatibleWith_koszulConnection` (+1 more) -- [**Theorem 363**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:levi-civita-exists-unique) — Existence and Uniqueness of the Levi-Civita Connection · `Physicslib4.Geometry.PseudoRiemannianMetric.exists_isLeviCivitaFor` (+3 more) -- [**Lemma 364**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:derivative-vanishes-along-path) — A Function Vanishing Along a Path has Zero Derivative Along It · `Physicslib4.Spacetime.SmoothPath.mfderiv_apply_tangent_eq_zero` -- [**Lemma 365**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:covariant-derivative-along-curve-local) — Covariant Derivative Along a Curve is Local · `Physicslib4.Spacetime.SmoothPath.covDeriv_eq_of_eventuallyEq` -- [**Lemma 366**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:local-vector-field-globalises) — Local Vector Fields Globalise · `Physicslib4.Spacetime.exists_contMDiff_vectorField_eventuallyEq` -- [**Lemma 367**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:path-local-left-inverse) — Local Left Inverse of a Smooth Path · `Physicslib4.Spacetime.SmoothPath.exists_localLeftInverse` -- [**Lemma 368**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:velocity-extends) — The Velocity of a Smooth Path Extends to a Vector Field · `Physicslib4.Spacetime.SmoothPath.exists_vectorField_eq_tangent` -- [**Definition 369**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:geodesic) — Geodesic · `Physicslib4.Spacetime.IsGeodesic` -- [**Definition 370**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:trip) — Trip · `Physicslib4.Spacetime.IsTripSegment` (+2 more) -- [**Definition 371**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causal-trip) — Causal Trip · `Physicslib4.Spacetime.IsCausalTripSegment` (+2 more) -- [**Theorem 372**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:precedence-transitive) — Transitivity of chronological and causal precedence · `Physicslib4.Spacetime.chronologicallyPrecedes_trans` (+1 more) -- [**Definition 373**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:no-closed-causal-curve) — No Closed Causal Curve (Causality Condition) · `Physicslib4.Spacetime.NoClosedCausalCurve` -- [**Theorem 374**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causal-order-refinements) — Irreflexivity and Antisymmetry under Causality · `Physicslib4.Spacetime.chronologicallyPrecedes_irrefl` (+2 more) -- [**Definition 375**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:chronological-future-and-chronological-past) — Chronological Future and Chronological Past · `Physicslib4.Spacetime.chronologicalFuture` (+3 more) -- [**Definition 376**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causal-future-and-causal-past) — Causal Future and Causal Past · `Physicslib4.Spacetime.causalFuture` (+3 more) -- [**Lemma 377**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:chronological-implies-causal) — Chronological Precedence Implies Causal Precedence · `Physicslib4.Spacetime.isCausal_of_isTimelike` (+3 more) -- [**Lemma 378**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:future-past-monotone) — Monotonicity of Futures and Pasts · `Physicslib4.Spacetime.chronologicalFutureSet_mono` (+3 more) +- [**Definition 336**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:lorentz-transformation) — Lorentz, Proper and Orthochronous Transformations · `Physicslib4.IsLorentz` (+2 more) +- [**Lemma 337**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:orthochronous-preserves-future-timelike) — Orthochronous Lorentz Transformations Preserve the Future Cone · `Physicslib4.isOrthochronous_pos_time_of_timelike` +- [**Lemma 338**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:lorentz-closure) — Closure of the Lorentz Conditions · `Physicslib4.isLorentz_refl` (+8 more) +- [**Definition 339**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:inhomogeneous-lorentz-group) — Inhomogeneous Lorentz Group · `Physicslib4.AQFT.HaagKastler.InhomogeneousLorentzGroup` +- [**Definition 340**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:time-orientable) — Time Orientation · `Physicslib4.Spacetime.TimeOrientation` (+1 more) +- [**Definition 341**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:future-and-past-pointing-vectors) — Future and Past Pointing Vectors · `Physicslib4.Spacetime.IsFuturePointing` (+1 more) +- [**Lemma 342**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pointing-orientation) — Orientation of Pointing Vectors · `Physicslib4.Spacetime.isTimelike_or_isNull_of_isFuturePointing` (+2 more) +- [**Lemma 343**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cone-sign-lemma) — Sign Lemma for the Future Cone · `Physicslib4.nonneg_of_orthogonal_timelike` (+4 more) +- [**Lemma 344**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:spacelike-complement-definite) — Definiteness of the Spacelike Complement · `Physicslib4.eq_zero_of_forall_bilin_eq_zero` (+2 more) +- [**Lemma 345**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:future-cone-convexity) — Convexity of the Future Cone · `Physicslib4.Spacetime.isFuturePointing_add` (+10 more) +- [**Definition 346**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:paths) — Paths · `Physicslib4.Spacetime.Path` (+1 more) +- [**Definition 347**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:curves) — Curves · `Physicslib4.Spacetime.Curve` (+1 more) +- [**Definition 348**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:timelike-and-causal-smooth-curves) — Timelike and Causal Smooth Curves · `Physicslib4.Spacetime.IsTimelikeSmoothCurve` (+1 more) +- [**Definition 349**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:future-and-past-oriented-smooth-curves) — Future and Past Oriented Smooth Curves · `Physicslib4.Spacetime.IsFutureOrientedSmoothCurve` (+1 more) +- [**Theorem 350**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:smooth-curve-causal-well-defined) — Reparametrisation-Invariance of the Causal Type · `Physicslib4.Spacetime.isTimelikeSmoothCurve_ofPath_iff` (+1 more) +- [**Definition 351**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:oriented-smooth-curve) — Oriented Smooth Curve · `Physicslib4.Spacetime.OrientedSmoothCurve` +- [**Theorem 352**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:oriented-curve-well-defined) — Reparametrisation-Invariance of Orientation · `Physicslib4.Spacetime.isFutureOrientedCurve_ofPath_iff` (+1 more) +- [**Theorem 353**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:oriented-curve-projection) — The Forgetful Projection of Oriented Curves · `Physicslib4.Spacetime.OrientedSmoothCurve.toSmoothCurve` (+1 more) +- [**Definition 354**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:endpoints) — Endpoints · `Physicslib4.Spacetime.IsEndpoint` (+2 more) +- [**Lemma 355**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:curve-endpoint-well-defined) — Endpoints of curves are well defined · `Physicslib4.Spacetime.IsCurveEndpoint` (+5 more) +- [**Lemma 356**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:extremal-parameter-mem-frontier) — An extremal parameter lies in the frontier · `Physicslib4.Spacetime.mem_frontier_of_isMin` (+3 more) +- [**Lemma 357**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:endpoint-parameter-space-eq-Icc) — Two endpoints force a compact parameter interval · `Physicslib4.Spacetime.parameterSpace_eq_Icc_of_endpoints` +- [**Definition 358**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:pseudo-riemannian-metric) — Pseudo-Riemannian Metric · `Physicslib4.Geometry.PseudoRiemannianMetric` (+1 more) +- [**Definition 359**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:metric-compatible-connection) — Covariant Derivative; Metric Compatibility · `Physicslib4.Geometry.CovariantDerivative.IsMetricCompatibleWith` +- [**Definition 360**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:levi-civita-connection) — Levi-Civita Connection · `Physicslib4.Geometry.CovariantDerivative.IsLeviCivitaFor` +- [**Lemma 361**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:musical-isomorphism) — Musical Isomorphism · `Physicslib4.Geometry.PseudoRiemannianMetric.bijective_val` +- [**Lemma 362**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:musical-inverse-smooth) — Smoothness of the Inverse Musical Isomorphism · `Physicslib4.Geometry.PseudoRiemannianMetric.contMDiff_flatEquiv_symm` (+1 more) +- [**Lemma 363**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:koszul-formula) — Koszul Formula · `Physicslib4.Geometry.CovariantDerivative.IsLeviCivitaFor.koszul` +- [**Lemma 364**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:koszul-expression-tensorial) — Tensoriality of the Koszul Expression · `Physicslib4.Geometry.PseudoRiemannianMetric.tensorialAt_koszulAux₁` (+2 more) +- [**Lemma 365**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:koszul-connection-is-covariant-derivative) — The Koszul Connection is a Covariant Derivative · `Physicslib4.Geometry.PseudoRiemannianMetric.isCovariantDerivativeOn_leviCivitaAux` +- [**Lemma 366**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:koszul-connection-is-levi-civita) — The Koszul Connection is Levi-Civita · `Physicslib4.Geometry.PseudoRiemannianMetric.isMetricCompatibleWith_koszulConnection` (+1 more) +- [**Theorem 367**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:levi-civita-exists-unique) — Existence and Uniqueness of the Levi-Civita Connection · `Physicslib4.Geometry.PseudoRiemannianMetric.exists_isLeviCivitaFor` (+3 more) +- [**Lemma 368**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:derivative-vanishes-along-path) — A Function Vanishing Along a Path has Zero Derivative Along It · `Physicslib4.Spacetime.SmoothPath.mfderiv_apply_tangent_eq_zero` +- [**Lemma 369**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:covariant-derivative-along-curve-local) — Covariant Derivative Along a Curve is Local · `Physicslib4.Spacetime.SmoothPath.covDeriv_eq_of_eventuallyEq` +- [**Lemma 370**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:local-vector-field-globalises) — Local Vector Fields Globalise · `Physicslib4.Spacetime.exists_contMDiff_vectorField_eventuallyEq` +- [**Lemma 371**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:path-local-left-inverse) — Local Left Inverse of a Smooth Path · `Physicslib4.Spacetime.SmoothPath.exists_localLeftInverse` +- [**Lemma 372**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:velocity-extends) — The Velocity of a Smooth Path Extends to a Vector Field · `Physicslib4.Spacetime.SmoothPath.exists_vectorField_eq_tangent` +- [**Definition 373**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:geodesic) — Geodesic · `Physicslib4.Spacetime.IsGeodesic` +- [**Definition 374**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:trip) — Trip · `Physicslib4.Spacetime.IsTripSegment` (+2 more) +- [**Definition 375**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causal-trip) — Causal Trip · `Physicslib4.Spacetime.IsCausalTripSegment` (+2 more) +- [**Theorem 376**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:precedence-transitive) — Transitivity of chronological and causal precedence · `Physicslib4.Spacetime.chronologicallyPrecedes_trans` (+1 more) +- [**Definition 377**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:no-closed-causal-curve) — No Closed Causal Curve (Causality Condition) · `Physicslib4.Spacetime.NoClosedCausalCurve` +- [**Theorem 378**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causal-order-refinements) — Irreflexivity and Antisymmetry under Causality · `Physicslib4.Spacetime.chronologicallyPrecedes_irrefl` (+2 more) +- [**Definition 379**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:chronological-future-and-chronological-past) — Chronological Future and Chronological Past · `Physicslib4.Spacetime.chronologicalFuture` (+3 more) +- [**Definition 380**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causal-future-and-causal-past) — Causal Future and Causal Past · `Physicslib4.Spacetime.causalFuture` (+3 more) +- [**Lemma 381**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:chronological-implies-causal) — Chronological Precedence Implies Causal Precedence · `Physicslib4.Spacetime.isCausal_of_isTimelike` (+3 more) +- [**Lemma 382**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:future-past-monotone) — Monotonicity of Futures and Pasts · `Physicslib4.Spacetime.chronologicalFutureSet_mono` (+3 more)
-§10.5.1 Causal diamonds, spacelike complement, and causal closure (p. 260) — 15 items - -- [**Definition 379**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causal-diamond) — Causal and chronological diamonds · `Physicslib4.Spacetime.causalDiamond` (+3 more) -- [**Lemma 380**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:causal-diamond-structure) — Structural properties of the causal diamond · `Physicslib4.Spacetime.causalDiamond_subset_of` (+2 more) -- [**Theorem 381**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causal-diamond-vs-chronological) — Chronological diamonds inside causal diamonds · `Physicslib4.Spacetime.chronologicalDiamond_subset_causalDiamond` (+1 more) -- [**Definition 382**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:spacelike-related) — Spacelike Related · `Physicslib4.Spacetime.IsSpacelikeRelated` -- [**Definition 383**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:completely-spacelike) — Completely Spacelike · `Physicslib4.Spacetime.IsCompletelySpacelike` -- [**Lemma 384**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:completely-spacelike-symm) — Symmetry of Spacelike Separation · `Physicslib4.Spacetime.isSpacelikeRelated_comm` (+1 more) -- [**Lemma 385**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:completely-spacelike-structural) — Structural Properties of Complete Spacelike Separation · `Physicslib4.Spacetime.isCompletelySpacelike_mono` (+9 more) -- [**Definition 386**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:spacelike-complement) — Spacelike Complement of a Region · `Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement` -- [**Lemma 387**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:spacelike-complement-order) — Order Structure of the Spacelike Complement · `Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement_antitone` (+3 more) -- [**Definition 388**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causal-closure) — Causal closure operator · `Physicslib4.Spacetime.LorentzianSpacetime.causalClosure` -- [**Lemma 389**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:causal-closure-is-closure-operator) — The Causal Closure is a Closure Operator · `Physicslib4.Spacetime.LorentzianSpacetime.causalClosure` (+1 more) -- [**Definition 390**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causally-complete-region) — Causally complete region · `Physicslib4.Spacetime.LorentzianSpacetime.IsCausallyComplete` -- [**Theorem 391**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causally-complete-lattice) — Lattice of causally complete regions · `Physicslib4.Spacetime.LorentzianSpacetime.CausallyCompleteRegion` (+8 more) -- [**Lemma 392**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:spacelike-complement-de-morgan) — De Morgan Laws for the Spacelike Complement · `Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement_union` (+1 more) -- [**Theorem 393**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causal-complement-de-morgan) — De Morgan Laws for the Causal Complement · `Physicslib4.Spacetime.LorentzianSpacetime.causalComplement_antitone` (+6 more) +§10.5.1 Causal diamonds, spacelike complement, and causal closure (p. 262) — 16 items + +- [**Definition 383**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causal-diamond) — Causal and chronological diamonds · `Physicslib4.Spacetime.causalDiamond` (+3 more) +- [**Lemma 384**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:causal-diamond-structure) — Structural properties of the causal diamond · `Physicslib4.Spacetime.causalDiamond_subset_of` (+2 more) +- [**Theorem 385**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causal-diamond-vs-chronological) — Chronological diamonds inside causal diamonds · `Physicslib4.Spacetime.chronologicalDiamond_subset_causalDiamond` (+1 more) +- [**Definition 386**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:spacelike-related) — Spacelike Related · `Physicslib4.Spacetime.IsSpacelikeRelated` +- [**Definition 387**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:completely-spacelike) — Completely Spacelike · `Physicslib4.Spacetime.IsCompletelySpacelike` +- [**Lemma 388**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:completely-spacelike-symm) — Symmetry of Spacelike Separation · `Physicslib4.Spacetime.isSpacelikeRelated_comm` (+1 more) +- [**Lemma 389**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:completely-spacelike-structural) — Structural Properties of Complete Spacelike Separation · `Physicslib4.Spacetime.isCompletelySpacelike_mono` (+9 more) +- [**Lemma 390**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:precedence-preserving-spacelike) — Precedence-Preserving Maps Preserve Spacelike Separation · `Physicslib4.Spacetime.isSpacelikeRelated_congr` (+1 more) +- [**Definition 391**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:spacelike-complement) — Spacelike Complement of a Region · `Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement` +- [**Lemma 392**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:spacelike-complement-order) — Order Structure of the Spacelike Complement · `Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement_antitone` (+3 more) +- [**Definition 393**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causal-closure) — Causal closure operator · `Physicslib4.Spacetime.LorentzianSpacetime.causalClosure` +- [**Lemma 394**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:causal-closure-is-closure-operator) — The Causal Closure is a Closure Operator · `Physicslib4.Spacetime.LorentzianSpacetime.causalClosure` (+1 more) +- [**Definition 395**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causally-complete-region) — Causally complete region · `Physicslib4.Spacetime.LorentzianSpacetime.IsCausallyComplete` +- [**Theorem 396**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causally-complete-lattice) — Lattice of causally complete regions · `Physicslib4.Spacetime.LorentzianSpacetime.CausallyCompleteRegion` (+8 more) +- [**Lemma 397**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:spacelike-complement-de-morgan) — De Morgan Laws for the Spacelike Complement · `Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement_union` (+1 more) +- [**Theorem 398**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causal-complement-de-morgan) — De Morgan Laws for the Causal Complement · `Physicslib4.Spacetime.LorentzianSpacetime.causalComplement_antitone` (+6 more)
-§10.5.2 Causal convexity (p. 265) — 3 items +§10.5.2 Causal convexity (p. 267) — 3 items -- [**Definition 394**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causally-convex-region) — Causally convex region · `Physicslib4.Spacetime.IsCausallyConvex` -- [**Lemma 395**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:causal-diamond-causally-convex) — Causal diamonds are causally convex · `Physicslib4.Spacetime.causalDiamond_isCausallyConvex` -- [**Theorem 396**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causally-complete-convex) — Causally complete regions are causally convex · `Physicslib4.Spacetime.spacelikeComplement_isCausallyConvex` (+2 more) +- [**Definition 399**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causally-convex-region) — Causally convex region · `Physicslib4.Spacetime.IsCausallyConvex` +- [**Lemma 400**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:causal-diamond-causally-convex) — Causal diamonds are causally convex · `Physicslib4.Spacetime.causalDiamond_isCausallyConvex` +- [**Theorem 401**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causally-complete-convex) — Causally complete regions are causally convex · `Physicslib4.Spacetime.spacelikeComplement_isCausallyConvex` (+2 more)
-§10.5.3 Causal convexity: closure structure, and the Alexandrov basis theorems (p. 265) — 24 items - -- [**Lemma 397**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:causally-convex-closure-ops) — Causally convex regions form a closure system · `Physicslib4.Spacetime.isCausallyConvex_univ` (+4 more) -- [**Definition 398**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causal-convex-hull) — Causal-convex hull · `Physicslib4.Spacetime.causalConvexHull` -- [**Lemma 399**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:causal-convex-hull-extensive) — The Causal-Convex Hull is Extensive and Causally Convex · `Physicslib4.Spacetime.subset_causalConvexHull` (+1 more) -- [**Theorem 400**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causal-convex-hull-closure) — The causal-convex hull is a closure operator · `Physicslib4.Spacetime.causalConvexHull_minimal` (+3 more) -- [**Definition 401**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:alexandrov-topology) — Alexandrov Topology · `Physicslib4.Spacetime.alexandrovTopology` -- [**Lemma 402**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:alexandrov-basis-open) — Basis Sets Are Alexandrov-Open · `Physicslib4.Spacetime.isOpen_alexandrov_of_mem_basis` -- [**Lemma 403**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:chronological-future-past-open) — Openness of Chronological Futures and Pasts · `Physicslib4.Spacetime.isOpen_chronologicalFuture_inter_chronologicalPast` (+2 more) -- [**Lemma 404**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-chronological-open) — Unconditional Openness of Chronological Futures and Pasts on Standard Minkowski · `Physicslib4.exists_chronologicalFuture_standardMinkowski` (+5 more) -- [**Lemma 405**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-directional-derivative-levi-civita) — The Directional Derivative is Levi-Civita on Standard Minkowski · `Physicslib4.Geometry.PseudoRiemannianMetric.isLeviCivitaFor_flatConnection` (+2 more) -- [**Lemma 406**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-levi-civita-flat) — The Minkowski Levi-Civita Connection is Flat · `Physicslib4.standardMinkowski_leviCivita_apply` (+1 more) -- [**Lemma 407**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-lines-are-geodesics) — Straight Lines are Geodesics in Standard Minkowski · `Physicslib4.standardMinkowskiLineSegmentPath_isGeodesic` (+1 more) -- [**Definition 408**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:minkowski-spacetime) — Minkowski Spacetime · `Physicslib4.MinkowskiSpacetime` -- [**Definition 409**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:lorentzian-spacetime) — Lorentzian Spacetime · `Physicslib4.Spacetime.LorentzianSpacetime` -- [**Lemma 410**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:lorentzian-causal-lifts) — Bundled Spacelike Separation and Basis Openness · `Physicslib4.Spacetime.LorentzianSpacetime.isCompletelySpacelike_comm` (+1 more) -- [**Lemma 411**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:alexandrov-nbhd-univ-of-no-diamond) — No-Diamond Points Have Only the Whole Space as Neighbourhood · `Physicslib4.Spacetime.alexandrov_nbhd_univ_of_no_diamond` -- [**Lemma 412**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:alexandrov-covering-hausdorff) — Covering from the Hausdorff Assumption · `Physicslib4.Spacetime.LorentzianSpacetime.sUnion_alexandrovBasis_eq_univ` -- [**Theorem 413**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:alexandrov-topological-basis) — The Alexandrov Diamonds Form a Topological Basis · `Physicslib4.Spacetime.LorentzianSpacetime.isTopologicalBasis_alexandrovBasis` -- [**Lemma 414**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-past-between) — Past Interpolation on Standard Minkowski · `Physicslib4.Spacetime.exists_past_between_standardMinkowski` -- [**Lemma 415**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-future-between) — Future Interpolation on Standard Minkowski · `Physicslib4.Spacetime.exists_future_between_standardMinkowski` -- [**Lemma 416**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-diamonds-downward-directed) — Standard Minkowski Diamonds Are Downward-Directed · `Physicslib4.Spacetime.alexandrovBasis_exists_subset_inter_standardMinkowski` -- [**Lemma 417**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-exists-common-past) — Common Chronological Predecessor on Standard Minkowski · `Physicslib4.Spacetime.exists_common_past` -- [**Lemma 418**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-exists-common-future) — Common Chronological Successor on Standard Minkowski · `Physicslib4.Spacetime.exists_common_future` -- [**Lemma 419**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-diamonds-upward-directed) — Standard Minkowski Diamonds Are Upward-Directed · `Physicslib4.Spacetime.alexandrovBasis_directed` -- [**Theorem 420**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:minkowski-alexandrov-basis) — The Alexandrov Diamonds are a Basis on Standard Minkowski · `Physicslib4.Spacetime.isTopologicalBasis_alexandrovBasis_standardMinkowski` +§10.5.3 Causal convexity: closure structure, and the Alexandrov basis theorems (p. 267) — 24 items + +- [**Lemma 402**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:causally-convex-closure-ops) — Causally convex regions form a closure system · `Physicslib4.Spacetime.isCausallyConvex_univ` (+4 more) +- [**Definition 403**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:causal-convex-hull) — Causal-convex hull · `Physicslib4.Spacetime.causalConvexHull` +- [**Lemma 404**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:causal-convex-hull-extensive) — The Causal-Convex Hull is Extensive and Causally Convex · `Physicslib4.Spacetime.subset_causalConvexHull` (+1 more) +- [**Theorem 405**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causal-convex-hull-closure) — The causal-convex hull is a closure operator · `Physicslib4.Spacetime.causalConvexHull_minimal` (+3 more) +- [**Definition 406**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:alexandrov-topology) — Alexandrov Topology · `Physicslib4.Spacetime.alexandrovTopology` +- [**Lemma 407**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:alexandrov-basis-open) — Basis Sets Are Alexandrov-Open · `Physicslib4.Spacetime.isOpen_alexandrov_of_mem_basis` +- [**Lemma 408**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:chronological-future-past-open) — Openness of Chronological Futures and Pasts · `Physicslib4.Spacetime.isOpen_chronologicalFuture_inter_chronologicalPast` (+2 more) +- [**Lemma 409**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-chronological-open) — Unconditional Openness of Chronological Futures and Pasts on Standard Minkowski · `Physicslib4.exists_chronologicalFuture_standardMinkowski` (+5 more) +- [**Lemma 410**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-directional-derivative-levi-civita) — The Directional Derivative is Levi-Civita on Standard Minkowski · `Physicslib4.Geometry.PseudoRiemannianMetric.isLeviCivitaFor_flatConnection` (+2 more) +- [**Lemma 411**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-levi-civita-flat) — The Minkowski Levi-Civita Connection is Flat · `Physicslib4.standardMinkowski_leviCivita_apply` (+1 more) +- [**Lemma 412**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-lines-are-geodesics) — Straight Lines are Geodesics in Standard Minkowski · `Physicslib4.standardMinkowskiLineSegmentPath_isGeodesic` (+1 more) +- [**Definition 413**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:minkowski-spacetime) — Minkowski Spacetime · `Physicslib4.MinkowskiSpacetime` +- [**Definition 414**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:lorentzian-spacetime) — Lorentzian Spacetime · `Physicslib4.Spacetime.LorentzianSpacetime` +- [**Lemma 415**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:lorentzian-causal-lifts) — Bundled Spacelike Separation and Basis Openness · `Physicslib4.Spacetime.LorentzianSpacetime.isCompletelySpacelike_comm` (+1 more) +- [**Lemma 416**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:alexandrov-nbhd-univ-of-no-diamond) — No-Diamond Points Have Only the Whole Space as Neighbourhood · `Physicslib4.Spacetime.alexandrov_nbhd_univ_of_no_diamond` +- [**Lemma 417**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:alexandrov-covering-hausdorff) — Covering from the Hausdorff Assumption · `Physicslib4.Spacetime.LorentzianSpacetime.sUnion_alexandrovBasis_eq_univ` +- [**Theorem 418**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:alexandrov-topological-basis) — The Alexandrov Diamonds Form a Topological Basis · `Physicslib4.Spacetime.LorentzianSpacetime.isTopologicalBasis_alexandrovBasis` +- [**Lemma 419**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-past-between) — Past Interpolation on Standard Minkowski · `Physicslib4.Spacetime.exists_past_between_standardMinkowski` +- [**Lemma 420**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-future-between) — Future Interpolation on Standard Minkowski · `Physicslib4.Spacetime.exists_future_between_standardMinkowski` +- [**Lemma 421**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-diamonds-downward-directed) — Standard Minkowski Diamonds Are Downward-Directed · `Physicslib4.Spacetime.alexandrovBasis_exists_subset_inter_standardMinkowski` +- [**Lemma 422**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-exists-common-past) — Common Chronological Predecessor on Standard Minkowski · `Physicslib4.Spacetime.exists_common_past` +- [**Lemma 423**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-exists-common-future) — Common Chronological Successor on Standard Minkowski · `Physicslib4.Spacetime.exists_common_future` +- [**Lemma 424**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-diamonds-upward-directed) — Standard Minkowski Diamonds Are Upward-Directed · `Physicslib4.Spacetime.alexandrovBasis_directed` +- [**Theorem 425**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:minkowski-alexandrov-basis) — The Alexandrov Diamonds are a Basis on Standard Minkowski · `Physicslib4.Spacetime.isTopologicalBasis_alexandrovBasis_standardMinkowski`
-§10.5.4 Dilations are causal automorphisms but not isometries (p. 272) — 3 items +§10.5.4 Dilations are causal automorphisms but not isometries (p. 274) — 3 items -- [**Lemma 421**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-dilation-cone) — Dilations Preserve the Minkowski Cones · `Physicslib4.minkowskiForwardCone_smul` (+1 more) -- [**Theorem 422**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:minkowski-dilation-causal-automorphism) — Dilations are Causal Automorphisms · `Physicslib4.alexandrovBasis_image_smul` -- [**Theorem 423**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:minkowski-dilation-not-isometry) — Dilations are Not Isometries · `Physicslib4.minkowskiForm_smul` (+1 more) +- [**Lemma 426**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:minkowski-dilation-cone) — Dilations Preserve the Minkowski Cones · `Physicslib4.minkowskiForwardCone_smul` (+1 more) +- [**Theorem 427**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:minkowski-dilation-causal-automorphism) — Dilations are Causal Automorphisms · `Physicslib4.alexandrovBasis_image_smul` +- [**Theorem 428**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:minkowski-dilation-not-isometry) — Dilations are Not Isometries · `Physicslib4.minkowskiForm_smul` (+1 more)
-§10.5.5 Isometries and basis-set preservation (p. 272) — 11 items - -- [**Lemma 424**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-preserves-classification) — Isometries Preserve the Causal Classification · `Physicslib4.Spacetime.Isometry.preserves_self` (+3 more) -- [**Lemma 425**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:path-parameter-unique-diff) — Unique Differentials Along a Path · `Physicslib4.Spacetime.Path.uniqueDiffOn_parameterSpace` -- [**Lemma 426**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pushforward-path) — Pushforward of a Path Under an Isometry · `Physicslib4.Spacetime.Isometry.pushforwardPath` (+5 more) -- [**Lemma 427**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-pullback-metric) — Pullback of Vector Fields and the Metric Under an Isometry · `Physicslib4.Spacetime.Isometry.val_mpullback` (+3 more) -- [**Lemma 428**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-pullback-metric-derivative) — Derivatives of Metric Pairings of Pulled-Back Fields · `Physicslib4.Spacetime.Isometry.mfderiv_val_mpullback` (+1 more) -- [**Lemma 429**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-pullback-lie-bracket) — Metric Pairings with Lie Brackets of Pulled-Back Fields · `Physicslib4.Spacetime.Isometry.val_mlieBracket_mpullback` (+1 more) -- [**Lemma 430**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-preserves-levi-civita) — Isometries Preserve the Levi-Civita Connection · `Physicslib4.Spacetime.Isometry.mfderiv_leviCivita_mpullback` (+1 more) -- [**Lemma 431**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-preserves-geodesics) — Isometries Preserve Geodesics · `Physicslib4.Spacetime.Isometry.pushforwardPath_isGeodesic` (+2 more) -- [**Lemma 432**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-preserves-chronology) — Isometries Preserve Chronology · `Physicslib4.Spacetime.Isometry.PreservesFutureOrientation` (+8 more) -- [**Lemma 433**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-preserves-basis-sets) — Isometries Preserve Basis Sets · `Physicslib4.Spacetime.Isometry.futureOrientationPreserving` (+8 more) -- [**Lemma 434**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:axiom5-basis-preservation) — Axiom 5 Basis-Set Preservation · `Physicslib4.Spacetime.LorentzianSpacetime.toAbstractIdentityComponent_isBasisSet_smul` +§10.5.5 Isometries and basis-set preservation (p. 274) — 15 items + +- [**Lemma 429**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-preserves-classification) — Isometries Preserve the Causal Classification · `Physicslib4.Spacetime.Isometry.preserves_self` (+3 more) +- [**Lemma 430**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:path-parameter-unique-diff) — Unique Differentials Along a Path · `Physicslib4.Spacetime.Path.uniqueDiffOn_parameterSpace` +- [**Lemma 431**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pushforward-path) — Pushforward of a Path Under an Isometry · `Physicslib4.Spacetime.Isometry.pushforwardPath` (+5 more) +- [**Lemma 432**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-pullback-metric) — Pullback of Vector Fields and the Metric Under an Isometry · `Physicslib4.Spacetime.Isometry.val_mpullback` (+3 more) +- [**Lemma 433**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-pullback-metric-derivative) — Derivatives of Metric Pairings of Pulled-Back Fields · `Physicslib4.Spacetime.Isometry.mfderiv_val_mpullback` (+1 more) +- [**Lemma 434**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-pullback-lie-bracket) — Metric Pairings with Lie Brackets of Pulled-Back Fields · `Physicslib4.Spacetime.Isometry.val_mlieBracket_mpullback` (+1 more) +- [**Lemma 435**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-preserves-levi-civita) — Isometries Preserve the Levi-Civita Connection · `Physicslib4.Spacetime.Isometry.mfderiv_leviCivita_mpullback` (+1 more) +- [**Lemma 436**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-preserves-geodesics) — Isometries Preserve Geodesics · `Physicslib4.Spacetime.Isometry.pushforwardPath_isGeodesic` (+2 more) +- [**Definition 437**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:lorentz-path) — Lorentz Pushforward of a Smooth Path · `Physicslib4.lorentzPath` +- [**Lemma 438**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:lorentz-path-causal) — Lorentz Pushforward of Causal Paths · `Physicslib4.lorentzPath_mfderivWithin` (+4 more) +- [**Theorem 439**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:causal-precedence-lorentz-invariant) — Lorentz Invariance of Causal Precedence · `Physicslib4.causalSegmentPrecedes_smul` (+2 more) +- [**Corollary 440**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#crllr:spacelike-lorentz-invariant) — Lorentz Invariance of Spacelike Separation · `Physicslib4.isSpacelikeRelated_smul_iff` (+1 more) +- [**Lemma 441**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-preserves-chronology) — Isometries Preserve Chronology · `Physicslib4.Spacetime.Isometry.PreservesFutureOrientation` (+8 more) +- [**Lemma 442**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isometry-preserves-basis-sets) — Isometries Preserve Basis Sets · `Physicslib4.Spacetime.Isometry.futureOrientationPreserving` (+8 more) +- [**Lemma 443**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:axiom5-basis-preservation) — Axiom 5 Basis-Set Preservation · `Physicslib4.Spacetime.LorentzianSpacetime.toAbstractIdentityComponent_isBasisSet_smul`
-§10.5.6 Pullback metrics and cross-metric isometries (p. 275) — 32 items - -- [**Definition 435**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:pullback-metric) — Pullback of a Spacetime Metric · `Physicslib4.Spacetime.bilinearPrecomp` (+3 more) -- [**Lemma 436**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:mfderiv-diffeo-linear-equiv) — The Differential of a Diffeomorphism is a Linear Equivalence · `Physicslib4.Spacetime.Diffeo` (+4 more) -- [**Lemma 437**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:mfderiv-symm-cancel-left) — Round-Trip Cancellation: $$d\psi$$ After $$d(\psi^{-1})$$ · `Physicslib4.Spacetime.mfderiv_symm_cancel_left` -- [**Lemma 438**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:mfderiv-symm-cancel-right) — Round-Trip Cancellation: $$d(\psi^{-1})$$ After $$d\psi$$ · `Physicslib4.Spacetime.mfderiv_symm_cancel_right` -- [**Lemma 439**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:mfderiv-inverse-eq-symm) — The Formal Inverse of $$d\psi_x$$ is the Inverse Equivalence · `Physicslib4.Spacetime.inverse_mfderiv_eq_symm` (+2 more) -- [**Lemma 440**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-metric-symm) — The Pullback Metric is Symmetric · `Physicslib4.Spacetime.pullbackVal_symm` -- [**Lemma 441**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-metric-nondegenerate) — The Pullback Metric is Non-Degenerate · `Physicslib4.Spacetime.pullbackVal_nondegenerate` -- [**Lemma 442**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-metric-lorentzian) — The Pullback Metric is Lorentzian · `Physicslib4.Spacetime.pullbackVal_lorentzian` -- [**Lemma 443**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-metric-smooth-in-charts) — The Pullback Metric is a Smooth Section of the Bilinear-Form Bundle · `Physicslib4.Spacetime.pullbackVal_contMDiff` -- [**Theorem 444**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pullback-is-spacetime) — The Pullback of a Spacetime is a Spacetime · `Physicslib4.Spacetime.pullback` (+4 more) -- [**Definition 445**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:preserves-future-orientation) — Two-Sided Preservation of Future Orientation · `Physicslib4.Spacetime.PreservesFutureOrientation` (+1 more) -- [**Lemma 446**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:mpullback-vectorField-contMDiff-of-diffeo) — The Pullback of a Bundle-Smooth Vector Field Along a Diffeomorphism is Bundle-Smooth · `Physicslib4.Spacetime.contMDiff_mpullback_vectorField` -- [**Lemma 447**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-time-orientation-ne-zero) — The Pullback Time Orientation is Nowhere Vanishing · `Physicslib4.Spacetime.mpullback_field_ne_zero` -- [**Lemma 448**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-time-orientation-timelike) — The Pullback Time Orientation is Everywhere Timelike · `Physicslib4.Spacetime.pullbackVal_mpullback_field_self` (+1 more) -- [**Lemma 449**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-time-orientation) — Pullback of a Time Orientation · `Physicslib4.Spacetime.pullbackTimeOrientation` (+1 more) -- [**Lemma 450**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-future-pointing-timelike) — Transport of Future-Pointing Timelike Vectors · `Physicslib4.Spacetime.pullbackVal_mpullback_field_apply` (+1 more) -- [**Lemma 451**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-future-pointing-null) — Transport of Future-Pointing Null Vectors · `Physicslib4.Spacetime.isFuturePointing_pullback_iff_of_isNull` -- [**Lemma 452**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-preserves-future-orientation) — The Pullback Preserves the Future Orientation Two-Sidedly · `Physicslib4.Spacetime.pullback_preservesFutureOrientationTwoSided` -- [**Definition 453**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:cross-metric-isometry) — Isometry Between Two Metrics on One Manifold · `Physicslib4.Spacetime.CrossIsometry` (+4 more) -- [**Lemma 454**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-isometry-symm) — The Inverse of a Cross-Metric Isometry is a Cross-Metric Isometry · `Physicslib4.Spacetime.CrossIsometry.symm_preserves` (+1 more) -- [**Lemma 455**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-isometry-preserves-classification) — Cross-Metric Isometries Preserve the Causal Classification · `Physicslib4.Spacetime.CrossIsometry.preserves_self` (+3 more) -- [**Lemma 456**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-pushforward-path-tangent) — The Tangent Chain Rule Along a Path · `Physicslib4.Spacetime.mfderivWithin_comp_diffeo` (+1 more) -- [**Lemma 457**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-pushforward-path) — Pushforward of a Path Under a Cross-Metric Isometry · `Physicslib4.Spacetime.pushforwardPath` (+2 more) -- [**Lemma 458**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-pushforward-path-causal) — The Pushforward Preserves the Timelike and Causal Conditions · `Physicslib4.Spacetime.CrossIsometry.pushforwardPath_isTimelike` (+1 more) -- [**Lemma 459**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-pushforward-path-endpoints) — The Pushforward Transports Endpoints · `Physicslib4.Spacetime.pushforwardPath_isPastEndpoint` (+1 more) -- [**Lemma 460**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-isometry-preserves-chronology) — Cross-Metric Isometries Transport Chronological Precedence · `Physicslib4.Spacetime.pushforwardPath_isFutureOriented` (+2 more) -- [**Lemma 461**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-chronological-future-image) — Image of the Chronological Future · `Physicslib4.Spacetime.CrossIsometry.chronologicalFuture_image` -- [**Lemma 462**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-chronological-past-image) — Image of the Chronological Past · `Physicslib4.Spacetime.CrossIsometry.chronologicalPast_image` -- [**Lemma 463**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-isometry-preserves-basis-sets) — Cross-Metric Isometries Preserve Basis Sets · `Physicslib4.Spacetime.CrossIsometry.alexandrovDiamond_image` (+1 more) -- [**Lemma 464**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bijection-generated-topology-homeomorphism) — A Bijection Matching Generating Families is a Homeomorphism · `Physicslib4.continuous_generateFrom_of_preimage_mem` (+4 more) -- [**Lemma 465**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-alexandrov-homeomorphism) — The Pullback Alexandrov Topology · `Physicslib4.Spacetime.pullback_alexandrovBasis_image` (+3 more) -- [**Theorem 466**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pullback-is-lorentzian-spacetime) — The Pullback of a Lorentzian Spacetime is a Lorentzian Spacetime · `Physicslib4.Spacetime.LorentzianSpacetime.pullback_alexandrov_t2` (+3 more) +§10.5.6 Pullback metrics and cross-metric isometries (p. 277) — 32 items + +- [**Definition 444**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:pullback-metric) — Pullback of a Spacetime Metric · `Physicslib4.Spacetime.bilinearPrecomp` (+3 more) +- [**Lemma 445**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:mfderiv-diffeo-linear-equiv) — The Differential of a Diffeomorphism is a Linear Equivalence · `Physicslib4.Spacetime.Diffeo` (+4 more) +- [**Lemma 446**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:mfderiv-symm-cancel-left) — Round-Trip Cancellation: $$d\psi$$ After $$d(\psi^{-1})$$ · `Physicslib4.Spacetime.mfderiv_symm_cancel_left` +- [**Lemma 447**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:mfderiv-symm-cancel-right) — Round-Trip Cancellation: $$d(\psi^{-1})$$ After $$d\psi$$ · `Physicslib4.Spacetime.mfderiv_symm_cancel_right` +- [**Lemma 448**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:mfderiv-inverse-eq-symm) — The Formal Inverse of $$d\psi_x$$ is the Inverse Equivalence · `Physicslib4.Spacetime.inverse_mfderiv_eq_symm` (+2 more) +- [**Lemma 449**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-metric-symm) — The Pullback Metric is Symmetric · `Physicslib4.Spacetime.pullbackVal_symm` +- [**Lemma 450**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-metric-nondegenerate) — The Pullback Metric is Non-Degenerate · `Physicslib4.Spacetime.pullbackVal_nondegenerate` +- [**Lemma 451**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-metric-lorentzian) — The Pullback Metric is Lorentzian · `Physicslib4.Spacetime.pullbackVal_lorentzian` +- [**Lemma 452**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-metric-smooth-in-charts) — The Pullback Metric is a Smooth Section of the Bilinear-Form Bundle · `Physicslib4.Spacetime.pullbackVal_contMDiff` +- [**Theorem 453**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pullback-is-spacetime) — The Pullback of a Spacetime is a Spacetime · `Physicslib4.Spacetime.pullback` (+4 more) +- [**Definition 454**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:preserves-future-orientation) — Two-Sided Preservation of Future Orientation · `Physicslib4.Spacetime.PreservesFutureOrientation` (+1 more) +- [**Lemma 455**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:mpullback-vectorField-contMDiff-of-diffeo) — The Pullback of a Bundle-Smooth Vector Field Along a Diffeomorphism is Bundle-Smooth · `Physicslib4.Spacetime.contMDiff_mpullback_vectorField` +- [**Lemma 456**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-time-orientation-ne-zero) — The Pullback Time Orientation is Nowhere Vanishing · `Physicslib4.Spacetime.mpullback_field_ne_zero` +- [**Lemma 457**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-time-orientation-timelike) — The Pullback Time Orientation is Everywhere Timelike · `Physicslib4.Spacetime.pullbackVal_mpullback_field_self` (+1 more) +- [**Lemma 458**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-time-orientation) — Pullback of a Time Orientation · `Physicslib4.Spacetime.pullbackTimeOrientation` (+1 more) +- [**Lemma 459**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-future-pointing-timelike) — Transport of Future-Pointing Timelike Vectors · `Physicslib4.Spacetime.pullbackVal_mpullback_field_apply` (+1 more) +- [**Lemma 460**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-future-pointing-null) — Transport of Future-Pointing Null Vectors · `Physicslib4.Spacetime.isFuturePointing_pullback_iff_of_isNull` +- [**Lemma 461**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-preserves-future-orientation) — The Pullback Preserves the Future Orientation Two-Sidedly · `Physicslib4.Spacetime.pullback_preservesFutureOrientationTwoSided` +- [**Definition 462**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:cross-metric-isometry) — Isometry Between Two Metrics on One Manifold · `Physicslib4.Spacetime.CrossIsometry` (+4 more) +- [**Lemma 463**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-isometry-symm) — The Inverse of a Cross-Metric Isometry is a Cross-Metric Isometry · `Physicslib4.Spacetime.CrossIsometry.symm_preserves` (+1 more) +- [**Lemma 464**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-isometry-preserves-classification) — Cross-Metric Isometries Preserve the Causal Classification · `Physicslib4.Spacetime.CrossIsometry.preserves_self` (+3 more) +- [**Lemma 465**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-pushforward-path-tangent) — The Tangent Chain Rule Along a Path · `Physicslib4.Spacetime.mfderivWithin_comp_diffeo` (+1 more) +- [**Lemma 466**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-pushforward-path) — Pushforward of a Path Under a Cross-Metric Isometry · `Physicslib4.Spacetime.pushforwardPath` (+2 more) +- [**Lemma 467**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-pushforward-path-causal) — The Pushforward Preserves the Timelike and Causal Conditions · `Physicslib4.Spacetime.CrossIsometry.pushforwardPath_isTimelike` (+1 more) +- [**Lemma 468**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-pushforward-path-endpoints) — The Pushforward Transports Endpoints · `Physicslib4.Spacetime.pushforwardPath_isPastEndpoint` (+1 more) +- [**Lemma 469**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-isometry-preserves-chronology) — Cross-Metric Isometries Transport Chronological Precedence · `Physicslib4.Spacetime.pushforwardPath_isFutureOriented` (+2 more) +- [**Lemma 470**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-chronological-future-image) — Image of the Chronological Future · `Physicslib4.Spacetime.CrossIsometry.chronologicalFuture_image` +- [**Lemma 471**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-chronological-past-image) — Image of the Chronological Past · `Physicslib4.Spacetime.CrossIsometry.chronologicalPast_image` +- [**Lemma 472**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cross-metric-isometry-preserves-basis-sets) — Cross-Metric Isometries Preserve Basis Sets · `Physicslib4.Spacetime.CrossIsometry.alexandrovDiamond_image` (+1 more) +- [**Lemma 473**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bijection-generated-topology-homeomorphism) — A Bijection Matching Generating Families is a Homeomorphism · `Physicslib4.continuous_generateFrom_of_preimage_mem` (+4 more) +- [**Lemma 474**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-alexandrov-homeomorphism) — The Pullback Alexandrov Topology · `Physicslib4.Spacetime.pullback_alexandrovBasis_image` (+3 more) +- [**Theorem 475**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pullback-is-lorentzian-spacetime) — The Pullback of a Lorentzian Spacetime is a Lorentzian Spacetime · `Physicslib4.Spacetime.LorentzianSpacetime.pullback_alexandrov_t2` (+3 more)
diff --git a/home_page/sec10-6-haag-kastler.md b/home_page/sec10-6-haag-kastler.md index 98f52a0..0d43660 100644 --- a/home_page/sec10-6-haag-kastler.md +++ b/home_page/sec10-6-haag-kastler.md @@ -1,5 +1,5 @@ --- -title: "§10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 287–328)" +title: "§10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 289–330)" usemathjax: true --- @@ -7,25 +7,25 @@ usemathjax: true ← [§10.5 Spacetime](sec10-5-spacetime.html) · [§10.7 Curved](sec10-7-curved.html) → -# §10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 287–328) - -The largest section by definition and theorem count (160 items, Definitions 467–626). Each axiom is stated as a definition so that it appears as a node in the declaration graph. - -- *The axioms and the quasilocal colimit (§10.6 and §10.6.1, pp. 287–305, items 467–495).* Axiom 1 (Local Algebras, Definition 467) assigns an abstract C\*-algebra to every Alexandrov-basis set, with $$\emptyset \mapsto \mathbb{C}1$$. Axiom 2 (Isotony, Definition 468) supplies the family of unital \*-monomorphisms $$i_{\mathbf{B}_1 \mathbf{B}_2}$$ **as chosen data**, subject to injectivity, an identity law, and a composition law—so that $$\mathbf{B} \mapsto \mathfrak{U}(\mathbf{B})$$ is a functor on the inclusion order of basis sets. The blueprint is explicit that the family cannot be an existence statement (the identity and composition conditions are equations between the maps themselves) and that the inclusion hypothesis is non-strict, matching the Lean. §10.6.1 then cashes this in: the diamonds are directed under inclusion (Lemma 469), the isotony family is a directed system in Mathlib's sense (Lemma 470), and the direct limit carries a well-defined norm (Lemmas 471–473) making it a normed \*-algebra (Lemma 474) satisfying the C\*-inequality (Lemma 475) but not, in general, complete. A block of results stated for an arbitrary normed \*-algebra under standing hypotheses (Definition 476) then carries the structure through completion—the involution and its extension (Definition 477, Lemma 478), the completion coercion as a bundled \*-algebra homomorphism (Lemma 479), and the passage of the C\*-inequality, the normed $$\mathbb{C}$$-algebra structure, and finally the C\*-algebra property to the completion (Lemmas 480–482)—yielding that the completion of the quasilocal colimit is a C\*-algebra (Lemma 483). The quasilocal algebra $$\mathfrak{U}$$ is defined as that colimit-then-completion (Definition 484); the blueprint records why the shortcut of realising $$\mathfrak{U}$$ as a closed \*-subalgebra of an ambient C\*-algebra is rejected—it would assume an ambient algebra containing copies of every $$\mathfrak{U}(\mathbf{B})$$, for which the physics supplies no justification. Axiom 3 (Local Commutativity, Definition 485) follows. It is stated, exactly as in curved spacetime, through the isotony maps: completely spacelike local algebras commute in every local algebra $$\mathfrak{U}(\mathbf{B})$$ containing both, and such a $$\mathbf{B}$$ always exists because the diamonds are directed. It then holds in every quasilocal algebra (Lemma 486), and it is equivalent to commutation in some quasilocal algebra (Lemma 487). The quasilocal observable (Definition 488) follows. Axiom 4 (Quasilocal Completeness, Definition 489) is presented as a bridge principle—the one axiom joining physical reality to the formalism, asserting the one-way inclusion that every physical observable corresponds to a quasilocal observable, and therefore having no mathematical consumers. The mathematical content is a separate theorem: every local net satisfying Axiom 2 admits a quasilocal algebra, with an injective cocone of canonical embeddings whose images are dense (Theorem 490, with the supporting Lemmas 491–495). The indexing over Alexandrov-basis sets only is essential and not stylistic, since the algebras attached to non-basis subsets are junk fibres about which the axioms say nothing. -- *§10.6.2 The von Neumann Density Theorem (pp. 305–307, items 496–506).* A subsection of general operator theory for a unital \*-homomorphism $$\pi : \mathfrak{A} \to \mathcal{B}(H)$$, whose only consumer is Theorem 506, the density result that keeps the bridge principle tenable in the presence of larger bicommutants: $$\pi_\omega(\mathfrak{U})$$ is strongly dense in $$\pi_\omega(\mathfrak{U})''$$. The density theorem is proved here following Murphy, Lemma 4.1.4. A closed subspace invariant under an operator and its adjoint has a commuting orthogonal projection (Lemma 496), so the closed cyclic subspace $$\overline{\pi(\mathfrak{A})\xi}$$ reduces the representation (Lemma 497) and every $$T \in \pi(\mathfrak{A})$$ maps $$\xi$$ into it (Lemma 498). The finite amplification $$\pi^\iota$$ on $$\ell^2(\iota; H)$$ (Definition 499) is a unital \*-homomorphism (Lemma 500); the block entries of an operator in its commutant lie in $$\pi(\mathfrak{A})'$$ (Lemma 501), an operator on $$\ell^2(\iota; H)$$ is recovered from its block entries (Lemma 502), and so $$\mathrm{diag}(T)$$ lies in the amplified bicommutant (Lemma 503). Applying the one-vector case to the amplification approximates $$T$$ on any finite set of vectors (Lemma 504), and a neighbourhood basis for the strong operator topology (Lemma 505) turns this into the density theorem (Theorem 506). The Kaplansky density theorem, which would also keep approximants self-adjoint, remains out of scope; nothing needs it. -- *§10.6.3 Lorentz Covariance and the Haag–Kastler Net (p. 308, items 507–508).* Axiom 5 (Lorentz Covariance, Definition 507), whose coherence condition is stated for the Axiom 2 isotony family itself, and the bundled `HaagKastlerNet` (Definition 508) close the axiomatic block. -- *§10.6.4 Einstein Causality (p. 308, item 509).* Derives the operator form of local commutativity in any \*-representation of the quasilocal algebra (Theorem 509). -- *§10.6.5 Local von Neumann Algebras (p. 308, items 510–522).* Defines the local von Neumann algebra $$R(\mathbf{B}) = \pi(\mathfrak{U}(\mathbf{B}))''$$ of a region in a representation (Definition 510), registers it as a first-class `VonNeumannAlgebra` via the bicommutant-of-a-self-adjoint-set lemma (Lemma 511, Definition 512), and proves Microcausality (Theorem 513) and Isotony of the von Neumann net (Theorem 514), together with their bundled forms (Theorem 515) and the region-indexed assignment $$\mathbf{B} \mapsto R(\mathbf{B})$$ as an order-preserving map—the net of von Neumann algebras itself (Definition 516). It then proves the Statistical Independence (Schlieder) property in set-level and bundled forms (Theorems 517–518): if the cyclic vector of one region is cyclic for its local observables, it is separating for the local von Neumann algebra of any spacelike-separated region. Additive-free locality (Theorem 519) expresses locality through the spacelike complement without attaching an algebra to the unbounded complement itself. Geometric Covariance (Theorem 520) shows conjugation by the implementing unitary carries $$R(\mathbf{B})$$ onto $$R(L \cdot \mathbf{B})$$; being a factor is therefore constant along the Lorentz orbit of a region (Theorem 521), and the set equality upgrades to a first-class \*-algebra isomorphism of the bundled algebras (Theorem 522). -- *§10.6.6 Relative Commutants of Nested Local Algebras (p. 311, items 523–530).* Organises the theory of inclusions $$R(\mathbf{B}_1) \subseteq R(\mathbf{B}_2)$$ around the relative commutant $$R(\mathbf{B}_1)' \cap R(\mathbf{B}_2)$$ (Definition 523). It proves antitonicity of the commutant (Theorem 524), that the relative commutant lies in the larger algebra and commutes with the smaller (Theorems 525–526), and that it always contains the centre of the ambient algebra (Theorem 527). An inclusion is irreducible when its relative commutant is trivial (Definition 528); an irreducible inclusion forces the ambient algebra to be a factor (Theorem 529), and the trivial self-inclusion is irreducible exactly when the algebra is a factor (Theorem 530). -- *§10.6.7 Irreducibility and Schur's Lemma (p. 312, items 531–553).* Introduces irreducible representations via their commutant (Definition 531) and establishes the Topological Schur Lemma for cyclic representations (Theorem 532) together with the operator-theoretic bridge identifying commutant scalars with coefficients proportional to the state (Theorem 533). Defines pure states (Definition 534) and proves "pure implies irreducible" directly (Theorem 535), then the full equivalence Pure $$\iff$$ Irreducible (Theorem 538) via a GNS Radon–Nikodym theorem realising every dominated positive functional as an operator in the commutant (Theorems 536–537). An irreducible representation generates a factor (Theorem 539) and, more sharply, all of $$\mathcal{B}(H)$$ (Theorem 540), with a bundled density form (Theorem 541); consequently the GNS representation of a pure state generates a factor (Theorem 542) and all of $$\mathcal{B}(H)$$ (Theorem 543). The norm of a positive linear functional on a unital C\*-algebra equals its value on the unit (Theorem 544), which supports Pure $$\iff$$ Extreme Point of the state space (Definition 545, Theorem 546), the underlying convexity and the bridge to Mathlib's extreme-points API (Theorem 547), and weak-\* compactness of the state space (Theorem 551), which supplies the existence of pure states via Krein–Milman. The pullback of a state along a unital \*-homomorphism is introduced as its own object (Definition 548) with functoriality (Theorem 549) and the invariance of purity under a \*-isomorphism (Theorem 550). The section closes by specialising to the quasilocal algebra (Theorems 552–553). -- *§10.6.8 Unitary Equivalence and Superselection (p. 316, items 554–555).* Defines unitary equivalence of representations (Definition 554) and shows irreducibility and factoriality are unitary invariants, transported by the cross-space conjugation induced by the implementing unitary (Theorem 555). -- *§10.6.9 GNS Covariance (p. 316, items 556–561).* Cyclicity pulls back along a surjective \*-homomorphism (Lemma 556), so GNS data transports covariantly along a \*-isomorphism of the algebras (Theorem 557), restated as a unitary equivalence (Theorem 558). Pullback along a surjection preserves the image algebra (Lemma 559), whence superselection type transports along a \*-isomorphism (Theorem 560): the whole sector structure is an invariant of the algebra, not of its presentation. Applied to the covariance equivalence $$\alpha_L : \mathfrak{U}(\mathbf{B}) \simeq \mathfrak{U}(L \cdot \mathbf{B})$$ supplied by Axiom 5, this says the superselection type of a local state is constant along the Lorentz orbit of its region (Theorem 561). -- *§10.6.10 Disjointness and Quasi-Equivalence (p. 318, items 562–579).* Defines disjointness via the vanishing of all intertwiners (Definition 562) and the coarser quasi-equivalence via a \*-isomorphism of generated von Neumann algebras (Definition 563). Proves Schur's Lemma in the form of the Irreducible Dichotomy—two irreducible representations are either disjoint or unitarily equivalent (Theorem 564)—together with Schur multiplicity (Lemma 565) and the triviality of the endomorphism algebra of an irreducible representation (Lemma 566). The commutant is packaged as the self-intertwiner (gauge) von Neumann algebra, trivial exactly when the representation is irreducible (Theorem 567), with double-commutant duality (Theorem 568) and the factor/triviality duality between an algebra and its commutant (Theorem 569). A supporting run develops the centre from scratch: abelian $$\iff$$ self-commuting (Lemma 570), the centre $$Z(R) = R \cap R'$$ (Definition 571) and its being a von Neumann algebra (Lemma 572), the general two-algebra intersection lemma that the relative commutants of §10.6.6 need (Lemma 573), the centre is abelian (Theorem 574), $$R$$ is abelian iff it equals its centre (Lemma 575), a factor is abelian iff it is the scalars (Theorem 576), an algebra and its commutant share a centre (Theorem 577), and factoriality is triviality of the centre (Theorem 578). The Pure-State Dichotomy underlying superselection sectors (Theorem 579) closes the block. -- *§10.6.11 Direct Sums, Amplification, and Reducibility (p. 321, items 580–583).* Defines the direct-sum representation on the $$\ell^2$$-direct sum (Definition 580), shows each summand embeds as a subrepresentation whose projection lies in the commutant of the sum (Theorem 581), defines the $$\iota$$-fold amplification (Definition 582), and proves a direct sum with at least two nonzero summands is reducible, so a multiply-amplified representation is never irreducible (Theorem 583). -- *§10.6.12 Covariant States and the Covariance Action (p. 321, items 584–610).* Defines covariant families of local states (Definition 584) with their composition law (Lemma 585), and the lift of the fibrewise covariance action to a \*-automorphism of the quasilocal algebra (Definition 586), with uniqueness (Lemma 587) and existence for every quasilocal algebra (Theorem 589), no covariance-compatibility hypothesis being needed because every quasilocal algebra is automatically covariance-compatible (Lemma 588), and the trivial net as a special case (Theorem 590). The covariant quasilocal algebra of a net is then simply its canonical quasilocal algebra together with the covariance action (Definition 591, realised in Lean as `HaagKastlerNet.action`, so the covariance dynamics are stated directly for the net), which is a genuine group action (Lemma 592). Invariant states are defined (Definition 593) and shown to be implemented by GNS unitaries (Theorem 594); a state that is both invariant and pure yields a GNS representation that is simultaneously covariant and irreducible (Theorem 595). The spectrum condition follows. Positive energy (Definition 596) asks that a one-parameter unitary group be strongly continuous with positive generator; the trivial group has positive energy (Lemma 597), the generator is unique (Lemma 598), positivity and positive energy are unitarily invariant (Lemmas 599–600), a positive-energy group is strongly continuous (Lemma 601), the bounded case $$\exp(itP)$$ is included (Lemma 602), and, by Stone's Theorem, positive energy says exactly that $$V(t) = e^{itA}$$ with $$A$$ positive and self-adjoint (Theorem 603), all collected in Theorem 604. A vacuum state (Definition 605) is an invariant state whose translation groups have positive energy, with the consequences of invariance alone (Theorem 606), the future-timelike translation subgroup (Definition 607), and the vacuum state with that concrete predicate substituted in, leaving no free parameter (Definition 608). Purity is preserved by any \*-automorphism and is therefore covariance-invariant (Theorem 609), and GNS data transports along the quasilocal covariance action (Theorem 610). -- *§10.6.13 The Separating Vector of a Faithful State (p. 326, item 611).* The cyclic vector of a faithful state is also separating for the image of the representation (Theorem 611)—the basic datum of Tomita–Takesaki modular theory. This holds in any representation reproducing a faithful state, not only the canonical GNS one. -- *§10.6.14 The KMS Condition and Thermal Equilibrium (p. 326, items 612–621).* Introduces one-parameter automorphism groups (Definition 612) and KMS states (Definition 613) as the algebraic characterisation of thermal equilibrium. The condition is phrased purely as an analyticity statement about correlation functions, so—unlike the spectrum condition—it needs no unbounded-operator theory. The KMS state set is convex (Theorem 614); a boundary-coincidence argument at $$a = 1$$ (Lemma 615) together with the Strip-Liouville Principle (Definition 616), proved at positive inverse temperature via an $$i\beta$$-periodic entire extension (Theorem 617) and Liouville's theorem (Theorem 618), yields that KMS states are automatically invariant under the time evolution (Theorem 619); uniqueness on the strip from boundary values (Theorem 620) gives uniqueness of the analytic completion of a KMS correlation function (Theorem 621). -- *§10.6.15 KMS States for the Covariance Flow (p. 328, items 622–626).* A one-parameter subgroup of the inhomogeneous Lorentz group induces a one-parameter automorphism group on the quasilocal algebra via the covariance lift (Definition 622, Lemma 623); KMS states for that flow are defined accordingly (Definition 624) and shown convex (Theorem 625). The zero-temperature ($$\beta \to \infty$$) counterpart—a ground state for a covariance flow, whose GNS-implementing unitary group has positive energy—is recorded alongside it (Definition 626). +# §10.6 Haag–Kastler Axioms in Minkowski spacetime (pp. 289–330) + +The largest section by definition and theorem count (162 items, Definitions 476–637). Each axiom is stated as a definition so that it appears as a node in the declaration graph. + +- *The axioms and the quasilocal colimit (§10.6 and §10.6.1, pp. 289–305, items 476–504).* Axiom 1 (Local Algebras, Definition 476) assigns an abstract C\*-algebra to every Alexandrov-basis set, with $$\emptyset \mapsto \mathbb{C}1$$. Axiom 2 (Isotony, Definition 477) supplies the family of unital \*-monomorphisms $$i_{\mathbf{B}_1 \mathbf{B}_2}$$ **as chosen data**, subject to injectivity, an identity law, and a composition law—so that $$\mathbf{B} \mapsto \mathfrak{U}(\mathbf{B})$$ is a functor on the inclusion order of basis sets. The blueprint is explicit that the family cannot be an existence statement (the identity and composition conditions are equations between the maps themselves) and that the inclusion hypothesis is non-strict, matching the Lean. §10.6.1 then cashes this in: the diamonds are directed under inclusion (Lemma 478), the isotony family is a directed system in Mathlib's sense (Lemma 479), and the direct limit carries a well-defined norm (Lemmas 480–482) making it a normed \*-algebra (Lemma 483) satisfying the C\*-inequality (Lemma 484) but not, in general, complete. A block of results stated for an arbitrary normed \*-algebra under standing hypotheses (Definition 485) then carries the structure through completion—the involution and its extension (Definition 486, Lemma 487), the completion coercion as a bundled \*-algebra homomorphism (Lemma 488), and the passage of the C\*-inequality, the normed $$\mathbb{C}$$-algebra structure, and finally the C\*-algebra property to the completion (Lemmas 489–491)—yielding that the completion of the quasilocal colimit is a C\*-algebra (Lemma 492). The quasilocal algebra $$\mathfrak{U}$$ is defined as that colimit-then-completion (Definition 493); the blueprint records why the shortcut of realising $$\mathfrak{U}$$ as a closed \*-subalgebra of an ambient C\*-algebra is rejected—it would assume an ambient algebra containing copies of every $$\mathfrak{U}(\mathbf{B})$$, for which the physics supplies no justification. Axiom 3 (Local Commutativity, Definition 494) follows. It is stated, exactly as in curved spacetime, through the isotony maps: completely spacelike local algebras commute in every local algebra $$\mathfrak{U}(\mathbf{B})$$ containing both, and such a $$\mathbf{B}$$ always exists because the diamonds are directed. It then holds in every quasilocal algebra (Lemma 495), and it is equivalent to commutation in some quasilocal algebra (Lemma 496). The quasilocal observable (Definition 497) follows. Axiom 4 (Quasilocal Completeness, Definition 498) is presented as a bridge principle—the one axiom joining physical reality to the formalism, asserting the one-way inclusion that every physical observable corresponds to a quasilocal observable, and therefore having no mathematical consumers. The mathematical content is a separate theorem: every local net satisfying Axiom 2 admits a quasilocal algebra, with an injective cocone of canonical embeddings whose images are dense (Theorem 499, with the supporting Lemmas 500–504). The indexing over Alexandrov-basis sets only is essential and not stylistic, since the algebras attached to non-basis subsets are junk fibres about which the axioms say nothing. +- *§10.6.2 The von Neumann Density Theorem (pp. 306–307, items 505–515).* A subsection of general operator theory for a unital \*-homomorphism $$\pi : \mathfrak{A} \to \mathcal{B}(H)$$, whose only consumer is Theorem 515, the density result that keeps the bridge principle tenable in the presence of larger bicommutants: $$\pi_\omega(\mathfrak{U})$$ is strongly dense in $$\pi_\omega(\mathfrak{U})''$$. The density theorem is proved here following Murphy, Lemma 4.1.4. A closed subspace invariant under an operator and its adjoint has a commuting orthogonal projection (Lemma 505), so the closed cyclic subspace $$\overline{\pi(\mathfrak{A})\xi}$$ reduces the representation (Lemma 506) and every $$T \in \pi(\mathfrak{A})$$ maps $$\xi$$ into it (Lemma 507). The finite amplification $$\pi^\iota$$ on $$\ell^2(\iota; H)$$ (Definition 508) is a unital \*-homomorphism (Lemma 509); the block entries of an operator in its commutant lie in $$\pi(\mathfrak{A})'$$ (Lemma 510), an operator on $$\ell^2(\iota; H)$$ is recovered from its block entries (Lemma 511), and so $$\mathrm{diag}(T)$$ lies in the amplified bicommutant (Lemma 512). Applying the one-vector case to the amplification approximates $$T$$ on any finite set of vectors (Lemma 513), and a neighbourhood basis for the strong operator topology (Lemma 514) turns this into the density theorem (Theorem 515). The Kaplansky density theorem, which would also keep approximants self-adjoint, remains out of scope; nothing needs it. +- *§10.6.3 Lorentz Covariance and the Haag–Kastler Net (p. 308, items 516–517).* Axiom 5 (Lorentz Covariance, Definition 516), whose coherence condition is stated for the Axiom 2 isotony family itself, and the bundled `HaagKastlerNet` (Definition 517) close the axiomatic block. +- *§10.6.4 Einstein Causality (p. 309, item 518).* Derives the operator form of local commutativity in any \*-representation of the quasilocal algebra (Theorem 518). +- *§10.6.5 Local von Neumann Algebras (p. 310, items 519–531).* Defines the local von Neumann algebra $$R(\mathbf{B}) = \pi(\mathfrak{U}(\mathbf{B}))''$$ of a region in a representation (Definition 519), registers it as a first-class `VonNeumannAlgebra` via the bicommutant-of-a-self-adjoint-set lemma (Lemma 520, Definition 521), and proves Microcausality (Theorem 522) and Isotony of the von Neumann net (Theorem 523), together with their bundled forms (Theorem 524) and the region-indexed assignment $$\mathbf{B} \mapsto R(\mathbf{B})$$ as an order-preserving map—the net of von Neumann algebras itself (Definition 525). It then proves the Statistical Independence (Schlieder) property in set-level and bundled forms (Theorems 526–527): if the cyclic vector of one region is cyclic for its local observables, it is separating for the local von Neumann algebra of any spacelike-separated region. Additive-free locality (Theorem 528) expresses locality through the spacelike complement without attaching an algebra to the unbounded complement itself. Geometric Covariance (Theorem 529) shows conjugation by the implementing unitary carries $$R(\mathbf{B})$$ onto $$R(L \cdot \mathbf{B})$$; being a factor is therefore constant along the Lorentz orbit of a region (Theorem 530), and the set equality upgrades to a first-class \*-algebra isomorphism of the bundled algebras (Theorem 531). +- *§10.6.6 Relative Commutants of Nested Local Algebras (p. 312, items 532–539).* Organises the theory of inclusions $$R(\mathbf{B}_1) \subseteq R(\mathbf{B}_2)$$ around the relative commutant $$R(\mathbf{B}_1)' \cap R(\mathbf{B}_2)$$ (Definition 532). It proves antitonicity of the commutant (Theorem 533), that the relative commutant lies in the larger algebra and commutes with the smaller (Theorems 534–535), and that it always contains the centre of the ambient algebra (Theorem 536). An inclusion is irreducible when its relative commutant is trivial (Definition 537); an irreducible inclusion forces the ambient algebra to be a factor (Theorem 538), and the trivial self-inclusion is irreducible exactly when the algebra is a factor (Theorem 539). +- *§10.6.7 Irreducibility and Schur's Lemma (p. 313, items 540–562).* Introduces irreducible representations via their commutant (Definition 540) and establishes the Topological Schur Lemma for cyclic representations (Theorem 541) together with the operator-theoretic bridge identifying commutant scalars with coefficients proportional to the state (Theorem 542). Defines pure states (Definition 543) and proves "pure implies irreducible" directly (Theorem 544), then the full equivalence Pure $$\iff$$ Irreducible (Theorem 547) via a GNS Radon–Nikodym theorem realising every dominated positive functional as an operator in the commutant (Theorems 545–546). An irreducible representation generates a factor (Theorem 548) and, more sharply, all of $$\mathcal{B}(H)$$ (Theorem 549), with a bundled density form (Theorem 550); consequently the GNS representation of a pure state generates a factor (Theorem 551) and all of $$\mathcal{B}(H)$$ (Theorem 552). The norm of a positive linear functional on a unital C\*-algebra equals its value on the unit (Theorem 553), which supports Pure $$\iff$$ Extreme Point of the state space (Definition 554, Theorem 555), the underlying convexity and the bridge to Mathlib's extreme-points API (Theorem 556), and weak-\* compactness of the state space (Theorem 560), which supplies the existence of pure states via Krein–Milman. The pullback of a state along a unital \*-homomorphism is introduced as its own object (Definition 557) with functoriality (Theorem 558) and the invariance of purity under a \*-isomorphism (Theorem 559). The section closes by specialising to the quasilocal algebra (Theorems 561–562). +- *§10.6.8 Unitary Equivalence and Superselection (p. 317, items 563–564).* Defines unitary equivalence of representations (Definition 563) and shows irreducibility and factoriality are unitary invariants, transported by the cross-space conjugation induced by the implementing unitary (Theorem 564). +- *§10.6.9 GNS Covariance (p. 317, items 565–570).* Cyclicity pulls back along a surjective \*-homomorphism (Lemma 565), so GNS data transports covariantly along a \*-isomorphism of the algebras (Theorem 566), restated as a unitary equivalence (Theorem 567). Pullback along a surjection preserves the image algebra (Lemma 568), whence superselection type transports along a \*-isomorphism (Theorem 569): the whole sector structure is an invariant of the algebra, not of its presentation. Applied to the covariance equivalence $$\alpha_L : \mathfrak{U}(\mathbf{B}) \simeq \mathfrak{U}(L \cdot \mathbf{B})$$ supplied by Axiom 5, this says the superselection type of a local state is constant along the Lorentz orbit of its region (Theorem 570). +- *§10.6.10 Disjointness and Quasi-Equivalence (p. 319, items 571–588).* Defines disjointness via the vanishing of all intertwiners (Definition 571) and the coarser quasi-equivalence via a \*-isomorphism of generated von Neumann algebras (Definition 572). Proves Schur's Lemma in the form of the Irreducible Dichotomy—two irreducible representations are either disjoint or unitarily equivalent (Theorem 573)—together with Schur multiplicity (Lemma 574) and the triviality of the endomorphism algebra of an irreducible representation (Lemma 575). The commutant is packaged as the self-intertwiner (gauge) von Neumann algebra, trivial exactly when the representation is irreducible (Theorem 576), with double-commutant duality (Theorem 577) and the factor/triviality duality between an algebra and its commutant (Theorem 578). A supporting run develops the centre from scratch: abelian $$\iff$$ self-commuting (Lemma 579), the centre $$Z(R) = R \cap R'$$ (Definition 580) and its being a von Neumann algebra (Lemma 581), the general two-algebra intersection lemma that the relative commutants of §10.6.6 need (Lemma 582), the centre is abelian (Theorem 583), $$R$$ is abelian iff it equals its centre (Lemma 584), a factor is abelian iff it is the scalars (Theorem 585), an algebra and its commutant share a centre (Theorem 586), and factoriality is triviality of the centre (Theorem 587). The Pure-State Dichotomy underlying superselection sectors (Theorem 588) closes the block. +- *§10.6.11 Direct Sums, Amplification, and Reducibility (p. 322, items 589–592).* Defines the direct-sum representation on the $$\ell^2$$-direct sum (Definition 589), shows each summand embeds as a subrepresentation whose projection lies in the commutant of the sum (Theorem 590), defines the $$\iota$$-fold amplification (Definition 591), and proves a direct sum with at least two nonzero summands is reducible, so a multiply-amplified representation is never irreducible (Theorem 592). +- *§10.6.12 Covariant States and the Covariance Action (p. 322, items 593–621).* Defines covariant families of local states (Definition 593) with their composition law (Lemma 594), and the lift of the fibrewise covariance action to a \*-automorphism of the quasilocal algebra (Definition 595), with uniqueness (Lemma 596) and existence for every quasilocal algebra (Theorem 598), no covariance-compatibility hypothesis being needed because every quasilocal algebra is automatically covariance-compatible (Lemma 597), and the trivial net as a special case (Theorem 599). The covariant quasilocal algebra of a net is then simply its canonical quasilocal algebra together with the covariance action (Definition 600, realised in Lean as `HaagKastlerNet.action`, so the covariance dynamics are stated directly for the net), which is a genuine group action (Lemma 601). Invariant states are defined (Definition 602) and shown to be implemented by GNS unitaries (Theorem 603); since weak continuity of a unitary representation implies strong continuity (Lemma 604), the GNS unitary representation is strongly continuous whenever its matrix coefficients are continuous (Theorem 605); a state that is both invariant and pure yields a GNS representation that is simultaneously covariant and irreducible (Theorem 606). The spectrum condition follows. Positive energy (Definition 607) asks that a one-parameter unitary group be strongly continuous with positive generator; the trivial group has positive energy (Lemma 608), the generator is unique (Lemma 609), positivity and positive energy are unitarily invariant (Lemmas 610–611), a positive-energy group is strongly continuous (Lemma 612), the bounded case $$\exp(itP)$$ is included (Lemma 613), and, by Stone's Theorem, positive energy says exactly that $$V(t) = e^{itA}$$ with $$A$$ positive and self-adjoint (Theorem 614), all collected in Theorem 615. A vacuum state (Definition 616) is an invariant state whose translation groups have positive energy, with the consequences of invariance alone (Theorem 617), the future-timelike translation subgroup (Definition 618), and the vacuum state with that concrete predicate substituted in, leaving no free parameter (Definition 619). Purity is preserved by any \*-automorphism and is therefore covariance-invariant (Theorem 620), and GNS data transports along the quasilocal covariance action (Theorem 621). +- *§10.6.13 The Separating Vector of a Faithful State (p. 328, item 622).* The cyclic vector of a faithful state is also separating for the image of the representation (Theorem 622)—the basic datum of Tomita–Takesaki modular theory. This holds in any representation reproducing a faithful state, not only the canonical GNS one. +- *§10.6.14 The KMS Condition and Thermal Equilibrium (p. 328, items 623–632).* Introduces one-parameter automorphism groups (Definition 623) and KMS states (Definition 624) as the algebraic characterisation of thermal equilibrium. The condition is phrased purely as an analyticity statement about correlation functions, so—unlike the spectrum condition—it needs no unbounded-operator theory. The KMS state set is convex (Theorem 625); a boundary-coincidence argument at $$a = 1$$ (Lemma 626) together with the Strip-Liouville Principle (Definition 627), proved at positive inverse temperature via an $$i\beta$$-periodic entire extension (Theorem 628) and Liouville's theorem (Theorem 629), yields that KMS states are automatically invariant under the time evolution (Theorem 630); uniqueness on the strip from boundary values (Theorem 631) gives uniqueness of the analytic completion of a KMS correlation function (Theorem 632). +- *§10.6.15 KMS States for the Covariance Flow (p. 329, items 633–637).* A one-parameter subgroup of the inhomogeneous Lorentz group induces a one-parameter automorphism group on the quasilocal algebra via the covariance lift (Definition 633, Lemma 634); KMS states for that flow are defined accordingly (Definition 635) and shown convex (Theorem 636). The zero-temperature ($$\beta \to \infty$$) counterpart—a ground state for a covariance flow, whose GNS-implementing unitary group has positive energy—is recorded alongside it (Definition 637). **Where the Lean lives:** `Physicslib4/AQFT/HaagKastler/`, `Physicslib4/GNS/` (`Irreducibility`, `Superselection`, `RadonNikodym`, `ExtremeState`, …), `Physicslib4/Operators/` (`ReducingSubspace`, `DensityTheorem`), `Physicslib4/AQFT/KMS.lean`, `Physicslib4/Analysis/StripPeriodicExtension.lean` @@ -34,258 +34,260 @@ The largest section by definition and theorem count (160 items, Definitions 467 Each blueprint subsection below is collapsed; click a heading to see its items. Every entry links to its node in the web blueprint and names the principal Lean declaration.
-§10.6 The axioms (p. 287) — 2 items +§10.6 The axioms (p. 289) — 2 items -- [**Definition 467**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-algebras) — Axiom 1: Local Algebras · `Physicslib4.AQFT.HaagKastler.LocalNet` -- [**Definition 468**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:isotony) — Axiom 2: Isotony · `Physicslib4.AQFT.HaagKastler.Isotony` +- [**Definition 476**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-algebras) — Axiom 1: Local Algebras · `Physicslib4.AQFT.HaagKastler.LocalNet` +- [**Definition 477**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:isotony) — Axiom 2: Isotony · `Physicslib4.AQFT.HaagKastler.Isotony`
-§10.6.1 The Quasilocal Colimit (p. 288) — 27 items - -- [**Lemma 469**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:alexandrov-diamonds-isDirected) — Alexandrov Diamonds are Directed under Inclusion · `Physicslib4.AQFT.HaagKastler.Diamond` (+2 more) -- [**Lemma 470**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isotony-directed-system) — The Isotony Family is a Directed System · `Physicslib4.AQFT.HaagKastler.transitionHom` (+1 more) -- [**Lemma 471**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-colimit-norm-well-defined) — The Colimit Norm is Well Defined · `Physicslib4.AQFT.HaagKastler.QuasilocalColimit` (+3 more) -- [**Lemma 472**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-colimit-common-representatives) — Common Representatives for Two Colimit Elements · `Physicslib4.AQFT.HaagKastler.exists_common_representatives` -- [**Lemma 473**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-colimit-norm-axioms) — The Colimit Norm is a Ring Norm and a Normed-Space Norm · `Physicslib4.AQFT.HaagKastler.instNonemptyDiamond` (+3 more) -- [**Lemma 474**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-union-normed-star-algebra) — The Quasilocal Union is a Normed \*-Algebra · `Physicslib4.AQFT.HaagKastler.norm_eq_colimitNorm` (+1 more) -- [**Lemma 475**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-colimit-cstar-identity) — The Colimit Satisfies the C\*-Inequality · `Physicslib4.AQFT.HaagKastler.colimitCStarRing` -- [**Definition 476**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:completion-standing-hypotheses) — Standing Hypotheses for the Completion Results · `Physicslib4.CStarCompletion` -- [**Definition 477**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:completion-star) — The Involution on a Completion · `Physicslib4.instStarCompletion` (+1 more) -- [**Lemma 478**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:star-extends-to-completion) — The Involution Extends to the Completion · `Physicslib4.instStarRingCompletion` (+1 more) -- [**Lemma 479**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:completion-coe-star-alg-hom) — The Completion Coercion as a Bundled \*-Algebra Homomorphism · `Physicslib4.coeStarAlgHom` -- [**Lemma 480**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:completion-cstar-identity) — The C\*-Inequality Passes to the Completion · `Physicslib4.instCStarRingCompletion` -- [**Lemma 481**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:completion-normed-algebra) — The Completion is a Normed $$\mathbb{C}$$-Algebra · `Physicslib4.instNormedAlgebraCompletion` -- [**Lemma 482**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:completion-of-cstar-normed-star-algebra) — The Completion of a C\*-Normed \*-Algebra is a C\*-Algebra · `Physicslib4.instCStarAlgebraCompletion` -- [**Lemma 483**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-completion-cstar) — The Completion of the Quasilocal Colimit is a C\*-Algebra · `Physicslib4.AQFT.HaagKastler.QuasilocalCompletion` (+1 more) -- [**Definition 484**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-algebra) — Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.QuasilocalAlgebra` -- [**Definition 485**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-commutativity) — Axiom 3: Local Commutativity · `Physicslib4.AQFT.HaagKastler.LocalCommutativity` -- [**Lemma 486**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:local-commutativity-any-quasilocal) — Local Commutativity Holds in Every Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.LocalCommutativity.commute_ι` (+1 more) -- [**Lemma 487**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:local-commutativity-iff-quasilocal) — Local Commutativity in the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.localCommutativity_iff_exists_commute_ι` -- [**Definition 488**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-observable) — Quasilocal Observable · `Physicslib4.AQFT.HaagKastler.IsQuasilocalObservable` -- [**Definition 489**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-completeness) — Axiom 4: Quasilocal Completeness · `Physicslib4.AQFT.HaagKastler.ObservableCorrespondence` -- [**Theorem 490**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-algebra-exists) — Existence of a Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.exists_quasilocalAlgebra` -- [**Lemma 491**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embedding) — The Canonical Embeddings into the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.colimitStarOf` (+1 more) -- [**Lemma 492**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embedding-injective) — The Canonical Embeddings are Injective · `Physicslib4.AQFT.HaagKastler.quasilocalEmbedding_injective` -- [**Lemma 493**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embedding-cocone) — The Canonical Embeddings Form a Cocone · `Physicslib4.AQFT.HaagKastler.quasilocalEmbedding_transitionHom` -- [**Lemma 494**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-colimit-union-of-insertions) — The Colimit is the Union of the Images of its Insertions · `Physicslib4.AQFT.HaagKastler.exists_eq_colimitStarOf` -- [**Lemma 495**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embeddings-dense) — The Images of the Canonical Embeddings are Dense · `Physicslib4.AQFT.HaagKastler.dense_iUnion_range_quasilocalEmbedding` +§10.6.1 The Quasilocal Colimit (p. 291) — 27 items + +- [**Lemma 478**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:alexandrov-diamonds-isDirected) — Alexandrov Diamonds are Directed under Inclusion · `Physicslib4.AQFT.HaagKastler.Diamond` (+2 more) +- [**Lemma 479**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:isotony-directed-system) — The Isotony Family is a Directed System · `Physicslib4.AQFT.HaagKastler.transitionHom` (+1 more) +- [**Lemma 480**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-colimit-norm-well-defined) — The Colimit Norm is Well Defined · `Physicslib4.AQFT.HaagKastler.QuasilocalColimit` (+3 more) +- [**Lemma 481**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-colimit-common-representatives) — Common Representatives for Two Colimit Elements · `Physicslib4.AQFT.HaagKastler.exists_common_representatives` +- [**Lemma 482**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-colimit-norm-axioms) — The Colimit Norm is a Ring Norm and a Normed-Space Norm · `Physicslib4.AQFT.HaagKastler.instNonemptyDiamond` (+3 more) +- [**Lemma 483**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-union-normed-star-algebra) — The Quasilocal Union is a Normed \*-Algebra · `Physicslib4.AQFT.HaagKastler.norm_eq_colimitNorm` (+1 more) +- [**Lemma 484**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-colimit-cstar-identity) — The Colimit Satisfies the C\*-Inequality · `Physicslib4.AQFT.HaagKastler.colimitCStarRing` +- [**Definition 485**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:completion-standing-hypotheses) — Standing Hypotheses for the Completion Results · `Physicslib4.CStarCompletion` +- [**Definition 486**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:completion-star) — The Involution on a Completion · `Physicslib4.instStarCompletion` (+1 more) +- [**Lemma 487**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:star-extends-to-completion) — The Involution Extends to the Completion · `Physicslib4.instStarRingCompletion` (+1 more) +- [**Lemma 488**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:completion-coe-star-alg-hom) — The Completion Coercion as a Bundled \*-Algebra Homomorphism · `Physicslib4.coeStarAlgHom` +- [**Lemma 489**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:completion-cstar-identity) — The C\*-Inequality Passes to the Completion · `Physicslib4.instCStarRingCompletion` +- [**Lemma 490**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:completion-normed-algebra) — The Completion is a Normed $$\mathbb{C}$$-Algebra · `Physicslib4.instNormedAlgebraCompletion` +- [**Lemma 491**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:completion-of-cstar-normed-star-algebra) — The Completion of a C\*-Normed \*-Algebra is a C\*-Algebra · `Physicslib4.instCStarAlgebraCompletion` +- [**Lemma 492**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-completion-cstar) — The Completion of the Quasilocal Colimit is a C\*-Algebra · `Physicslib4.AQFT.HaagKastler.QuasilocalCompletion` (+1 more) +- [**Definition 493**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-algebra) — Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.QuasilocalAlgebra` +- [**Definition 494**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-commutativity) — Axiom 3: Local Commutativity · `Physicslib4.AQFT.HaagKastler.LocalCommutativity` +- [**Lemma 495**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:local-commutativity-any-quasilocal) — Local Commutativity Holds in Every Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.LocalCommutativity.commute_ι` (+1 more) +- [**Lemma 496**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:local-commutativity-iff-quasilocal) — Local Commutativity in the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.localCommutativity_iff_exists_commute_ι` +- [**Definition 497**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-observable) — Quasilocal Observable · `Physicslib4.AQFT.HaagKastler.IsQuasilocalObservable` (+1 more) +- [**Definition 498**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-completeness) — Axiom 4: Quasilocal Completeness · `Physicslib4.AQFT.HaagKastler.ObservableCorrespondence` +- [**Theorem 499**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-algebra-exists) — Existence of a Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.exists_quasilocalAlgebra` +- [**Lemma 500**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embedding) — The Canonical Embeddings into the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.colimitStarOf` (+1 more) +- [**Lemma 501**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embedding-injective) — The Canonical Embeddings are Injective · `Physicslib4.AQFT.HaagKastler.quasilocalEmbedding_injective` +- [**Lemma 502**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embedding-cocone) — The Canonical Embeddings Form a Cocone · `Physicslib4.AQFT.HaagKastler.quasilocalEmbedding_transitionHom` +- [**Lemma 503**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-colimit-union-of-insertions) — The Colimit is the Union of the Images of its Insertions · `Physicslib4.AQFT.HaagKastler.exists_eq_colimitStarOf` +- [**Lemma 504**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-embeddings-dense) — The Images of the Canonical Embeddings are Dense · `Physicslib4.AQFT.HaagKastler.dense_iUnion_range_quasilocalEmbedding`
-§10.6.2 The von Neumann Density Theorem (p. 305) — 11 items - -- [**Lemma 496**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:reducing-subspace-projection-commutes) — Reducing Subspaces Give Commuting Projections · `Physicslib4.commute_starProjection_of_invariant` -- [**Lemma 497**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cyclic-subspace-reduces) — Cyclic Subspaces Reduce the Representation · `Physicslib4.starProjection_cyclic_mem_centralizer` -- [**Lemma 498**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bicommutant-single-vector) — The Bicommutant on a Single Vector · `Physicslib4.apply_mem_closure_of_mem_bicommutant` -- [**Definition 499**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:finite-amplification) — Finite Amplification · `Physicslib4.diagAmplification` (+2 more) -- [**Lemma 500**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:finite-amplification-star-hom) — The Finite Amplification is a Unital \*-Homomorphism · `Physicslib4.diagAmplification` -- [**Lemma 501**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:commutant-of-amplification-entries) — Block Entries of the Amplified Commutant · `Physicslib4.blockEntry_mem_centralizer` -- [**Lemma 502**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:amplification-block-expansion) — Block Expansion on the Amplified Space · `Physicslib4.apply_eq_sum_blockEntry` -- [**Lemma 503**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:diagonal-in-amplified-bicommutant) — Diagonal Operators in the Amplified Bicommutant · `Physicslib4.ampDiag_mem_bicommutant` -- [**Lemma 504**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bicommutant-finite-vectors) — The Bicommutant on Finitely Many Vectors · `Physicslib4.exists_forall_norm_sub_lt_of_mem_bicommutant` -- [**Lemma 505**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:strong-neighbourhood-basis) — A Neighbourhood Basis for the Strong Operator Topology · `Physicslib4.AQFT.HaagKastler.hasBasis_nhds_ofFun` -- [**Theorem 506**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-strongly-dense) — Quasilocal Observables are Strongly Dense in the Bicommutant · `Physicslib4.AQFT.HaagKastler.dense_range_in_bicommutant` — the von Neumann density theorem, proved locally from Lemmas 448–457 (see [Formalisation status](./#formalisation-status)) +§10.6.2 The von Neumann Density Theorem (p. 306) — 11 items + +- [**Lemma 505**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:reducing-subspace-projection-commutes) — Reducing Subspaces Give Commuting Projections · `Physicslib4.commute_starProjection_of_invariant` +- [**Lemma 506**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cyclic-subspace-reduces) — Cyclic Subspaces Reduce the Representation · `Physicslib4.starProjection_cyclic_mem_centralizer` +- [**Lemma 507**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bicommutant-single-vector) — The Bicommutant on a Single Vector · `Physicslib4.apply_mem_closure_of_mem_bicommutant` +- [**Definition 508**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:finite-amplification) — Finite Amplification · `Physicslib4.diagAmplification` (+2 more) +- [**Lemma 509**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:finite-amplification-star-hom) — The Finite Amplification is a Unital \*-Homomorphism · `Physicslib4.diagAmplification` +- [**Lemma 510**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:commutant-of-amplification-entries) — Block Entries of the Amplified Commutant · `Physicslib4.blockEntry_mem_centralizer` +- [**Lemma 511**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:amplification-block-expansion) — Block Expansion on the Amplified Space · `Physicslib4.apply_eq_sum_blockEntry` +- [**Lemma 512**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:diagonal-in-amplified-bicommutant) — Diagonal Operators in the Amplified Bicommutant · `Physicslib4.ampDiag_mem_bicommutant` +- [**Lemma 513**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bicommutant-finite-vectors) — The Bicommutant on Finitely Many Vectors · `Physicslib4.exists_forall_norm_sub_lt_of_mem_bicommutant` +- [**Lemma 514**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:strong-neighbourhood-basis) — A Neighbourhood Basis for the Strong Operator Topology · `Physicslib4.AQFT.HaagKastler.hasBasis_nhds_ofFun` +- [**Theorem 515**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-strongly-dense) — Quasilocal Observables are Strongly Dense in the Bicommutant · `Physicslib4.AQFT.HaagKastler.dense_range_in_bicommutant` — the von Neumann density theorem, proved locally from the lemmas above (see [Formalisation status](./#formalisation-status))
-§10.6.3 Lorentz Covariance and the Haag–Kastler Net (p. 307) — 2 items +§10.6.3 Lorentz Covariance and the Haag–Kastler Net (p. 308) — 2 items -- [**Definition 507**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:lorentz-covariance) — Axiom 5: Lorentz Covariance · `Physicslib4.AQFT.HaagKastler.LorentzCovariance` -- [**Definition 508**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:haag-kastler-net) — Haag–Kastler Net · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet` +- [**Definition 516**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:lorentz-covariance) — Axiom 5: Lorentz Covariance · `Physicslib4.AQFT.HaagKastler.LorentzCovariance` +- [**Definition 517**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:haag-kastler-net) — Haag–Kastler Net · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet`
-§10.6.4 Einstein Causality (p. 308) — 1 item +§10.6.4 Einstein Causality (p. 309) — 1 item -- [**Theorem 509**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:einstein-causality) — Einstein Causality in a Representation · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.einstein_causality` (+1 more) +- [**Theorem 518**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:einstein-causality) — Einstein Causality in a Representation · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.einstein_causality` (+1 more)
-§10.6.5 Local von Neumann Algebras (p. 309) — 13 items - -- [**Definition 510**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann) — Local von Neumann Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localOperators` (+1 more) -- [**Lemma 511**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bicommutant-of-selfadjoint-is-von-neumann) — The Bicommutant of a Self-Adjoint Set is a von Neumann Algebra · `Physicslib4.GNS.vonNeumannOfSelfAdjoint` -- [**Definition 512**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann-algebra) — $$R(\mathbf{B})$$ as a von Neumann Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra` -- [**Theorem 513**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-microcausality) — Microcausality at the von Neumann Level · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_subset_centralizer` -- [**Theorem 514**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-isotony) — Isotony of the von Neumann Net · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_mono` -- [**Theorem 515**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-bundled-order) — Bundled von Neumann Microcausality and Isotony · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra_le_commutant` (+1 more) -- [**Definition 516**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:von-neumann-net) — The Net of von Neumann Algebras · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.vonNeumannNet` -- [**Theorem 517**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence) — Statistical Independence (Schlieder Property) · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_separating` (+1 more) -- [**Theorem 518**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence-bundled) — Statistical Independence, bundled · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra_separating` -- [**Theorem 519**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:additive-free-locality) — Additive-Free Locality via the Spacelike Complement · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra_le_commutant_of_subset_spacelikeComplement` -- [**Theorem 520**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-geometric-covariance) — Geometric Covariance of the von Neumann Net · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.lieConj_image_localVonNeumann` (+1 more) -- [**Theorem 521**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-factor-orbit) — Orbit-Invariance of Factoriality · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_isFactor_smul` -- [**Theorem 522**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-covariance-iso) — Geometric Covariance as a von Neumann Algebra Isomorphism · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannEquiv` +§10.6.5 Local von Neumann Algebras (p. 310) — 13 items + +- [**Definition 519**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann) — Local von Neumann Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localOperators` (+1 more) +- [**Lemma 520**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:bicommutant-of-selfadjoint-is-von-neumann) — The Bicommutant of a Self-Adjoint Set is a von Neumann Algebra · `Physicslib4.GNS.vonNeumannOfSelfAdjoint` +- [**Definition 521**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann-algebra) — $$R(\mathbf{B})$$ as a von Neumann Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra` +- [**Theorem 522**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-microcausality) — Microcausality at the von Neumann Level · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_subset_centralizer` +- [**Theorem 523**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-isotony) — Isotony of the von Neumann Net · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_mono` +- [**Theorem 524**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-bundled-order) — Bundled von Neumann Microcausality and Isotony · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra_le_commutant` (+1 more) +- [**Definition 525**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:von-neumann-net) — The Net of von Neumann Algebras · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.vonNeumannNet` +- [**Theorem 526**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence) — Statistical Independence (Schlieder Property) · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_separating` (+1 more) +- [**Theorem 527**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence-bundled) — Statistical Independence, bundled · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra_separating` +- [**Theorem 528**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:additive-free-locality) — Additive-Free Locality via the Spacelike Complement · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannAlgebra_le_commutant_of_subset_spacelikeComplement` +- [**Theorem 529**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-geometric-covariance) — Geometric Covariance of the von Neumann Net · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.lieConj_image_localVonNeumann` (+1 more) +- [**Theorem 530**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-factor-orbit) — Orbit-Invariance of Factoriality · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumann_isFactor_smul` +- [**Theorem 531**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-covariance-iso) — Geometric Covariance as a von Neumann Algebra Isomorphism · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.localVonNeumannEquiv`
-§10.6.6 Relative Commutants of Nested Local Algebras (p. 311) — 8 items +§10.6.6 Relative Commutants of Nested Local Algebras (p. 312) — 8 items -- [**Definition 523**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:relative-commutant) — Relative Commutant of a Nested Pair · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.relativeCommutant` -- [**Theorem 524**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-antitone) — Antitonicity of the Commutant · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.commutant_le_commutant_of_le` -- [**Theorem 525**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-le-right) — Relative Commutant Lies in the Larger Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.relativeCommutant_le_right` -- [**Theorem 526**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-coe-commutant) — Relative Commutant Commutes with the Smaller Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.relativeCommutant_coe_subset_commutant` -- [**Theorem 527**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-center) — Relative Commutant Contains the Centre · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.center_le_relativeCommutant` -- [**Definition 528**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:irreducible-inclusion) — Irreducible Inclusion · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsIrreducibleInclusion` -- [**Theorem 529**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-inclusion-factor) — An Irreducible Inclusion has Factor Ambient · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.isFactor_of_isIrreducibleInclusion` -- [**Theorem 530**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:self-inclusion-factor) — Self-Inclusion is Irreducible iff Factor · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.isIrreducibleInclusion_self_iff_isFactor` +- [**Definition 532**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:relative-commutant) — Relative Commutant of a Nested Pair · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.relativeCommutant` +- [**Theorem 533**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-antitone) — Antitonicity of the Commutant · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.commutant_le_commutant_of_le` +- [**Theorem 534**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-le-right) — Relative Commutant Lies in the Larger Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.relativeCommutant_le_right` +- [**Theorem 535**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-coe-commutant) — Relative Commutant Commutes with the Smaller Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.relativeCommutant_coe_subset_commutant` +- [**Theorem 536**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-center) — Relative Commutant Contains the Centre · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.center_le_relativeCommutant` +- [**Definition 537**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:irreducible-inclusion) — Irreducible Inclusion · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsIrreducibleInclusion` +- [**Theorem 538**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-inclusion-factor) — An Irreducible Inclusion has Factor Ambient · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.isFactor_of_isIrreducibleInclusion` +- [**Theorem 539**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:self-inclusion-factor) — Self-Inclusion is Irreducible iff Factor · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.isIrreducibleInclusion_self_iff_isFactor`
-§10.6.7 Irreducibility and Schur's Lemma (p. 312) — 23 items - -- [**Definition 531**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:irreducible-representation) — Irreducible Representation · `Physicslib4.GNS.IsIrreducible` -- [**Theorem 532**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:schur-lemma) — Topological Schur Lemma · `Physicslib4.GNS.eq_smul_one_of_commute_of_cyclic` -- [**Theorem 533**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-scalar-iff) — Commutant Scalar iff Proportional Coefficient · `Physicslib4.GNS.isScalar_iff_coeff_proportional` -- [**Definition 534**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:pure-state) — Pure State · `Physicslib4.GNS.IsPure` -- [**Theorem 535**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-implies-irreducible) — Pure Implies Irreducible · `Physicslib4.GNS.isIrreducible_of_isPure` -- [**Theorem 536**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-form-bound) — The GNS Radon–Nikodym Form is Bounded · `Physicslib4.GNS.gns_form_norm_le` (+1 more) -- [**Theorem 537**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-radon-nikodym-operator) — The GNS Radon–Nikodym Operator · `Physicslib4.GNS.rnOp` (+3 more) -- [**Theorem 538**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-irreducible) — Pure $$\iff$$ Irreducible · `Physicslib4.GNS.isPure_iff_isIrreducible` -- [**Theorem 539**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-factor) — An Irreducible Representation Generates a Factor · `Physicslib4.GNS.center_gnsVonNeumann_eq_of_isIrreducible` -- [**Theorem 540**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-generates-all) — Irreducibility $$\iff$$ Generating $$\mathcal{B}(H)$$ · `Physicslib4.GNS.isIrreducible_iff_gnsVonNeumann_eq_univ` (+1 more) -- [**Theorem 541**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-generates-all-bundled) — Bundled Density Form of Irreducibility · `Physicslib4.GNS.coe_gnsVonNeumannAlgebra_eq_univ_of_isIrreducible` (+1 more) -- [**Theorem 542**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-factor) — The GNS Representation of a Pure State is a Factor · `Physicslib4.GNS.exists_gns_factor_of_isPure` -- [**Theorem 543**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-generates-all) — The GNS Representation of a Pure State Generates $$\mathcal{B}(H)$$ · `Physicslib4.GNS.exists_gns_generates_all_of_isPure` -- [**Theorem 544**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:norm-positive-functional) — Norm of a Positive Functional · `Physicslib4.GNS.norm_eq_re_apply_one_of_positive` -- [**Definition 545**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:extreme-state) — Extreme Point of the State Space · `Physicslib4.GNS.State.IsExtremePoint` -- [**Theorem 546**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-extreme) — Pure $$\iff$$ Extreme Point · `Physicslib4.GNS.isPure_iff_isExtremePoint` -- [**Theorem 547**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:state-space-convex-bridge) — State-Space Convexity and the Extreme-Point Bridge · `Physicslib4.GNS.convex_stateSpace` (+1 more) -- [**Definition 548**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:state-pullback) — Pullback of a State · `Physicslib4.GNS.State.comp` -- [**Theorem 549**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:state-pullback-functorial) — Functoriality of the State Pullback · `Physicslib4.GNS.State.comp_id` (+1 more) -- [**Theorem 550**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-pullback-invariant) — Purity is Invariant under a \*-Isomorphism · `Physicslib4.GNS.isPure_comp_iff` -- [**Theorem 551**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:state-space-weak-compact) — Weak-\* Compactness of the State Space · `Physicslib4.GNS.isCompact_weakStateSet` -- [**Theorem 552**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-extreme-quasilocal) — Pure $$\iff$$ Extreme Point on the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.pure_iff_extreme` -- [**Theorem 553**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-irreducible-quasilocal) — Pure $$\iff$$ Irreducible GNS on the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.exists_gns_pure_iff_irreducible` +§10.6.7 Irreducibility and Schur's Lemma (p. 313) — 23 items + +- [**Definition 540**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:irreducible-representation) — Irreducible Representation · `Physicslib4.GNS.IsIrreducible` +- [**Theorem 541**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:schur-lemma) — Topological Schur Lemma · `Physicslib4.GNS.eq_smul_one_of_commute_of_cyclic` +- [**Theorem 542**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-scalar-iff) — Commutant Scalar iff Proportional Coefficient · `Physicslib4.GNS.isScalar_iff_coeff_proportional` +- [**Definition 543**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:pure-state) — Pure State · `Physicslib4.GNS.IsPure` +- [**Theorem 544**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-implies-irreducible) — Pure Implies Irreducible · `Physicslib4.GNS.isIrreducible_of_isPure` +- [**Theorem 545**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-form-bound) — The GNS Radon–Nikodym Form is Bounded · `Physicslib4.GNS.gns_form_norm_le` (+1 more) +- [**Theorem 546**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-radon-nikodym-operator) — The GNS Radon–Nikodym Operator · `Physicslib4.GNS.rnOp` (+3 more) +- [**Theorem 547**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-irreducible) — Pure $$\iff$$ Irreducible · `Physicslib4.GNS.isPure_iff_isIrreducible` +- [**Theorem 548**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-factor) — An Irreducible Representation Generates a Factor · `Physicslib4.GNS.center_gnsVonNeumann_eq_of_isIrreducible` (+1 more) +- [**Theorem 549**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-generates-all) — Irreducibility $$\iff$$ Generating $$\mathcal{B}(H)$$ · `Physicslib4.GNS.isIrreducible_iff_gnsVonNeumann_eq_univ` (+1 more) +- [**Theorem 550**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-generates-all-bundled) — Bundled Density Form of Irreducibility · `Physicslib4.GNS.coe_gnsVonNeumannAlgebra_eq_univ_of_isIrreducible` (+1 more) +- [**Theorem 551**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-factor) — The GNS Representation of a Pure State is a Factor · `Physicslib4.GNS.exists_gns_factor_of_isPure` +- [**Theorem 552**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-generates-all) — The GNS Representation of a Pure State Generates $$\mathcal{B}(H)$$ · `Physicslib4.GNS.exists_gns_generates_all_of_isPure` +- [**Theorem 553**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:norm-positive-functional) — Norm of a Positive Functional · `Physicslib4.GNS.norm_eq_re_apply_one_of_positive` +- [**Definition 554**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:extreme-state) — Extreme Point of the State Space · `Physicslib4.GNS.State.IsExtremePoint` +- [**Theorem 555**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-extreme) — Pure $$\iff$$ Extreme Point · `Physicslib4.GNS.isPure_iff_isExtremePoint` +- [**Theorem 556**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:state-space-convex-bridge) — State-Space Convexity and the Extreme-Point Bridge · `Physicslib4.GNS.convex_stateSpace` (+1 more) +- [**Definition 557**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:state-pullback) — Pullback of a State · `Physicslib4.GNS.State.comp` +- [**Theorem 558**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:state-pullback-functorial) — Functoriality of the State Pullback · `Physicslib4.GNS.State.comp_id` (+1 more) +- [**Theorem 559**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-pullback-invariant) — Purity is Invariant under a \*-Isomorphism · `Physicslib4.GNS.isPure_comp_iff` +- [**Theorem 560**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:state-space-weak-compact) — Weak-\* Compactness of the State Space · `Physicslib4.GNS.isCompact_weakStateSet` +- [**Theorem 561**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-extreme-quasilocal) — Pure $$\iff$$ Extreme Point on the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.pure_iff_extreme` +- [**Theorem 562**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-irreducible-quasilocal) — Pure $$\iff$$ Irreducible GNS on the Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.exists_gns_pure_iff_irreducible`
-§10.6.8 Unitary Equivalence and Superselection (p. 316) — 2 items +§10.6.8 Unitary Equivalence and Superselection (p. 317) — 2 items -- [**Definition 554**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:unitary-equivalence) — Unitary Equivalence of Representations · `Physicslib4.GNS.UnitaryEquiv` -- [**Theorem 555**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:unitary-equiv-invariants) — Irreducibility and Factoriality are Unitary Invariants · `Physicslib4.GNS.UnitaryEquiv.isIrreducible_iff` (+1 more) +- [**Definition 563**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:unitary-equivalence) — Unitary Equivalence of Representations · `Physicslib4.GNS.UnitaryEquiv` +- [**Theorem 564**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:unitary-equiv-invariants) — Irreducibility and Factoriality are Unitary Invariants · `Physicslib4.GNS.UnitaryEquiv.isIrreducible_iff` (+1 more)
-§10.6.9 GNS Covariance (p. 316) — 6 items +§10.6.9 GNS Covariance (p. 317) — 6 items -- [**Lemma 556**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cyclic-pullback-surjective) — Cyclicity pulls back along a surjective \*-homomorphism · `Physicslib4.GNS.isCyclicVector_comp_of_surjective` -- [**Theorem 557**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance) — GNS covariance under a \*-isomorphism · `Physicslib4.GNS.exists_unitary_of_gns_comp` -- [**Theorem 558**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-unitary-equiv) — The GNS representation of a pullback state · `Physicslib4.GNS.unitaryEquiv_comp_of_gns` -- [**Lemma 559**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-image-invariants) — Pullback along a surjection preserves the image algebra · `Physicslib4.GNS.range_comp_of_surjective` (+2 more) -- [**Theorem 560**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-sector-transport) — Superselection type transports along a \*-isomorphism · `Physicslib4.GNS.isIrreducible_iff_of_gns_comp` (+1 more) -- [**Theorem 561**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-local) — GNS covariance for local algebras · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.unitaryEquiv_gns_covEquiv` (+2 more) +- [**Lemma 565**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:cyclic-pullback-surjective) — Cyclicity pulls back along a surjective \*-homomorphism · `Physicslib4.GNS.isCyclicVector_comp_of_surjective` +- [**Theorem 566**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance) — GNS covariance under a \*-isomorphism · `Physicslib4.GNS.exists_unitary_of_gns_comp` +- [**Theorem 567**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-unitary-equiv) — The GNS representation of a pullback state · `Physicslib4.GNS.unitaryEquiv_comp_of_gns` +- [**Lemma 568**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:pullback-image-invariants) — Pullback along a surjection preserves the image algebra · `Physicslib4.GNS.range_comp_of_surjective` (+2 more) +- [**Theorem 569**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-sector-transport) — Superselection type transports along a \*-isomorphism · `Physicslib4.GNS.isIrreducible_iff_of_gns_comp` (+1 more) +- [**Theorem 570**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-local) — GNS covariance for local algebras · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.unitaryEquiv_gns_covEquiv` (+2 more)
-§10.6.10 Disjointness and Quasi-Equivalence (p. 318) — 18 items - -- [**Definition 562**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:disjoint-representations) — Disjoint Representations · `Physicslib4.GNS.AreDisjoint` -- [**Definition 563**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasi-equivalence) — Quasi-Equivalence of Representations · `Physicslib4.GNS.QuasiEquiv` -- [**Theorem 564**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-dichotomy) — Schur's Lemma and the Irreducible Dichotomy · `Physicslib4.GNS.UnitaryEquiv.of_intertwines_of_isIrreducible` (+1 more) -- [**Lemma 565**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:schur-multiplicity) — Schur Multiplicity · `Physicslib4.GNS.eq_smul_of_intertwines_of_isIrreducible` -- [**Lemma 566**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:endomorphism-scalar) — Endomorphism Algebra of an Irreducible Representation · `Physicslib4.GNS.intertwines_self_iff_isScalar` -- [**Theorem 567**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-von-neumann) — The commutant (self-intertwiner) von Neumann algebra · `Physicslib4.GNS.commutantVonNeumann` (+2 more) -- [**Theorem 568**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:double-commutant-duality) — Double-Commutant Duality · `Physicslib4.GNS.commutant_gnsVonNeumannAlgebra` (+1 more) -- [**Theorem 569**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-factor-duality) — A Factor and its Commutant; Triviality Duality · `Physicslib4.GNS.isFactor_gnsVonNeumann_iff_isFactor_commutant` (+1 more) -- [**Lemma 570**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-abelian-self-commuting) — Abelian $$\iff$$ Self-Commuting · `Physicslib4.GNS.isAbelian_iff_le_commutant` -- [**Definition 571**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:von-neumann-center) — Centre of a von Neumann algebra · `Physicslib4.GNS.vonNeumannCenter` -- [**Lemma 572**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-inter-is-von-neumann) — The centre is a von Neumann algebra · `Physicslib4.GNS.bicommutant_inter_commutant_eq` -- [**Lemma 573**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-inter-general) — The intersection of two von Neumann algebras is a von Neumann algebra · `Physicslib4.GNS.bicommutant_inter_eq` -- [**Theorem 574**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-center-abelian) — The centre of a von Neumann algebra is abelian · `Physicslib4.GNS.vonNeumannCenter_isAbelian` -- [**Lemma 575**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-center-eq-self-iff-abelian) — $$R$$ is abelian iff it equals its centre · `Physicslib4.GNS.vonNeumannCenter_eq_self_iff_isAbelian` -- [**Theorem 576**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:factor-abelian-iff-scalars) — A factor is abelian iff it is the scalars · `Physicslib4.GNS.isAbelian_iff_eq_scalars_of_isFactor` -- [**Theorem 577**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:center-eq-commutant-center) — A von Neumann algebra and its commutant share a centre · `Physicslib4.GNS.vonNeumannCenter_eq_commutant` -- [**Theorem 578**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:factor-iff-center-scalars) — A von Neumann algebra is a factor iff its centre is the scalars · `Physicslib4.GNS.isFactor_iff_center_eq_scalars` -- [**Theorem 579**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-state-dichotomy) — The Pure-State Dichotomy · `Physicslib4.GNS.exists_gns_areDisjoint_or_unitaryEquiv_of_isPure` +§10.6.10 Disjointness and Quasi-Equivalence (p. 319) — 18 items + +- [**Definition 571**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:disjoint-representations) — Disjoint Representations · `Physicslib4.GNS.AreDisjoint` +- [**Definition 572**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasi-equivalence) — Quasi-Equivalence of Representations · `Physicslib4.GNS.QuasiEquiv` +- [**Theorem 573**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-dichotomy) — Schur's Lemma and the Irreducible Dichotomy · `Physicslib4.GNS.UnitaryEquiv.of_intertwines_of_isIrreducible` (+1 more) +- [**Lemma 574**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:schur-multiplicity) — Schur Multiplicity · `Physicslib4.GNS.eq_smul_of_intertwines_of_isIrreducible` +- [**Lemma 575**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:endomorphism-scalar) — Endomorphism Algebra of an Irreducible Representation · `Physicslib4.GNS.intertwines_self_iff_isScalar` +- [**Theorem 576**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-von-neumann) — The commutant (self-intertwiner) von Neumann algebra · `Physicslib4.GNS.commutantVonNeumann` (+2 more) +- [**Theorem 577**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:double-commutant-duality) — Double-Commutant Duality · `Physicslib4.GNS.commutant_gnsVonNeumannAlgebra` (+1 more) +- [**Theorem 578**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:commutant-factor-duality) — A Factor and its Commutant; Triviality Duality · `Physicslib4.GNS.isFactor_gnsVonNeumann_iff_isFactor_commutant` (+1 more) +- [**Lemma 579**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-abelian-self-commuting) — Abelian $$\iff$$ Self-Commuting · `Physicslib4.GNS.isAbelian_iff_le_commutant` +- [**Definition 580**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:von-neumann-center) — Centre of a von Neumann algebra · `Physicslib4.GNS.vonNeumannCenter` +- [**Lemma 581**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-inter-is-von-neumann) — The centre is a von Neumann algebra · `Physicslib4.GNS.bicommutant_inter_commutant_eq` +- [**Lemma 582**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-inter-general) — The intersection of two von Neumann algebras is a von Neumann algebra · `Physicslib4.GNS.bicommutant_inter_eq` +- [**Theorem 583**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-center-abelian) — The centre of a von Neumann algebra is abelian · `Physicslib4.GNS.vonNeumannCenter_isAbelian` +- [**Lemma 584**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:von-neumann-center-eq-self-iff-abelian) — $$R$$ is abelian iff it equals its centre · `Physicslib4.GNS.vonNeumannCenter_eq_self_iff_isAbelian` +- [**Theorem 585**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:factor-abelian-iff-scalars) — A factor is abelian iff it is the scalars · `Physicslib4.GNS.isAbelian_iff_eq_scalars_of_isFactor` +- [**Theorem 586**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:center-eq-commutant-center) — A von Neumann algebra and its commutant share a centre · `Physicslib4.GNS.vonNeumannCenter_eq_commutant` +- [**Theorem 587**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:factor-iff-center-scalars) — A von Neumann algebra is a factor iff its centre is the scalars · `Physicslib4.GNS.isFactor_iff_center_eq_scalars` +- [**Theorem 588**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-state-dichotomy) — The Pure-State Dichotomy · `Physicslib4.GNS.exists_gns_areDisjoint_or_unitaryEquiv_of_isPure`
-§10.6.11 Direct Sums, Amplification, and Reducibility (p. 321) — 4 items +§10.6.11 Direct Sums, Amplification, and Reducibility (p. 322) — 4 items -- [**Definition 580**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:direct-sum-representation) — Direct-Sum Representation · `Physicslib4.GNS.directSum` -- [**Theorem 581**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:direct-sum-subrepresentation) — Subrepresentations and Commutant of a Direct Sum · `Physicslib4.GNS.intertwines_single` (+1 more) -- [**Definition 582**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:amplification) — Amplification · `Physicslib4.GNS.amplification` -- [**Theorem 583**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:direct-sum-reducible) — Reducibility of a Direct Sum · `Physicslib4.GNS.not_isIrreducible_directSum` +- [**Definition 589**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:direct-sum-representation) — Direct-Sum Representation · `Physicslib4.GNS.directSum` +- [**Theorem 590**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:direct-sum-subrepresentation) — Subrepresentations and Commutant of a Direct Sum · `Physicslib4.GNS.intertwines_single` (+1 more) +- [**Definition 591**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:amplification) — Amplification · `Physicslib4.GNS.amplification` +- [**Theorem 592**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:direct-sum-reducible) — Reducibility of a Direct Sum · `Physicslib4.GNS.not_isIrreducible_directSum`
-§10.6.12 Covariant States and the Covariance Action (p. 321) — 27 items - -- [**Definition 584**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:covariant-state-family) — Covariant Family of Local States · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsCovariantFamily` -- [**Lemma 585**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:covariant-state-family-compose) — Composition of Covariance · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsCovariantFamily.comp` -- [**Definition 586**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-lift) — Quasilocal Covariance Automorphism · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.QuasilocalLift` -- [**Lemma 587**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-lift-unique) — Uniqueness of the Quasilocal Lift · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.QuasilocalLift.unique` -- [**Lemma 588**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-covariance-compatible) — Every Quasilocal Algebra is Covariance-Compatible · `Physicslib4.AQFT.HaagKastler.isCovariantQuasilocal` -- [**Theorem 589**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-lift-exists) — Existence of the Quasilocal Lift · `Physicslib4.AQFT.HaagKastler.nonempty_quasilocalLift` -- [**Theorem 590**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-lift-trivial) — Existence for the Trivial Net · `Physicslib4.AQFT.HaagKastler.nonempty_trivialQuasilocalLift` -- [**Definition 591**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:covariant-quasilocal-algebra) — Covariant Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.action` -- [**Lemma 592**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-action-coherence) — Group-Action Coherence of the Covariance Automorphism · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.action_one` (+1 more) -- [**Definition 593**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:invariant-state) — Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState` -- [**Theorem 594**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:invariant-state-gns-unitary) — GNS Unitary Implementation of an Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_unitary` -- [**Theorem 595**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-covariant-representation) — Irreducible Covariant Representation of a Pure Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_irreducible_covariant` -- [**Definition 596**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:positive-energy) — Positive Energy · `Physicslib4.AQFT.IsPositiveEnergy` -- [**Lemma 597**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-const) — The Trivial Group has Positive Energy · `Physicslib4.AQFT.isPositiveEnergy_const_refl` -- [**Lemma 598**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:exp-generator-unique) — Uniqueness of the Generator · `Physicslib4.AQFT.generator_eq_of_eq_expUnitary` (+1 more) -- [**Lemma 599**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-conj) — Unitary Conjugation Preserves Positivity · `Physicslib4.Spectral.Unbounded.IsPositive.of_conj` -- [**Lemma 600**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-conj) — Positive Energy is Unitarily Invariant · `Physicslib4.AQFT.IsPositiveEnergy.conj` -- [**Lemma 601**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-strong-continuous) — A Positive-Energy Group is Strongly Continuous · `Physicslib4.AQFT.IsPositiveEnergy.strongContinuous` -- [**Lemma 602**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-exp-bounded) — The Bounded Case · `Physicslib4.AQFT.isPositiveEnergy_exp` -- [**Theorem 603**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:positive-energy-iff-exp) — Positive Energy in Exponential Form · `Physicslib4.AQFT.isPositiveEnergy_iff_exists_exp` -- [**Theorem 604**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:positive-energy-api) — Properties of Positive Energy · `Physicslib4.AQFT.isPositiveEnergy_const_refl` (+5 more) -- [**Definition 605**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:vacuum-state) — Vacuum State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumState` -- [**Theorem 606**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:vacuum-invariance-consequences) — Consequences of Invariance for a Vacuum State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumState.invariant` (+1 more) -- [**Definition 607**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:future-timelike-translation) — Future-Timelike Translation Subgroup · `Physicslib4.AQFT.HaagKastler.translationSub` (+3 more) -- [**Definition 608**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:vacuum-state-concrete) — Vacuum State with the Concrete Spectrum Condition · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumStateConcrete` -- [**Theorem 609**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:purity-covariance-invariant) — Purity is Covariance-Invariant · `Physicslib4.GNS.isPure_precomp_iff` (+1 more) -- [**Theorem 610**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-quasilocal-action) — GNS covariance along the quasilocal action · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.unitaryEquiv_gns_action` (+2 more) +§10.6.12 Covariant States and the Covariance Action (p. 322) — 29 items + +- [**Definition 593**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:covariant-state-family) — Covariant Family of Local States · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsCovariantFamily` +- [**Lemma 594**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:covariant-state-family-compose) — Composition of Covariance · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsCovariantFamily.comp` +- [**Definition 595**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:quasilocal-lift) — Quasilocal Covariance Automorphism · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.QuasilocalLift` +- [**Lemma 596**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-lift-unique) — Uniqueness of the Quasilocal Lift · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.QuasilocalLift.unique` +- [**Lemma 597**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-covariance-compatible) — Every Quasilocal Algebra is Covariance-Compatible · `Physicslib4.AQFT.HaagKastler.isCovariantQuasilocal` +- [**Theorem 598**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-lift-exists) — Existence of the Quasilocal Lift · `Physicslib4.AQFT.HaagKastler.nonempty_quasilocalLift` +- [**Theorem 599**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:quasilocal-lift-trivial) — Existence for the Trivial Net · `Physicslib4.AQFT.HaagKastler.nonempty_trivialQuasilocalLift` +- [**Definition 600**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:covariant-quasilocal-algebra) — Covariant Quasilocal Algebra · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.action` +- [**Lemma 601**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:quasilocal-action-coherence) — Group-Action Coherence of the Covariance Automorphism · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.action_one` (+1 more) +- [**Definition 602**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:invariant-state) — Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState` +- [**Theorem 603**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:invariant-state-gns-unitary) — GNS Unitary Implementation of an Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_unitary` (+1 more) +- [**Lemma 604**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:weak-continuity-implies-strong) — Weak Continuity Implies Strong Continuity · `Physicslib4.strongContinuous_of_weak` +- [**Theorem 605**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:invariant-state-gns-unitary-strongly-continuous) — Strongly Continuous GNS Unitary of an Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_unitary_strongContinuous` (+1 more) +- [**Theorem 606**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-covariant-representation) — Irreducible Covariant Representation of a Pure Invariant State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsInvariantState.exists_gns_irreducible_covariant` +- [**Definition 607**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:positive-energy) — Positive Energy · `Physicslib4.AQFT.IsPositiveEnergy` +- [**Lemma 608**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-const) — The Trivial Group has Positive Energy · `Physicslib4.AQFT.isPositiveEnergy_const_refl` +- [**Lemma 609**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:exp-generator-unique) — Uniqueness of the Generator · `Physicslib4.AQFT.generator_eq_of_eq_expUnitary` +- [**Lemma 610**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-conj) — Unitary Conjugation Preserves Positivity · `Physicslib4.Spectral.Unbounded.IsPositive.of_conj` +- [**Lemma 611**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-conj) — Positive Energy is Unitarily Invariant · `Physicslib4.AQFT.IsPositiveEnergy.conj` +- [**Lemma 612**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-strong-continuous) — A Positive-Energy Group is Strongly Continuous · `Physicslib4.AQFT.IsPositiveEnergy.strongContinuous` +- [**Lemma 613**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:positive-energy-exp-bounded) — The Bounded Case · `Physicslib4.AQFT.isPositiveEnergy_exp` +- [**Theorem 614**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:positive-energy-iff-exp) — Positive Energy in Exponential Form · `Physicslib4.AQFT.isPositiveEnergy_iff_exists_exp` +- [**Theorem 615**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:positive-energy-api) — Properties of Positive Energy · `Physicslib4.AQFT.isPositiveEnergy_const_refl` (+5 more) +- [**Definition 616**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:vacuum-state) — Vacuum State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumState` +- [**Theorem 617**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:vacuum-invariance-consequences) — Consequences of Invariance for a Vacuum State · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumState.invariant` (+1 more) +- [**Definition 618**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:future-timelike-translation) — Future-Timelike Translation Subgroup · `Physicslib4.AQFT.HaagKastler.translationSub` (+3 more) +- [**Definition 619**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:vacuum-state-concrete) — Vacuum State with the Concrete Spectrum Condition · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsVacuumStateConcrete` +- [**Theorem 620**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:purity-covariance-invariant) — Purity is Covariance-Invariant · `Physicslib4.GNS.isPure_precomp_iff` (+1 more) +- [**Theorem 621**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-quasilocal-action) — GNS covariance along the quasilocal action · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.unitaryEquiv_gns_action` (+2 more)
-§10.6.13 The Separating Vector of a Faithful State (p. 326) — 1 item +§10.6.13 The Separating Vector of a Faithful State (p. 328) — 1 item -- [**Theorem 611**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:separating-faithful) — Separating Vector of a Faithful State · `Physicslib4.GNS.separating_of_faithful` (+1 more) +- [**Theorem 622**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:separating-faithful) — Separating Vector of a Faithful State · `Physicslib4.GNS.separating_of_faithful` (+1 more)
-§10.6.14 The KMS Condition and Thermal Equilibrium (p. 326) — 10 items - -- [**Definition 612**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:one-parameter-aut) — One-Parameter Automorphism Group · `Physicslib4.AQFT.IsOneParameterAut` -- [**Definition 613**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:kms-state) — KMS State · `Physicslib4.AQFT.IsKMSState` -- [**Theorem 614**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-convex) — The KMS State Set is Convex · `Physicslib4.AQFT.IsKMSState.convexCombo` -- [**Lemma 615**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:kms-correlation-one) — Boundary Coincidence for $$a = 1$$ · `Physicslib4.AQFT.IsKMSState.correlationOne` -- [**Definition 616**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:strip-liouville) — Strip-Liouville Principle · `Physicslib4.AQFT.StripLiouville` -- [**Theorem 617**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:strip-periodic-extension) — $$i\beta$$-Periodic Entire Extension (Strip Schwarz Reflection) · `Physicslib4.exists_bounded_entire_extension_of_strip_periodic` -- [**Theorem 618**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:strip-liouville-pos) — Strip-Liouville Holds for $$\beta > 0$$ · `Physicslib4.AQFT.stripLiouville_of_pos` -- [**Theorem 619**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-invariance) — KMS States are Invariant · `Physicslib4.AQFT.IsKMSState.invariant_of_pos` -- [**Theorem 620**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:strip-uniqueness) — Uniqueness on the Strip from Boundary Values · `Physicslib4.eqOn_strip_of_eq_boundary` -- [**Theorem 621**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-correlation-unique) — Uniqueness of the KMS Correlation Function · `Physicslib4.AQFT.IsKMSState.correlation_eqOn` +§10.6.14 The KMS Condition and Thermal Equilibrium (p. 328) — 10 items + +- [**Definition 623**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:one-parameter-aut) — One-Parameter Automorphism Group · `Physicslib4.AQFT.IsOneParameterAut` +- [**Definition 624**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:kms-state) — KMS State · `Physicslib4.AQFT.IsKMSState` +- [**Theorem 625**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-convex) — The KMS State Set is Convex · `Physicslib4.AQFT.IsKMSState.convexCombo` +- [**Lemma 626**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:kms-correlation-one) — Boundary Coincidence for $$a = 1$$ · `Physicslib4.AQFT.IsKMSState.correlationOne` +- [**Definition 627**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:strip-liouville) — Strip-Liouville Principle · `Physicslib4.AQFT.StripLiouville` +- [**Theorem 628**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:strip-periodic-extension) — $$i\beta$$-Periodic Entire Extension (Strip Schwarz Reflection) · `Physicslib4.exists_bounded_entire_extension_of_strip_periodic` +- [**Theorem 629**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:strip-liouville-pos) — Strip-Liouville Holds for $$\beta > 0$$ · `Physicslib4.AQFT.stripLiouville_of_pos` +- [**Theorem 630**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-invariance) — KMS States are Invariant · `Physicslib4.AQFT.IsKMSState.invariant_of_pos` +- [**Theorem 631**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:strip-uniqueness) — Uniqueness on the Strip from Boundary Values · `Physicslib4.eqOn_strip_of_eq_boundary` +- [**Theorem 632**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-correlation-unique) — Uniqueness of the KMS Correlation Function · `Physicslib4.AQFT.IsKMSState.correlation_eqOn`
-§10.6.15 KMS States for the Covariance Flow (p. 328) — 5 items +§10.6.15 KMS States for the Covariance Flow (p. 329) — 5 items -- [**Definition 622**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:flow-aut-covariance) — Covariance-Flow Automorphism Family · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.flowAut` -- [**Lemma 623**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:one-parameter-aut-flow-covariance) — A Lorentz One-Parameter Subgroup Induces a One-Parameter Group · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.isOneParameterAut_flowAut` -- [**Definition 624**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:kms-state-for-flow-covariance) — KMS State for the Covariance Flow · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsKMSStateForFlow` -- [**Theorem 625**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-convex-for-flow-covariance) — Convexity of the Covariance-Flow KMS States · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsKMSStateForFlow.convexCombo` -- [**Definition 626**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:ground-state-for-flow-covariance) — Ground State for a Covariance Flow · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsGroundStateForFlow` (+2 more) +- [**Definition 633**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:flow-aut-covariance) — Covariance-Flow Automorphism Family · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.flowAut` +- [**Lemma 634**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:one-parameter-aut-flow-covariance) — A Lorentz One-Parameter Subgroup Induces a One-Parameter Group · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.isOneParameterAut_flowAut` +- [**Definition 635**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:kms-state-for-flow-covariance) — KMS State for the Covariance Flow · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsKMSStateForFlow` +- [**Theorem 636**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-convex-for-flow-covariance) — Convexity of the Covariance-Flow KMS States · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsKMSStateForFlow.convexCombo` +- [**Definition 637**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:ground-state-for-flow-covariance) — Ground State for a Covariance Flow · `Physicslib4.AQFT.HaagKastler.HaagKastlerNet.IsGroundStateForFlow` (+2 more)
diff --git a/home_page/sec10-7-curved.md b/home_page/sec10-7-curved.md index a6e7627..8862ca1 100644 --- a/home_page/sec10-7-curved.md +++ b/home_page/sec10-7-curved.md @@ -1,5 +1,5 @@ --- -title: "§10.7 Haag–Kastler Axioms in curved spacetime (pp. 328–338)" +title: "§10.7 Haag–Kastler Axioms in curved spacetime (pp. 330–340)" usemathjax: true --- @@ -7,17 +7,17 @@ usemathjax: true ← [§10.6 Minkowski](sec10-6-haag-kastler.html) · [§10.8 Covariance](sec10-8-general-covariance.html) → -# §10.7 Haag–Kastler Axioms in curved spacetime (pp. 328–338) +# §10.7 Haag–Kastler Axioms in curved spacetime (pp. 330–340) -49 items, Definitions 627–675. The axioms are restated for a Lorentzian spacetime: Local Algebras (Definition 627), Isotony (Definition 628, again supplying the family as chosen data with identity and composition laws), Local Commutativity (Definition 629, which uses the Axiom 2 family rather than choosing its own witnesses), local observables (Definition 630), Local Completeness (Definition 631), and Isometric Covariance (Definition 632, whose coherence condition, as in Minkowski, is stated for the Axiom 2 isotony family), bundled into a `HaagKastlerNet` in curved spacetime (Definition 633). This form of Axioms 2 and 3 has a visible consequence downstream: statements about nested regions have to factor a three-fold inclusion $$\mathbf{B}_1 \subseteq \mathbf{B}_2 \subseteq \mathbf{B}$$ inside a common containing algebra, and that factorisation is the composition law of Axiom 2, so **no coherence hypothesis is needed at any such site**—curved isotony (Theorem 638) and the curved von Neumann net (Definition 640) are unconditional. An explicit hypothesis remains only where the abstract interface genuinely cannot supply it, namely basis-set preservation in Geometric Covariance (Theorem 644), discharged for nets arising from a concrete geometric spacetime; the coherence of the stabiliser action with the isotony embeddings is condition (3) of Axiom 5 transported along $$g \cdot \mathbf{B} = \mathbf{B}$$. +51 items, Definitions 638–688. The axioms are restated for a Lorentzian spacetime: Local Algebras (Definition 638), Isotony (Definition 639, again supplying the family as chosen data with identity and composition laws), Local Commutativity (Definition 640, which uses the Axiom 2 family rather than choosing its own witnesses), local observables (Definition 641), Local Completeness (Definition 642), and Isometric Covariance (Definition 643, whose coherence condition, as in Minkowski, is stated for the Axiom 2 isotony family), bundled into a `HaagKastlerNet` in curved spacetime (Definition 644), and the axioms are jointly satisfiable (Theorem 645). This form of Axioms 2 and 3 has a visible consequence downstream: statements about nested regions have to factor a three-fold inclusion $$\mathbf{B}_1 \subseteq \mathbf{B}_2 \subseteq \mathbf{B}$$ inside a common containing algebra, and that factorisation is the composition law of Axiom 2, so **no coherence hypothesis is needed at any such site**—curved isotony (Theorem 651) and the curved von Neumann net (Definition 653) are unconditional. An explicit hypothesis remains only where the abstract interface genuinely cannot supply it, namely basis-set preservation in Geometric Covariance (Theorem 657), discharged for nets arising from a concrete geometric spacetime; the coherence of the stabiliser action with the isotony embeddings is condition (3) of Axiom 5 transported along $$g \cdot \mathbf{B} = \mathbf{B}$$. -- *§10.7.1 Einstein Causality in Curved Spacetime (p. 331, item 634).* The operator form of local commutativity, expressed in a representation of a common containing local algebra rather than of a global quasilocal algebra (Theorem 634). -- *§10.7.2 Local von Neumann Algebras in Curved Spacetime (p. 331, items 635–646).* The Minkowski development of §10.6.5 mirrored relative to a containing region: the local von Neumann algebra of a subregion and its bundled registration (Definitions 635–636), Microcausality (Theorem 637), unconditional Isotony (Theorem 638), the bundled form (Theorem 639), the net as an order-preserving map on the poset of subregions (Definition 640), Statistical Independence in set-level and bundled forms (Theorems 641–642), additive-free locality (Theorem 643), and Geometric Covariance via the stabiliser GNS representation (Theorem 644) with orbit-invariance of factoriality (Theorem 645) and the upgrade to a \*-algebra isomorphism (Theorem 646). -- *§10.7.3 Relative Commutants of Nested Local Algebras in Curved Spacetime (p. 333, items 647–655).* The curved counterpart of §10.6.6: the relative commutant of a nested pair inside a containing region (Definition 647), its containment in the larger algebra, commutation with the smaller, and containment of the centre (Theorems 648–650), irreducible inclusions (Definition 651) and their consequences (Theorems 652–653), plus the abelian/centre facts specialised to curved local algebras (Theorems 654–655). -- *§10.7.4 Purity of States on Local Algebras in Curved Spacetime (p. 334, items 656–660).* Each local algebra is a unital C\*-algebra with its own state space, so the abstract purity characterisations are registered per region: Pure $$\iff$$ Extreme Point (Theorem 656), Pure $$\iff$$ Irreducible GNS (Theorem 657), the GNS representation of a pure state generates a factor and indeed all of $$\mathcal{B}(H)$$ (Theorem 658), the Irreducible Dichotomy for a curved local algebra (Theorem 659), and GNS covariance for curved local algebras (Theorem 660). -- *§10.7.5 Covariant States in Curved Spacetime (p. 335, items 661–662).* Covariant families of local states in curved spacetime (Definition 661) and their composition law (Lemma 662). -- *§10.7.6 The Stabilizer GNS Unitary in Curved Spacetime (p. 335, items 663–669).* Since no quasilocal algebra exists on a generic Lorentzian spacetime, the covariance action restricts to the stabiliser subgroup $$\mathrm{Stab}(\mathbf{B})$$ of a region, giving automorphisms of the single local algebra $$\mathfrak{U}(\mathbf{B})$$ (Definition 663) that form a genuine group action (Lemma 664). A stabiliser-invariant state carries a unitary GNS representation of $$\mathrm{Stab}(\mathbf{B})$$ (Theorem 665), strongly continuous when the matrix coefficients are continuous (Theorem 666); a state both stabiliser-invariant and pure yields an irreducible covariant representation (Theorem 667); purity is invariant under the stabiliser action (Theorem 668), and GNS data transports along it (Theorem 669). -- *§10.7.7 KMS States for a Killing Flow (p. 337, items 670–675).* Killing flows are identified as one-parameter subgroups of the stabiliser of a region, inducing a one-parameter automorphism group on $$\mathfrak{U}(\mathbf{B})$$ (Definition 670, Lemma 671). KMS states for a Killing flow (Definition 672) are the precise algebraic sense in which the Hartle–Hawking and Gibbons–Hawking states are thermal. Such a state at positive inverse temperature automatically carries a strongly continuous one-parameter unitary group on its GNS Hilbert space implementing the flow, yielding the curved-spacetime thermal representation—the analogue of the Minkowski vacuum representation (Theorem 673); these states form a convex set (Theorem 674), and the corresponding ground state is recorded alongside them (Definition 675). +- *§10.7.1 Einstein Causality in Curved Spacetime (p. 333, items 646–647).* The operator form of local commutativity, expressed in a representation of a common containing local algebra rather than of a global quasilocal algebra (Theorem 646), and its restriction to subregions of completely spacelike-separated regions on a concrete spacetime (Lemma 647). +- *§10.7.2 Local von Neumann Algebras in Curved Spacetime (p. 333, items 648–659).* The Minkowski development of §10.6.5 mirrored relative to a containing region: the local von Neumann algebra of a subregion and its bundled registration (Definitions 648–649), Microcausality (Theorem 650), unconditional Isotony (Theorem 651), the bundled form (Theorem 652), the net as an order-preserving map on the poset of subregions (Definition 653), Statistical Independence in set-level and bundled forms (Theorems 654–655), additive-free locality (Theorem 656), and Geometric Covariance via the stabiliser GNS representation (Theorem 657) with orbit-invariance of factoriality (Theorem 658) and the upgrade to a \*-algebra isomorphism (Theorem 659). +- *§10.7.3 Relative Commutants of Nested Local Algebras in Curved Spacetime (p. 335, items 660–668).* The curved counterpart of §10.6.6: the relative commutant of a nested pair inside a containing region (Definition 660), its containment in the larger algebra, commutation with the smaller, and containment of the centre (Theorems 661–663), irreducible inclusions (Definition 664) and their consequences (Theorems 665–666), plus the abelian/centre facts specialised to curved local algebras (Theorems 667–668). +- *§10.7.4 Purity of States on Local Algebras in Curved Spacetime (p. 337, items 669–673).* Each local algebra is a unital C\*-algebra with its own state space, so the abstract purity characterisations are registered per region: Pure $$\iff$$ Extreme Point (Theorem 669), Pure $$\iff$$ Irreducible GNS (Theorem 670), the GNS representation of a pure state generates a factor and indeed all of $$\mathcal{B}(H)$$ (Theorem 671), the Irreducible Dichotomy for a curved local algebra (Theorem 672), and GNS covariance for curved local algebras (Theorem 673). +- *§10.7.5 Covariant States in Curved Spacetime (p. 337, items 674–675).* Covariant families of local states in curved spacetime (Definition 674) and their composition law (Lemma 675). +- *§10.7.6 The Stabilizer GNS Unitary in Curved Spacetime (p. 338, items 676–682).* Since no quasilocal algebra exists on a generic Lorentzian spacetime, the covariance action restricts to the stabiliser subgroup $$\mathrm{Stab}(\mathbf{B})$$ of a region, giving automorphisms of the single local algebra $$\mathfrak{U}(\mathbf{B})$$ (Definition 676) that form a genuine group action (Lemma 677). A stabiliser-invariant state carries a unitary GNS representation of $$\mathrm{Stab}(\mathbf{B})$$ (Theorem 678), strongly continuous when the matrix coefficients are continuous (Theorem 679); a state both stabiliser-invariant and pure yields an irreducible covariant representation (Theorem 680); purity is invariant under the stabiliser action (Theorem 681), and GNS data transports along it (Theorem 682). +- *§10.7.7 KMS States for a Killing Flow (p. 339, items 683–688).* Killing flows are identified as one-parameter subgroups of the stabiliser of a region, inducing a one-parameter automorphism group on $$\mathfrak{U}(\mathbf{B})$$ (Definition 683, Lemma 684). KMS states for a Killing flow (Definition 685) are the precise algebraic sense in which the Hartle–Hawking and Gibbons–Hawking states are thermal. Such a state at positive inverse temperature automatically carries a strongly continuous one-parameter unitary group on its GNS Hilbert space implementing the flow, yielding the curved-spacetime thermal representation—the analogue of the Minkowski vacuum representation (Theorem 686); these states form a convex set (Theorem 687), and the corresponding ground state is recorded alongside them (Definition 688). **Where the Lean lives:** `Physicslib4/AQFT/HaagKastlerCurved/` (`LocalVonNeumann`, `StabilizerAction`, `StabilizerKMS`, `Purity`, `GeometricCovariance`, …) @@ -26,78 +26,80 @@ usemathjax: true Every entry links to its node in the web blueprint and names the principal Lean declaration. -### §10.7 The axioms (p. 328) - -- [**Definition 627**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-algebras-in-curved-spacetime) — Axiom 1: Local Algebras · `Physicslib4.AQFT.HaagKastlerCurved.LocalNet` -- [**Definition 628**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:isotony-in-curved-spacetime) — Axiom 2: Isotony · `Physicslib4.AQFT.HaagKastlerCurved.Isotony` -- [**Definition 629**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-commutativity-in-curved-spacetime) — Axiom 3: Local Commutativity · `Physicslib4.AQFT.HaagKastlerCurved.LocalCommutativity` -- [**Definition 630**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-observable) — Local Observable · `Physicslib4.AQFT.HaagKastlerCurved.IsLocalObservable` -- [**Definition 631**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-completeness-in-curved-spacetime) — Axiom 4: Local Completeness · `Physicslib4.AQFT.HaagKastlerCurved.LocalAlgebra` -- [**Definition 632**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:isometric-covariance-in-curved-spacetime) — Axiom 5: Isometric Covariance · `Physicslib4.AQFT.HaagKastlerCurved.IsometricCovariance` *(implemented via the explicitly orientation-preserving subgroup — see [Formalisation status](./#formalisation-status))* -- [**Definition 633**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:haag-kastler-net-in-curved-spacetime) — Haag–Kastler Net in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet` - -### §10.7.1 Einstein Causality in Curved Spacetime (p. 331) - -- [**Theorem 634**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:einstein-causality-in-curved-spacetime) — Einstein Causality in a Representation (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.einstein_causality` (+1 more) - -### §10.7.2 Local von Neumann Algebras in Curved Spacetime (p. 331) - -- [**Definition 635**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann-in-curved-spacetime) — Local von Neumann Algebra in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localOperators` (+1 more) -- [**Definition 636**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann-algebra-in-curved-spacetime) — $$R(\mathbf{B}')$$ as a von Neumann Algebra (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra` -- [**Theorem 637**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-microcausality-in-curved-spacetime) — Microcausality at the von Neumann Level (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_subset_centralizer` -- [**Theorem 638**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-isotony-in-curved-spacetime) — Isotony of the von Neumann Net (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_mono` -- [**Theorem 639**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-bundled-order-in-curved-spacetime) — Bundled von Neumann Microcausality and Isotony (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_le_commutant` (+1 more) -- [**Definition 640**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:von-neumann-net-in-curved-spacetime) — The Net of von Neumann Algebras in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.vonNeumannNet` -- [**Theorem 641**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence-in-curved-spacetime) — Statistical Independence (Schlieder Property) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_separating` (+1 more) -- [**Theorem 642**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence-bundled-in-curved-spacetime) — Statistical Independence, bundled (curved spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_separating` -- [**Theorem 643**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:additive-free-locality-in-curved-spacetime) — Additive-Free Locality via the Spacelike Complement (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_le_commutant_of_subset_spacelikeComplement_geometric` -- [**Theorem 644**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-geometric-covariance-in-curved-spacetime) — Geometric Covariance of the von Neumann Net (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localVonNeumann` (+2 more) -- [**Theorem 645**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-factor-orbit-in-curved-spacetime) — Orbit-Invariance of Factoriality (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_isFactor_smul` -- [**Theorem 646**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-covariance-iso-in-curved-spacetime) — Geometric Covariance as a von Neumann Algebra Isomorphism (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannEquiv` - -### §10.7.3 Relative Commutants of Nested Local Algebras in Curved Spacetime (p. 333) - -- [**Definition 647**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:relative-commutant-in-curved-spacetime) — Relative Commutant of a Nested Pair (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant` -- [**Theorem 648**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-le-right-in-curved-spacetime) — Relative Commutant Lies in the Larger Algebra (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant_le_right` -- [**Theorem 649**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-coe-commutant-in-curved-spacetime) — Relative Commutant Commutes with the Smaller Algebra (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant_coe_subset_commutant` -- [**Theorem 650**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-center-in-curved-spacetime) — Relative Commutant Contains the Centre (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.center_le_relativeCommutant` -- [**Definition 651**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:irreducible-inclusion-in-curved-spacetime) — Irreducible Inclusion (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsIrreducibleInclusion` -- [**Theorem 652**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-inclusion-factor-in-curved-spacetime) — An Irreducible Inclusion has Factor Ambient (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isFactor_of_isIrreducibleInclusion` -- [**Theorem 653**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:self-inclusion-factor-in-curved-spacetime) — Self-Inclusion is Irreducible iff Factor (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isIrreducibleInclusion_self_iff_isFactor` -- [**Theorem 654**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:abelian-local-von-neumann-in-curved-spacetime) — Abelian Local von Neumann Algebras (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_center_isAbelian` (+1 more) -- [**Theorem 655**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:center-duality-local-von-neumann-in-curved-spacetime) — Centre Duality for Local von Neumann Algebras (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_center_eq_commutant` (+1 more) - -### §10.7.4 Purity of States on Local Algebras in Curved Spacetime (p. 334) - -- [**Theorem 656**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-extreme-in-curved-spacetime) — Pure $$\iff$$ Extreme Point on a Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.pure_iff_extreme` -- [**Theorem 657**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-irreducible-in-curved-spacetime) — Pure $$\iff$$ Irreducible GNS on a Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_pure_iff_irreducible` -- [**Theorem 658**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-factor-generates-in-curved-spacetime) — Pure GNS on a Local Algebra is a Factor Generating $$\mathcal{B}(H)$$ · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_factor_of_isPure` (+1 more) -- [**Theorem 659**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-dichotomy-in-curved-spacetime) — The Irreducible Dichotomy for a Curved Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.areDisjoint_or_unitaryEquiv_of_isIrreducible` -- [**Theorem 660**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-local-in-curved-spacetime) — GNS covariance for curved local algebras · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.unitaryEquiv_gns_covEquiv` (+2 more) - -### §10.7.5 Covariant States in Curved Spacetime (p. 335) - -- [**Definition 661**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:covariant-state-family-in-curved-spacetime) — Covariant Family of Local States in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsCovariantFamily` -- [**Lemma 662**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:covariant-state-family-compose-in-curved-spacetime) — Composition of Covariance in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsCovariantFamily.comp` - -### §10.7.6 The Stabiliser GNS Unitary in Curved Spacetime (p. 336) - -- [**Definition 663**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:stabilizer-action-in-curved-spacetime) — Stabiliser Action on a Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut` -- [**Lemma 664**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:stabilizer-action-laws-in-curved-spacetime) — The Stabiliser Action is a Group Action · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut_one` (+1 more) -- [**Theorem 665**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-unitary-stabilizer-in-curved-spacetime) — GNS Unitary Representation of the Stabiliser · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer` -- [**Theorem 666**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-unitary-stabilizer-strongly-continuous-in-curved-spacetime) — Strongly Continuous Stabiliser GNS Unitary · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer_strongContinuous` -- [**Theorem 667**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-covariant-representation-in-curved-spacetime) — Irreducible Covariant Representation of a Pure Invariant State (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_irreducible_covariant_stabilizer` -- [**Theorem 668**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:purity-covariance-invariant-in-curved-spacetime) — Purity is Invariant under the Stabiliser Action · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isPure_precomp_stabAut_iff` -- [**Theorem 669**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-stabilizer-action) — GNS covariance along the stabiliser action · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.unitaryEquiv_gns_stabAut` (+2 more) - -### §10.7.7 KMS States for a Killing Flow (p. 337) - -- [**Definition 670**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:flow-aut-in-curved-spacetime) — Killing-Flow Automorphism Family · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.flowAut` -- [**Lemma 671**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:one-parameter-aut-flow-in-curved-spacetime) — A Killing Flow Induces a One-Parameter Group · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isOneParameterAut_flowAut` -- [**Definition 672**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:kms-state-for-flow-in-curved-spacetime) — KMS State for a Killing Flow · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsKMSStateForFlow` -- [**Theorem 673**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-thermal-representation-in-curved-spacetime) — The Killing-Flow KMS Thermal Representation · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsKMSStateForFlow.exists_gns_unitary_strongContinuous` -- [**Theorem 674**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-convex-for-flow-in-curved-spacetime) — Convexity of the Killing-Flow KMS States · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsKMSStateForFlow.convexCombo` -- [**Definition 675**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:ground-state-for-flow-in-curved-spacetime) — Ground State for a Killing Flow · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsGroundStateForFlow` (+2 more) +### §10.7 The axioms (p. 330) + +- [**Definition 638**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-algebras-in-curved-spacetime) — Axiom 1: Local Algebras · `Physicslib4.AQFT.HaagKastlerCurved.LocalNet` +- [**Definition 639**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:isotony-in-curved-spacetime) — Axiom 2: Isotony · `Physicslib4.AQFT.HaagKastlerCurved.Isotony` +- [**Definition 640**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-commutativity-in-curved-spacetime) — Axiom 3: Local Commutativity · `Physicslib4.AQFT.HaagKastlerCurved.LocalCommutativity` +- [**Definition 641**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-observable) — Local Observable · `Physicslib4.AQFT.HaagKastlerCurved.IsLocalObservable` (+1 more) +- [**Definition 642**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-completeness-in-curved-spacetime) — Axiom 4: Local Completeness · `Physicslib4.AQFT.HaagKastlerCurved.LocalAlgebra` +- [**Definition 643**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:isometric-covariance-in-curved-spacetime) — Axiom 5: Isometric Covariance · `Physicslib4.AQFT.HaagKastlerCurved.IsometricCovariance` *(implemented via the explicitly orientation-preserving subgroup — see [Formalisation status](./#formalisation-status))* +- [**Definition 644**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:haag-kastler-net-in-curved-spacetime) — Haag–Kastler Net in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet` +- [**Theorem 645**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:haag-kastler-net-in-curved-spacetime-nonempty) — The Curved Haag–Kastler Axioms are Jointly Satisfiable · `Physicslib4.Spacetime.LorentzianSpacetime.nonempty_haagKastlerNet_identityComponent` (+2 more) + +### §10.7.1 Einstein Causality in Curved Spacetime (p. 333) + +- [**Theorem 646**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:einstein-causality-in-curved-spacetime) — Einstein Causality in a Representation (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.einstein_causality` (+1 more) +- [**Lemma 647**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:local-commutativity-mono-in-curved-spacetime) — Local Commutativity for Subregions (Concrete Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.commute_of_spacelike_mono_geometric` + +### §10.7.2 Local von Neumann Algebras in Curved Spacetime (p. 333) + +- [**Definition 648**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann-in-curved-spacetime) — Local von Neumann Algebra in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localOperators` (+1 more) +- [**Definition 649**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:local-von-neumann-algebra-in-curved-spacetime) — $$R(\mathbf{B}')$$ as a von Neumann Algebra (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra` +- [**Theorem 650**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-microcausality-in-curved-spacetime) — Microcausality at the von Neumann Level (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_subset_centralizer` +- [**Theorem 651**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-isotony-in-curved-spacetime) — Isotony of the von Neumann Net (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_mono` +- [**Theorem 652**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-bundled-order-in-curved-spacetime) — Bundled von Neumann Microcausality and Isotony (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_le_commutant` (+1 more) +- [**Definition 653**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:von-neumann-net-in-curved-spacetime) — The Net of von Neumann Algebras in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.vonNeumannNet` +- [**Theorem 654**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence-in-curved-spacetime) — Statistical Independence (Schlieder Property) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_separating` (+1 more) +- [**Theorem 655**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:statistical-independence-bundled-in-curved-spacetime) — Statistical Independence, bundled (curved spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_separating` +- [**Theorem 656**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:additive-free-locality-in-curved-spacetime) — Additive-Free Locality via the Spacelike Complement (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_le_commutant_of_subset_spacelikeComplement_geometric` +- [**Theorem 657**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-geometric-covariance-in-curved-spacetime) — Geometric Covariance of the von Neumann Net (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localVonNeumann` (+3 more) +- [**Theorem 658**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-factor-orbit-in-curved-spacetime) — Orbit-Invariance of Factoriality (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_isFactor_smul` +- [**Theorem 659**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:von-neumann-covariance-iso-in-curved-spacetime) — Geometric Covariance as a von Neumann Algebra Isomorphism (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannEquiv` + +### §10.7.3 Relative Commutants of Nested Local Algebras in Curved Spacetime (p. 335) + +- [**Definition 660**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:relative-commutant-in-curved-spacetime) — Relative Commutant of a Nested Pair (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant` +- [**Theorem 661**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-le-right-in-curved-spacetime) — Relative Commutant Lies in the Larger Algebra (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant_le_right` +- [**Theorem 662**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-coe-commutant-in-curved-spacetime) — Relative Commutant Commutes with the Smaller Algebra (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant_coe_subset_commutant` +- [**Theorem 663**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:relative-commutant-center-in-curved-spacetime) — Relative Commutant Contains the Centre (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.center_le_relativeCommutant` +- [**Definition 664**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:irreducible-inclusion-in-curved-spacetime) — Irreducible Inclusion (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsIrreducibleInclusion` +- [**Theorem 665**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-inclusion-factor-in-curved-spacetime) — An Irreducible Inclusion has Factor Ambient (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isFactor_of_isIrreducibleInclusion` +- [**Theorem 666**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:self-inclusion-factor-in-curved-spacetime) — Self-Inclusion is Irreducible iff Factor (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isIrreducibleInclusion_self_iff_isFactor` +- [**Theorem 667**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:abelian-local-von-neumann-in-curved-spacetime) — Abelian Local von Neumann Algebras (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_center_isAbelian` (+1 more) +- [**Theorem 668**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:center-duality-local-von-neumann-in-curved-spacetime) — Centre Duality for Local von Neumann Algebras (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_center_eq_commutant` (+1 more) + +### §10.7.4 Purity of States on Local Algebras in Curved Spacetime (p. 337) + +- [**Theorem 669**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-extreme-in-curved-spacetime) — Pure $$\iff$$ Extreme Point on a Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.pure_iff_extreme` +- [**Theorem 670**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-iff-irreducible-in-curved-spacetime) — Pure $$\iff$$ Irreducible GNS on a Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_pure_iff_irreducible` +- [**Theorem 671**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:pure-factor-generates-in-curved-spacetime) — Pure GNS on a Local Algebra is a Factor Generating $$\mathcal{B}(H)$$ · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_factor_of_isPure` (+1 more) +- [**Theorem 672**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-dichotomy-in-curved-spacetime) — The Irreducible Dichotomy for a Curved Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.areDisjoint_or_unitaryEquiv_of_isIrreducible` +- [**Theorem 673**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-local-in-curved-spacetime) — GNS covariance for curved local algebras · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.unitaryEquiv_gns_covEquiv` (+2 more) + +### §10.7.5 Covariant States in Curved Spacetime (p. 337) + +- [**Definition 674**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:covariant-state-family-in-curved-spacetime) — Covariant Family of Local States in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsCovariantFamily` +- [**Lemma 675**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:covariant-state-family-compose-in-curved-spacetime) — Composition of Covariance in Curved Spacetime · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsCovariantFamily.comp` + +### §10.7.6 The Stabiliser GNS Unitary in Curved Spacetime (p. 338) + +- [**Definition 676**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:stabilizer-action-in-curved-spacetime) — Stabiliser Action on a Local Algebra · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut` +- [**Lemma 677**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:stabilizer-action-laws-in-curved-spacetime) — The Stabiliser Action is a Group Action · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut_one` (+1 more) +- [**Theorem 678**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-unitary-stabilizer-in-curved-spacetime) — GNS Unitary Representation of the Stabiliser · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer` (+1 more) +- [**Theorem 679**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-unitary-stabilizer-strongly-continuous-in-curved-spacetime) — Strongly Continuous Stabiliser GNS Unitary · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer_strongContinuous` (+1 more) +- [**Theorem 680**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:irreducible-covariant-representation-in-curved-spacetime) — Irreducible Covariant Representation of a Pure Invariant State (Curved Spacetime) · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_irreducible_covariant_stabilizer` +- [**Theorem 681**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:purity-covariance-invariant-in-curved-spacetime) — Purity is Invariant under the Stabiliser Action · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isPure_precomp_stabAut_iff` +- [**Theorem 682**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:gns-covariance-stabilizer-action) — GNS covariance along the stabiliser action · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.unitaryEquiv_gns_stabAut` (+2 more) + +### §10.7.7 KMS States for a Killing Flow (p. 339) + +- [**Definition 683**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:flow-aut-in-curved-spacetime) — Killing-Flow Automorphism Family · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.flowAut` +- [**Lemma 684**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#lmm:one-parameter-aut-flow-in-curved-spacetime) — A Killing Flow Induces a One-Parameter Group · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isOneParameterAut_flowAut` +- [**Definition 685**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:kms-state-for-flow-in-curved-spacetime) — KMS State for a Killing Flow · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsKMSStateForFlow` +- [**Theorem 686**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-thermal-representation-in-curved-spacetime) — The Killing-Flow KMS Thermal Representation · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsKMSStateForFlow.exists_gns_unitary_strongContinuous` +- [**Theorem 687**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#thrm:kms-convex-for-flow-in-curved-spacetime) — Convexity of the Killing-Flow KMS States · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsKMSStateForFlow.convexCombo` +- [**Definition 688**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:ground-state-for-flow-in-curved-spacetime) — Ground State for a Killing Flow · `Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsGroundStateForFlow` (+2 more) [Home](./) · [§10.1 GNS](sec10-1-gns.html) · [§10.2 Spectral](sec10-2-spectral.html) · [§10.3 Unbounded](sec10-3-unbounded.html) · [§10.4 Stone](sec10-4-stone.html) · [§10.5 Spacetime](sec10-5-spacetime.html) · [§10.6 Minkowski](sec10-6-haag-kastler.html) · **§10.7 Curved** · [§10.8 Covariance](sec10-8-general-covariance.html) · [Axioms](axioms.html) · [Guide](guide.html) diff --git a/home_page/sec10-8-general-covariance.md b/home_page/sec10-8-general-covariance.md index 20cc73a..136f0dc 100644 --- a/home_page/sec10-8-general-covariance.md +++ b/home_page/sec10-8-general-covariance.md @@ -1,5 +1,5 @@ --- -title: "§10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 338–339)" +title: "§10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 340–343)" usemathjax: true --- @@ -7,9 +7,9 @@ usemathjax: true ← [§10.7 Curved](sec10-7-curved.html) -# §10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 338–339) +# §10.8 General Covariance: Nets on Pullback-Related Metrics (pp. 340–343) -Two definitions. The gauge group of general relativity is the full diffeomorphism group of $$M$$, so the physical content of a spacetime is its diffeomorphism-equivalence class and not the pair $$(M, g)$$; the hole argument shows that treating a relabelling as physical would destroy determinism, and Leibniz equivalence resolves it by declaring diffeomorphic models to represent the same physical situation. Accordingly, an equivalence of Haag–Kastler nets is defined (Definition 676) as a chosen family of unital \*-isomorphisms $$\Theta_\mathbf{B} : \mathfrak{U}_1(\mathbf{B}) \to \mathfrak{U}_2(e(\mathbf{B}))$$ along a basis-set-preserving bijection $$e$$ of carriers, natural with respect to the isotony embeddings. The carriers are related by data rather than by a type equality, deliberately: an equality of carrier types cannot be transported along and would force every comparison through a cast. A net theory is then a section assigning a net to every Lorentzian spacetime, and it is generally covariant when the nets over $$L$$ and over its pullback $$\psi^* L$$ are equivalent along $$\psi$$ (Definition 677). In Lean the nets are indexed over the identity-component bridge `toAbstractIdentityComponent`, so Axiom 5 of each net asks only for isometries connected to the identity, as in the blueprint. Four points are worth carrying away: this is a **postulate, not a theorem**—nothing forces the two nets to be isomorphic, and it says nothing about backgrounds that are not diffeomorphism-related; the morphism is specified geometrically rather than causally, because a purely causal morphism would admit the dilations (Theorems 422–423) and thereby demand a scale covariance that is false for a massive theory; the relabelling must be $$\psi$$ and not the identity, since a basis set of one metric is in general not a basis set of the other; and general covariance is a property of the section $$L \mapsto \mathfrak{U}_L$$, not a sixth field of the net structure, so no restriction to diffeomorphisms connected to the identity is needed here. +Two definitions. The gauge group of general relativity is the full diffeomorphism group of $$M$$, so the physical content of a spacetime is its diffeomorphism-equivalence class and not the pair $$(M, g)$$; the hole argument shows that treating a relabelling as physical would destroy determinism, and Leibniz equivalence resolves it by declaring diffeomorphic models to represent the same physical situation. Accordingly, an equivalence of Haag–Kastler nets is defined (Definition 689) as a chosen family of unital \*-isomorphisms $$\Theta_\mathbf{B} : \mathfrak{U}_1(\mathbf{B}) \to \mathfrak{U}_2(e(\mathbf{B}))$$ along a basis-set-preserving bijection $$e$$ of carriers, natural with respect to the isotony embeddings. The carriers are related by data rather than by a type equality, deliberately: an equality of carrier types cannot be transported along and would force every comparison through a cast. A net theory is then a section assigning a net to every Lorentzian spacetime, and it is generally covariant when the nets over $$L$$ and over its pullback $$\psi^* L$$ are equivalent along $$\psi$$ (Definition 690). In Lean the nets are indexed over the identity-component bridge `toAbstractIdentityComponent`, so Axiom 5 of each net asks only for isometries connected to the identity, as in the blueprint. Four points are worth carrying away: this is a **postulate, not a theorem**—nothing forces the two nets to be isomorphic, and it says nothing about backgrounds that are not diffeomorphism-related; the morphism is specified geometrically rather than causally, because a purely causal morphism would admit the dilations (Theorems 427–428) and thereby demand a scale covariance that is false for a massive theory; the relabelling must be $$\psi$$ and not the identity, since a basis set of one metric is in general not a basis set of the other; and general covariance is a property of the section $$L \mapsto \mathfrak{U}_L$$, not a sixth field of the net structure, so no restriction to diffeomorphisms connected to the identity is needed here. **Where the Lean lives:** `Physicslib4/AQFT/HaagKastlerCurved/GeneralCovariance.lean` @@ -19,10 +19,10 @@ Two definitions. The gauge group of general relativity is the full diffeomorphis Every entry links to its node in the web blueprint and names the principal Lean declaration. -### §10.8 General Covariance: Nets on Pullback-Related Metrics (p. 338) +### §10.8 General Covariance: Nets on Pullback-Related Metrics (p. 340) -- [**Definition 676**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:net-equivalence-in-curved-spacetime) — Equivalence of Haag–Kastler Nets · `Physicslib4.AQFT.HaagKastlerCurved.NetEquivalence` -- [**Definition 677**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:general-covariance-in-curved-spacetime) — General Covariance · `Physicslib4.AQFT.HaagKastlerCurved.NetTheory` (+4 more) +- [**Definition 689**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:net-equivalence-in-curved-spacetime) — Equivalence of Haag–Kastler Nets · `Physicslib4.AQFT.HaagKastlerCurved.NetEquivalence` +- [**Definition 690**](https://physicslib.github.io/physicslib4/blueprint/chptr-haag-kastler-axioms-blueprint.html#def:general-covariance-in-curved-spacetime) — General Covariance · `Physicslib4.AQFT.HaagKastlerCurved.NetTheory` (+4 more) [Home](./) · [§10.1 GNS](sec10-1-gns.html) · [§10.2 Spectral](sec10-2-spectral.html) · [§10.3 Unbounded](sec10-3-unbounded.html) · [§10.4 Stone](sec10-4-stone.html) · [§10.5 Spacetime](sec10-5-spacetime.html) · [§10.6 Minkowski](sec10-6-haag-kastler.html) · [§10.7 Curved](sec10-7-curved.html) · **§10.8 Covariance** · [Axioms](axioms.html) · [Guide](guide.html) diff --git a/notes/agents/audit-review-1.0-fixes-2026-10-14.md b/notes/agents/audit-review-1.0-fixes-2026-10-14.md new file mode 100644 index 0000000..1e45b00 --- /dev/null +++ b/notes/agents/audit-review-1.0-fixes-2026-10-14.md @@ -0,0 +1,125 @@ +# Audit: review-1.0 fix plan, stages 1-7 (40a55f6..working tree) + +Scope: `git diff 40a55f6 -- Physicslib4 blueprint/src` plus uncommitted `home_page/`. +Commits baf635a..309b89d (stages 1-7 of `notes/agents/plan-review-1.0-fixes-2026-10-11.md`). +`scratch_*` ignored. + +## 1. Protected changes + +`python3 ~/.claude/lean-kit/lean_guard.py check --base 40a55f6` reports 12 protected changes: +9 blueprint-text edits (all stage-4/5/6 prose: `gns-construction-details.tex`, +`haag-kastler-axioms-in-curved-spacetime.tex`, `haag-kastler-axioms.tex`, `spacetime.tex`, +`spectral-theorems.tex`, `unbounded-spectral-theorems.tex`, `isotony.tex`, +`quasilocal-algebra.tex`) plus `.github/workflows/{blueprint,lean_action_ci}.yml` +(stage 7, R16-2/R16-3) and `.claude/{CLAUDE.md,lean-policy.json}` (new protected files on +this branch, unmodified in this diff scope). None unexpected. + +No Lean declaration appears in this list because all four Lean signature changes the plan +calls out are covered by recorded approvals in `.git/lean-kit/approvals.json`: +`VacuumState.lean` (D1), `QuasilocalKMS.lean` (D1), `PureStateExists.lean` (D7), +`LorentzCauchySchwarz.lean` (D7). I read the diffs directly to confirm each matches its +decision: + +- D1: `HaagKastlerNet.IsVacuumState (N : HaagKastlerNet)` -> `(N : HaagKastlerNet.{u})`, + `∃ (K : Type) ...` -> `∃ (K : Type u) ...` (`VacuumState.lean:122-128`). Same pattern for + `IsGroundStateForFlow` and `IsGroundStateForFlow.exists_strongContinuous_unitary`, both now + take `N : HaagKastlerNet.{u}` explicitly and use `Type u` (`QuasilocalKMS.lean:96-131`). + Matches "D1 universes" exactly. +- D7 (`isCompact_weakStateSet`): `[Nontrivial A]` hypothesis dropped; proof adds a + `subsingleton_or_nontrivial A` case split, trivial subcase shown empty hence compact + (`PureStateExists.lean:83-95`). Matches. +- D7 (`reverse_cauchy_schwarz_of_lorentzianAt`): `(_hv : B v v < 0)` hypothesis removed; + statement now holds for arbitrary `v` given only `w` timelike (`LorentzCauchySchwarz.lean:60`). + The blueprint-facing wrapper `Spacetime.reverse_cauchy_schwarz` keeps both hypotheses + (`hv` renamed `_hv`, now unused) to match `lmm:reverse-cauchy-schwarz`'s symmetric + statement, as the plan records. Matches. + +No other changed Lean signature found outside these four files when diffing the full +`Physicslib4/` tree against `40a55f6`. **Pass.** + +## 2. sorry / admit / native_decide / axioms / set_option maxHeartbeats + +`git diff 40a55f6 -- Physicslib4` grepped (case-insensitive) for `sorry`, `admit`, +`native_decide`, `axiom `, `set_option maxHeartbeats`: no additions, no removals. Consistent +with stage 2's note that all 11 `set_option`s were kept (none removable) and stage 3's `erw` +removal being unrelated to heartbeats. **Pass.** + +## 3. Changed docstrings / new comments + +Spot-checked the added docstrings and notes (`Net.lean:341-342,347-348` two new one-line +theorem docstrings; `Concrete.lean:38-68` module docstring + new "Implementation note" on +`toAbstractIdentityComponent`; the stage-4/5 blueprint "Formalization note" / prose additions +in `haag-kastler-axioms-in-curved-spacetime.tex`, `haag-kastler-axioms.tex`, +`gns-construction-details.tex`, `unbounded-spectral-theorems.tex`). All present tense, no +"no longer" / "previously" / "now proved" style narration (confirmed also by the grep for +those patterns over all `+` lines in the Lean diff: no hits). Content checks: +- `Concrete.lean`'s rewritten module-docstring line still asserts `orientedIdentityComponent` + "is the blueprint's group of 'isometries connected to the identity'" in the same breath as + pointing to the implementation note; the note itself (and the mirrored blueprint + Formalization note) correctly state the equality is *not* proved, only asserted by + definition, pending a Myers-Steenrod-type rigidity theorem. Matches D3's content; the + surrounding sentence is slightly less hedged than the note it points to, but not inaccurate + given D3 explicitly decided the two descriptions "name the same group" and chose a note + over a Restriction line. No independent Restriction marker was added for D2 or D3, matching + the recorded decisions (both ended up "no Lean change / no Restriction line"). **Pass**, + one minor wording observation above, not a finding. + +## 4. Blueprint `\lean{}` tags and statement spot-checks + +Collected every name inside an added `\lean{...}` tag in `git diff 40a55f6 -- blueprint/src` +(55 names, stage 4/5 additions). All 55 resolve in `blueprint/lean_decls`. The one name that +disappears from a `\lean{}` tag (`Physicslib4.AQFT.exp_generator_unique`) is on a deletion +line only (R06-2, stage 4), not a dangling reference. + +Spot-checked declarations exist with matching signatures: +- `IsLorentz`, `IsProper`, `IsOrthochronous` (`Minkowski.lean:206-216`) against + `def:lorentz-transformation` (`spacetime.tex`): exact match (Minkowski-form preservation, + det = 1, positive time component of `L e_0`). +- `isOrthochronous_pos_time_of_timelike`, `isLorentz_refl`/`_trans`/... exist and are used + under the matching new nodes `lmm:orthochronous-preserves-future-timelike`, + `lmm:lorentz-closure`. +- `nonempty_haagKastlerNet_identityComponent`, `isometricCovariance_trivial_identityComponent`, + `identityComponent_faithfulSMul` (`IdentityComponent.lean:33-55`) exist and their types + line up with the new theorem `thrm:haag-kastler-net-in-curved-spacetime-nonempty`'s prose + (trivial net over the concrete oriented-identity-component spacetime satisfies isometric + covariance and is nonempty; faithfulness stated for the full identity component, with the + blueprint text noting faithfulness passes to the Axiom-5 subgroup rather than Lean proving + a second faithfulness instance directly for it). +- `commute_of_spacelike_mono_geometric` exists in `HaagKastlerCurved/` and backs + `lmm:local-commutativity-mono-in-curved-spacetime`. + +No mismatches found among the 13 new nodes' spot-checked statements. **Pass** (statement +match verified by reading Lean source for ~6 of 13 new nodes plus the GNS/Formalization-note +additions by inspection; the remaining new nodes were only checked for `\lean{}` name +resolution, per the task's "focus on obvious mismatches" instruction given they were already +reviewed). + +## 5. Home page vs `blueprint/web` + +`blueprint/web/*.html` timestamp (07:56) is after the last commit (07:49), consistent with +stage-8-in-progress regeneration (`notes/agents/stage8-progress.md`, uncommitted). Checked 5 +entries from `home_page/sec10-5-spacetime.md` and `home_page/sec10-7-curved.md` against +`blueprint/web/chptr-haag-kastler-axioms-blueprint.html`: + +| Entry | MD number/title | HTML `id` / label / title | Match | +|---|---|---|---| +| Definition 336 | Lorentz, Proper and Orthochronous Transformations | `def:lorentz-transformation`, 336, same title | yes | +| Lemma 337 | Orthochronous Lorentz Transformations Preserve the Future Cone | `lmm:orthochronous-preserves-future-timelike`, 337, same | yes | +| Lemma 338 | Closure of the Lorentz Conditions | `lmm:lorentz-closure`, 338, same | yes | +| Theorem 645 | The Curved Haag-Kastler Axioms are Jointly Satisfiable | `thrm:haag-kastler-net-in-curved-spacetime-nonempty`, 645, same | yes | +| Lemma 647 | Local Commutativity for Subregions (Concrete Spacetime) | `lmm:local-commutativity-mono-in-curved-spacetime`, 647, same | yes | + +All 5 match (number, title, anchor). **Pass.** + +## Summary + +All five checks pass. No unapproved protected Lean signature changes, no sorry/admit/ +native_decide/new-axiom/set_option additions, docstrings read present-tense and accurate +(one minor non-finding wording note on `Concrete.lean`'s module docstring), all 55 new +`\lean{}` names resolve and spot-checked statements match, and the 5 sampled home-page +entries agree with the freshly rebuilt `blueprint/web` HTML. + +Verified by tool: `lean_guard.py check`, `grep`/`git diff` greps for sorry/admit/axioms/ +set_option, `blueprint/lean_decls` membership check for all 55 added `\lean{}` names. +Judged by reading: docstring wording/accuracy, statement-vs-Lean matches for the new nodes, +home-page entry correctness. diff --git a/notes/agents/plan-review-1.0-fixes-2026-10-11.md b/notes/agents/plan-review-1.0-fixes-2026-10-11.md new file mode 100644 index 0000000..549962a --- /dev/null +++ b/notes/agents/plan-review-1.0-fixes-2026-10-11.md @@ -0,0 +1,202 @@ +# Staged plan: fixes from the 1.0 pre-release review + +Source: `notes/agents/review-1.0/SUMMARY.md` and reports R01–R16. Finding IDs refer to those reports. + +Execution rule for every stage: one subagent at a time, each writing progress to disk; one commit per +stage on `numina/aqft-in-lean` (or a stage branch); `lake build` + `lake lint` + `leanblueprint checkdecls` ++ `scripts/check_temporal_wording.py` before each commit. + +## Decisions recorded (2026-10-11) + +No stage has started; work resumes after the weekly budget reset. + +- D1: approved as recommended, option (a): `Type u`. +- D2: option (c). Reword the blueprint so `def:geodesic` is stated for spacetimes only; no Lean change and + no Restriction line. +- D3: no `**Restriction:**` line. The two descriptions name the same group, so the Lean group is not + narrower in content. Record the implementation choice as a note instead: + - Blueprint: a "Formalization note" under Axiom 5 (curved) saying the Lean defines the group as the + identity component intersected with the future-orientation-preserving isometries; that this equals + the isometries connected to the identity; and that the Lean doesn't prove the equality because it + needs a Myers–Steenrod-type rigidity result Mathlib lacks. + - Lean: an "Implementation note" in the docstring of `toAbstractIdentityComponent` with the same content, + replacing the bare "matching the blueprint's ..." claim in the module docstring. + - Plan stage 1 drops the D3 Restriction item and instead carries these two notes. +- D4: approved as recommended, option (a): new Chapter 10 nodes for headline results. +- D5: approved as recommended: `\uses{}` follows the informal proof; GNS option (a) with a + Formalization note. +- D6: default approved. +- D7: default approved. + +## Stage 1 progress (2026-10-13) + +- Done, uncommitted: D2 rewording (`spacetime.tex` lead-in and `def:geodesic`, `home_page/sec10-5-spacetime.md`); + D3 notes (Formalization note under curved Axiom 5; implementation notes in `Concrete.lean` and on + `orientedIdentityComponent`, whose docstring wrongly said the C⁰ topology "does not give" orientation + preservation); docstring on `isCompact_weakStateSet`. +- After the user's guard approvals: D1 applied (`IsVacuumState`, `IsGroundStateForFlow`, + `IsGroundStateForFlow.exists_strongContinuous_unitary` now use `Type u`); D7 applied (`[Nontrivial A]` + removed from `isCompact_weakStateSet`). Build and lint pass. Stage 1 complete, uncommitted. +- (Earlier) blocked by the lean-kit guard, waiting on user approval: D1 for `IsVacuumState`, `IsGroundStateForFlow` + and `IsGroundStateForFlow.exists_strongContinuous_unitary` (Minkowski, `VacuumState.lean`, `QuasilocalKMS.lean`); + D7 removal of `[Nontrivial A]` (`PureStateExists.lean`). +- D1 scope correction: the curved declarations in R13-7 need no change. The curved `LocalNet.algebra` + lands in `Type` (`HaagKastlerCurved/LocalAlgebras.lean:62`), so `∃ (H : Type)` already matches. The + Minkowski `LocalNet.algebra` uses `Type*` (`HaagKastler/LocalAlgebras.lean:82`). New protected question + D8: should the curved net also be universe-polymorphic? + +## Stage 2 progress (2026-10-13) + +Done, uncommitted; details in `notes/agents/stage2-progress.md`. All items applied. No `set_option` could +be removed (each is needed; the guard also treats removing one as a definition change), so all 11 carry +a comment. One `erw` became `rw`; two stay with comments. Build and lint pass. + +## Stage 3 progress (2026-10-13) + +Details in `notes/agents/stage3-progress.md`. R15-3 done: the shared estimate is `private lemma +norm_mul_sub_mul_lt` in `OperatorIntegral.lean` (29 lines fewer). R01-3 was not a real duplicate: +`lmm2` already uses `nullSubmodule_isClosed`. R07-1 done after the user's guard approval (blueprint left as is; the blueprint-named wrapper +`Spacetime.reverse_cauchy_schwarz` keeps both hypotheses, the first unused). Before that it waited on +`LorentzCauchySchwarz.lean`; callers are `LorentzCone.lean:73` and `LorentzOrthogonal.lean:57,219-220`; +blueprint node `lmm:reverse-cauchy-schwarz` (`spacetime.tex:67-86`) assumes both vectors timelike. + +## Stage 4 progress (2026-10-14) + +Done, uncommitted. Formalization notes: GNS existence (Mathlib's `PositiveLinearMap.GNS`), `prpstn:hall-10.3` +(bounded resolvent), `prpstn:integral-norm-bound` (the preliminary 6-sup bound via `lmm:lemma-1`; per D5 a +note rather than a `\uses{}` edge). `\lean{}`: added `isSelfAdjoint_toOperator` and `integral_one`; removed +`exp_generator_unique` from `lmm:exp-generator-unique` (bounded special case; its docstring says so). +R14-1: blueprint statement narrowed to the derivative claim. R03-1: proof `\uses{}` of `prpstn:hall-8.4`, +`thrm:hall-8.10` and `thrm:spectral-theorem-for-bounded-operators` trimmed to the nodes each proof cites +(19→5, 47→6, 75→2). `blueprint_validate` shows only the existing `conv:` label errors. + +## Stage 5 progress (2026-10-14) + +Done, uncommitted; details in `notes/agents/stage5-progress.md`. Three batches, each written then reviewed, +review fixes applied. 5a (`spacetime.tex`): 9 new nodes for the Lorentz group, the inhomogeneous Lorentz +group and the Lorentz invariance of the causal relations. 5b (`haag-kastler-axioms.tex`): 2 new nodes +(weak-to-strong continuity, strongly continuous GNS unitaries) and 3 extended `\lean{}` tags. 5c (curved): +2 new nodes (joint satisfiability of the curved axioms; local commutativity under inclusion) and 4 extended +tags. All new Lean names resolve. Not done: the D4 "list remaining helpers in module docstrings" part, and +triage of the rest of R13's 43-item uncited list (`chronologicallyPrecedes_smul_iff`, +`isAlexandrovBasisSet_smul` and smaller API lemmas). + +## Stage 6 progress (2026-10-14) + +Done, uncommitted. R09-1: the density-theorem entry says "the lemmas above" instead of a number range, so +it survives renumbering. R05-1: `FunctionalCalculus` listed on the §10.3 page. R12-1: stray `*` replaced. +R12-2: chapter and section numbers in `sec5/quasilocal-algebra.tex` and `sec10/haag-kastler-axioms.tex` +("Chapter 9", "Chapter 6.2", "Chapter 6.3") replaced by `\ref`s. R02-3: contraction removed. R15-5: +non-emptiness dropped from the Boundedness Theorem. R14-3: no change; every Lean subscript in prose uses +`\ensuremath{_n}` consistently, and raw Unicode appears only inside `\lean{}`. + +## Stage 7 progress (2026-10-14) + +Done, uncommitted. D6: `check_temporal_wording.py` flags "no longer", "previously" (except positional uses +such as "as we previously proved", stripped by a `POSITIONAL` regex), and history-telling "now" phrases +("is now proved/proven/formalized/available", "now formalized"); bare "now" is not flagged because proofs +use it positionally. A `temporal-ok` marker skips a line. The stricter check flagged two prose lines, +reworded. R16-2: `issues`/`pull-requests` permissions removed from `blueprint.yml` (takes effect on the +next push to `main`). R16-3: both workflows use `actions/checkout@v6` and the same pinned `lean-action` +SHA; `free-disk-space` pinned to v1.3.2. + +## Stage 8 progress (2026-10-14) + +Done, home page uncommitted. `lake build` and `lake lint` pass; `leanblueprint pdf`, `web` and `checkdecls` +pass (no undefined references in the final LaTeX pass). Home page renumbered from the rebuilt blueprint +(details in `notes/agents/stage8-progress.md`): 675 declarations, 1,188 distinct Lean names, 345 PDF pages, +13 new entries, all 681 links resolve. Verify receipt refreshed (`lean_guard.py verify`). Re-audit of the +plan's changes passes (`notes/agents/audit-review-1.0-fixes-2026-10-14.md`). The plan is complete. + +## Decisions needed first (original options, kept for reference) + +Protected (definitions, axiom-structure fields, or blueprint specification): + +- **D1. Universe of GNS witnesses** (R10-1, R13-7). About 14 declarations state `∃ (H : Type) ...` although + `gns_construction` gives `H : Type u`. Two of them are Prop-valued definitions used as hypotheses: + `HaagKastlerNet.IsVacuumState` and the two `IsGroundStateForFlow`. For a net in `Type u` with `u > 0`, + a `Type 0` witness need not exist, so these definitions can be unsatisfiable there. + - (a) `Type u`, matching the net's algebra universe. **Recommended.** + - (b) A separate universe `Type v`: more general, but the definitions get an extra universe parameter + that callers must fix. + - (c) Keep `Type` and add a `**Restriction:**` line. +- **D2. Scope of `IsGeodesic`** (R14-2). The blueprint states `def:geodesic` for a general + pseudo-Riemannian manifold; Lean states it only for `M : Spacetime`, because `SmoothPath` exists only + for `Spacetime`. + - (a) Keep Lean and add a `**Restriction:**` line (waiting on `SmoothPath` for general manifolds), + plus a sentence in the blueprint. **Recommended.** + - (b) Generalise `SmoothPath` and `IsGeodesic` to `PseudoRiemannianMetric`. Larger, and touches + every curve declaration. + - (c) Reword the blueprint to state it for spacetimes only. +- **D3. Axiom 5 and the oriented identity component** (R11-1). `toAbstractIdentityComponent` uses + orientation-preserving isometries connected to the identity; the blueprint says "isometries connected + to the identity". The reason (needs a Myers–Steenrod-type result) appears only on the home page. + - (a) Add a `**Restriction:**` line in `Concrete.lean` and a remark in the blueprint, keeping the + blueprint definition as is. **Recommended.** + - (b) Change the blueprint definition to the oriented component. +- **D4. Blueprint coverage of Lean without nodes** (R07-2, R07-4, R06-1, R10-2, R13-6). This covers the + orthochronous Lorentz section of `Minkowski.lean`, most of `LorentzCausality.lean` (imported by the + Axiom 5 and quasilocal-action files), `strongContinuous_of_weak`, the stronger `exists_gns_unitary*` + variants, and about 10 theorems such as `exists_isLocalObservable`. + - (a) Add Chapter 10 nodes for the headline results (new blueprint specification), and list the + remaining helpers in the module docstrings. **Recommended.** + - (b) Only add names to existing nodes' `\lean{}` tags, with no new nodes. + - (c) Delete Lean that nothing uses and no node needs. +- **D5. `\uses{}` convention and the GNS proof route** (R01-1, R01-2, R04-1, R03-1). + `thrm:gns-construction-theorem` cites `lmm1`, `lmm2` and `cauchy_schwarz_inequality`, but the Lean + proof uses Mathlib's GNS. `prpstn:hall-10.3` cites the direct-sum lemmas, but the Lean proof uses a + bounded resolvent. + - Convention: `\uses{}` in a proof lists the informal proof's direct dependencies; a note records a + different Lean route. **Recommended.** The alternative is to have the tags follow the Lean proof. + - GNS: (a) keep the blueprint's five-part proof and its edges, and add a "Formalization note" that + Lean uses Mathlib's GNS; `lmm1`/`lmm2`/`cauchy_schwarz_inequality` stay as stated lemmas. + **Recommended.** (b) Rewrite the blueprint proof along the Mathlib route and drop the edges. + - Under the convention, the broad `\uses{}` on the capstone nodes (R03-1) is checked against the + informal proofs, not the one-line Lean proofs. + +Not protected; defaults apply unless you say otherwise: + +- **D6. Present-tense check** (R16-1). Add patterns for bare "no longer" and "previously", and for "now" + only in history-telling phrases ("is now", "are now", "now proved", "now uses"), since bare "now" + matches far too much. A per-line allow comment covers legitimate uses. +- **D7. Hypotheses to remove instead of record.** Remove `[Nontrivial A]` from `isCompact_weakStateSet` + (R01-5): for trivial `A` the state set is empty, so it is compact. Drop the unneeded non-emptiness + hypothesis from the blueprint's Boundedness Theorem (R15-5). Drop the unused timelike-`v` hypothesis + (R07-1) if no caller needs it. + +## Stages + +1. **Restrictions and universes** (after D1–D3). Apply D1 to the 14 declarations + (mechanical: their proofs already use `gns_construction`), with matching blueprint wording if any + statement names the universe. Add the D2 and D3 restriction lines and blueprint remarks. Do D7's + `[Nontrivial A]` removal. Check: the declarations that consume `IsVacuumState` and `IsGroundStateForFlow` + still build. +2. **Small Lean fixes** (no decisions). Remove the leaked comment `OperatorIntegral.lean:963-965` (R15-2). + Move the misplaced docstring in `Minkowski.lean:592-608` (R13-3). Add docstrings to the + substantive undocumented theorems (R13-5). Fix the `Basic.lean` claim about `IsCyclicVector` (R01-4) + and the KMS wording (R10-3). Explain or remove each unexplained `set_option` (R07-3, R08-1, R13-1, + R13-2, R13-4, R03-2) and `erw` (R04-2), testing removal first. +3. **Code tidying.** Remove the duplicated ~70-line proof (R15-3) by reordering or extracting a lemma. + Resolve `nullSubmodule_isClosed` versus `lmm2` (R01-3). Remove the unused hypothesis (R07-1). + Run `fuse:golfer` on the touched proofs only. +4. **`\uses{}` and `\lean{}` tags** (after D5). Add the GNS and `prpstn:hall-10.3` formalization notes. + Add the missing edge `lmm:lemma-1` (R15-1). Complete the `\lean{}` lists: `isSelfAdjoint_toOperator` + (R02-1), `integral_one` (R15-4); reassess `exp_generator_unique` (R06-2). Add the other half of + `mfderiv_apply_tangent_eq_zero`, or narrow the blueprint statement (R14-1). Then `blueprint_validate`. +5. **Blueprint coverage** (after D4). `fuse:blueprint` + `fuse:blueprint-reviewer` draft the new nodes + for Lorentz/orthochronous, Lorentz covariance of causal relations, the weak-to-strong continuity + upgrade, and the uncited theorems. Link them with `blueprint_set_declaration_status`. Update the + affected module docstrings' "Main results" lists. +6. **Prose and home page.** Fix the lemma range in `home_page/sec10-6-haag-kastler.md:90` (R09-1, likely + renumbered by stage 5 anyway), list `FunctionalCalculus.lean` on the §10.3 page (R05-1), fix the stray + `*` (R12-1), use `\cref` for chapter references (R12-2), the LaTeX markup (R14-3), the contraction + (R02-3) and the Boundedness hypothesis (R15-5). +7. **CI and infrastructure.** Apply D6 to `check_temporal_wording.py` and fix whatever it then flags. + Drop the unused `issues`/`pull-requests` permissions (R16-2). Align checkout versions and pinning + (R16-3). +8. **Rebuild and verify.** Full build and lint, `checkdecls`, rebuild the blueprint and `lean_decls`, + regenerate the home-page numbers and links (node numbers shift after stage 5), refresh the verify + receipt (R03-3), then a short re-audit of the touched nodes only. + +Order: stages 2, 3, 6 and 7 need no decisions and can run while D1–D5 are open; 1, 4 and 5 wait on +their decisions; 8 is last. diff --git a/notes/agents/stage2-progress.md b/notes/agents/stage2-progress.md new file mode 100644 index 0000000..62fb742 --- /dev/null +++ b/notes/agents/stage2-progress.md @@ -0,0 +1,104 @@ +# Stage 2 progress (small Lean fixes, 2026-10-13) + +Working through `notes/agents/plan-review-1.0-fixes-2026-10-11.md` stage 2 items. + +## 1. R15-2 — leaked process comment (OperatorIntegral.lean:963-965) +Replaced "proved by another prover in the real file" with "proved above in this file". +Comment-only change; no proof/statement touched, so no diagnostics risk. Will be covered +by the final `lake build`. + +## 2. R13-3 — misplaced docstring (Minkowski.lean:592-608) +Moved the three paragraphs describing `MinkowskiSpacetime` (carrier, convention, retopologisation) +into `MinkowskiSpacetime`'s own docstring at line ~1729 (merged, no duplication). Left +`standardMinkowskiTimeOrientation`'s docstring as a short standalone description (the `∂_0` +vector field and why it's needed, pointing to `MinkowskiSpacetime` for the Alexandrov-topology +context). Verified with `lake env lean Physicslib4/Spacetime/Minkowski.lean`: no output (no +errors/warnings). + +## 3. R13-5 — docstrings on substantive undocumented theorems +Added one-line docstrings to: `isCausallyComplete_iff_isClosed`, `causalComplement_antitone` +(CausalComplement.lean); `isClosed_weakStateSet` (GNS/PureStateExists.lean; +`isCompact_weakStateSet` already had one, skipped as instructed); +`extendHom_continuous`, `nonempty_trivialQuasilocalLift` (QuasilocalIntertwiner.lean); +`trivialLocalNet_localCommutativity`, `trivialLocalNet_lorentzCovariance` (HaagKastler/Net.lean). +Verified each touched file with `lake env lean `: no output (no errors/warnings) for all four +files. + +## 4. R01-4 — GNS/Basic.lean docstring correctness +Corrected the "Notes" paragraph: it had claimed `IsCyclicVector` keeps `A` "general enough" for a +plain algebra per `def:cyclic-vector`, but `A` is fixed to `[CStarAlgebra A]` via the file's +ambient `variable`. Reworded to say Lean requires a unital C*-algebra (matching every call site) +and that the generality is kept on the `H` side instead. Docstring-only change. +Verified with `lake env lean Physicslib4/GNS/Basic.lean`: no output. + +## 5. R10-3 — KMS.lean module docstring wording +Replaced "unconditionally for `β > 0`" (ambiguous: reads as "with no hypotheses") with an explicit +statement that `IsKMSState.invariant` needs a `StripLiouville β` hypothesis and that +`invariant_of_pos` discharges it automatically whenever `β > 0`. Confirmed against the actual +signatures (`IsKMSState.invariant` takes `hSL : StripLiouville β`; `invariant_of_pos` takes +`hβ : 0 < β`). Verified with `lake env lean Physicslib4/AQFT/KMS.lean`: no output. + +## 6. Unexplained `set_option` uses +All six `respectTransparency` declarations (`StandardMinkowskiSpacetime`, +`standardMinkowskiTimeOrientation`, `MinkowskiSpacetime` in Minkowski.lean; `pullbackVal` and its +six siblings in Pullback.lean; one site in CrossMetricIsometry.lean) are committed, so removing the +`set_option` line itself is a guard-protected change to the declaration (confirmed: the guard +blocked a direct test-removal on `StandardMinkowskiSpacetime`, citing "changes the definition of +the existing def"). Tested removal anyway via a throwaway `/tmp` copy with `sed` + `lake env lean` +(never touching the tracked files): with all three Minkowski.lean occurrences stripped, five +distinct elaboration failures appear across the file (`Application type mismatch` between +`TangentSpace 𝓘(ℝ, SpacetimeModel) x` and `SpacetimeModel`/`WithLp 2 (...)`, and a failed +`ContMDiffAt` rewrite) — confirmed load-bearing. Kept every occurrence (R07-3, R13-1, R13-2, +R08-1's 7 sites) and added a one-line explanatory comment above each `set_option` line (a +comment-only edit, not a guard-blocked change): the reason is that `TangentSpace I x` is only +semireducibly, not reducibly, defeq to the model space `SpacetimeModel`, which several +`val`/`symm`/`contMDiff` fields and lemmas need across the transparency boundary. The first +Pullback.lean occurrence and the CrossMetricIsometry.lean occurrence point to the existing +explanation already on `bilinearPrecomp`'s docstring; the other five in Pullback.lean just +cross-reference the first. +For R13-4 (`GNS/Amplification.lean:47`, `linter.unusedDecidableInType false` on +`not_isIrreducible_directSum`): tested removal first with `lake env lean` on the single file +(clean, no output), so removed it — but the whole-project `lake build` then reintroduced the +warning the option was suppressing (`'not_isIrreducible_directSum' does not use ... [DecidableEq +ι]`), i.e. single-file `lake env lean` does not run this linter the same way the full build does. +Restored the `set_option` with a one-line comment explaining the hypothesis is bound by the +ambient `variable` but only used irrelevantly (through `lp.single`/instance resolution). +Re-verified with `lake build Physicslib4.GNS.Amplification`: warning gone, build succeeds. +Set_options removed: 0. Set_options kept with an added comment: 9 (`respectTransparency` × 3 in +Minkowski.lean, × 6 in Pullback.lean [one site also carries `maxHeartbeats`, already commented], +× 1 in CrossMetricIsometry.lean; `linter.unusedDecidableInType false` × 1 in Amplification.lean). + +## Final verification +`lake build` (whole project): succeeds, 3358 jobs. All warnings present are pre-existing and in +files/lines untouched by stage 2 (`GNS/Construction.lean`, `GNS/ExtremeState.lean`, +`GNS/NullSpace.lean`, `GNS/Superselection.lean`, `Operators/Conjugation.lean`, +`Spacetime/CrossMetricIsometry.lean:633/636/638` [pre-existing, not the line I commented], +`Spectral/Basic.lean`, `Spectral/BorelCalculus.lean`, `Spectral/ContinuousCalculus.lean`, +`Spectral/Forms.lean`, `Spectral/OperatorIntegral.lean` [pre-existing, not the lines I edited], +`Spectral/ProjectionValuedMeasure.lean`, `Spectral/Spectrum.lean`, `Analysis/CStarCompletion.lean`) +— none of the eight files edited in this stage (CausalComplement.lean, PureStateExists.lean, +QuasilocalIntertwiner.lean, Net.lean, GNS/Basic.lean, KMS.lean, BorelClasses.lean, DirectSum.lean, +Minkowski.lean, Pullback.lean, Amplification.lean, OperatorIntegral.lean) contribute a new warning. +`lake lint`: "Linting passed for Physicslib4." Stage 2 complete. + +## 7. R03-2 — `@[nolint unusedArguments]` on `L0` (BorelClasses.lean:279) +Added a one-line comment above the attribute: `MeasurableSpace X` is unused in the body (only +`TopologicalSpace X` is needed via `C(X, ℝ)`), kept for parity with the file's other definitions, +and pointed back at the docstring for why the measurability conjunct is dropped. Comment-only +edit (the `def` itself untouched). Verified with +`lake env lean Physicslib4/Spectral/BorelClasses.lean`: no output. + +## 8. R04-2 — unexplained `erw` in DirectSum.lean (lines 262, 294, 467 in the original; shifted by +docstring edits elsewhere in this stage) +Tested `rw` in place of each `erw`, individually, via the Edit tool (these are proof-body edits on +committed `private theorem`s, not statement changes, so not guard-protected): +- `single_mem_range_subSmul` (`apply_single_of_isDirectSumOperator hT i ξ`): `rw` fails with an + `Application type mismatch` between `T.domain` and `(subSmul T c).domain` membership proofs + under the subtype — a genuine semireducible-defeq gap. Kept `erw`, added a one-line comment. +- `sum_single_mem_graph` (`apply_single_of_isDirectSumOperator hT i (η i)` ... `at this`): `rw` + **succeeds** here (no defeq obstruction in this context, unlike the `subSmul`-domain sites). + Replaced `erw` with `rw` — a genuine simplification, no comment needed. +- `summand_mem_range_subSmul` (`apply_of_mem_summand hT m ξ`): same `subSmul`-domain mismatch as + the first site; `rw` fails the same way. Kept `erw`, added a one-line comment. +Verified with `lake env lean Physicslib4/Spectral/Unbounded/DirectSum.lean`: no output (no new +errors/warnings) after all three changes together. diff --git a/notes/agents/stage3-progress.md b/notes/agents/stage3-progress.md new file mode 100644 index 0000000..4e25221 --- /dev/null +++ b/notes/agents/stage3-progress.md @@ -0,0 +1,91 @@ +# Stage 3 progress log + +## Item 1 (R15-3): extract shared diagonal/product estimate in OperatorIntegral.lean + +- File: `Physicslib4/Spectral/OperatorIntegral.lean`. +- Added a private lemma `norm_mul_sub_mul_lt` (placed in the `Physicslib4.Spectral` namespace, + right before `namespace ProjectionValuedMeasure` opens, so it is visible both inside that + namespace and later at the `Spectral` level) capturing the shared pointwise ε/2-splitting + estimate: given bounds `Mf` on `‖f‖`, `Mr` on `‖r‖`, and closeness `‖f - s‖ < ε/(2*Mr)`, + `‖g - r‖ < ε/(2*Mf)`, concludes `‖f*g - s*r‖ < ε`. +- `integral_bddMeasurable_mul`'s `hdiag` block now derives its final estimate from + `norm_mul_sub_mul_lt` (instantiated with `Mf := Mf'+1`, `Mr := Mg'+1`) instead of reproving the + calc chain inline; removed the stray "cannot be cited here" comment and replaced it with an + accurate one explaining the diagonal vs. product-filter distinction. +- `tendstoUniformly_mul_of_bounded`'s final `intro i hi j hj x` branch now calls the same lemma + (`Mf := M_f`, `Mr := M_r`) instead of re-deriving `hterm1`/`hterm2`/`hnorm`/`hsum`. +- Net diff: 46 insertions, 75 deletions (file shrinks by ~29 lines) despite adding a new + documented lemma, since both call sites shrank by much more than the lemma's size. +- Verified: `lake env lean Physicslib4/Spectral/OperatorIntegral.lean` compiles with the same two + pre-existing `unnecessarySimpa` warnings as before the change (at `simpa`s unrelated to this + edit, confirmed by diffing against the pre-edit file via `git stash`). No new warnings or + errors. + +## Item 2 (R01-3): dedupe `nullSubmodule_isClosed` / `lmm2` + +- File: `Physicslib4/GNS/NullSpace.lean`. +- Checked the current state: `lmm2` (lines 163-165) already derives its closedness clause + directly from `nullSubmodule_isClosed`: + `⟨nullSubmodule ω, nullSubmodule_coe ω, nullSubmodule_isClosed ω⟩`. There is no independent + re-derivation of the closedness argument inside `lmm2` to remove; the helper lemma is already + the single source of that proof, matching exactly what R01-3 (and this stage's item 2) asks for. + (R01-3's own verdict was "plausible," not "confirmed," and noted the split is documented as + deliberate in the module docstring.) No edit made; no diagnostics run needed since no code + changed. + +## Item 3 (R07-1): drop unused `_hv` in `reverse_cauchy_schwarz_of_lorentzianAt` + +- SKIPPED. Checked `.git/lean-kit/approvals.json` for an approval covering + `Physicslib4/Spacetime/LorentzCauchySchwarz.lean`; the only "D7"-noted entry present is for + `Physicslib4/GNS/PureStateExists.lean` (a different decision also labelled D7 in this project's + numbering), not for `LorentzCauchySchwarz.lean`. No approval is listed for this file, so per the + task's instruction this item is skipped rather than applied. +- No edit made to `Physicslib4/Spacetime/LorentzCauchySchwarz.lean`; the unused `_hv : B v v < 0` + hypothesis in `reverse_cauchy_schwarz_of_lorentzianAt` (lines 46-93) remains as committed. +- Blueprint node: `lmm:reverse-cauchy-schwarz` in + `blueprint/src/sections/sec10/spacetime.tex:67-86`, tagged + `\lean{Physicslib4.reverse_cauchy_schwarz_of_lorentzianAt, Physicslib4.Spacetime.reverse_cauchy_schwarz}`. + Its statement explicitly requires both vectors timelike: "let $v, w$ be timelike, that is + $g(v,v) < 0$ and $g(w,w) < 0$," so the blueprint's stated hypotheses match the Lean signature + (symmetric in `v`/`w`) even though the proof only needs `w` timelike. +- If the user approves this file later (`lean_guard.py approve --file + Physicslib4/Spacetime/LorentzCauchySchwarz.lean`), the callers that pass the now-unused argument + and would need updating are: `Physicslib4/Spacetime/LorentzCone.lean:73`, + `Physicslib4/Spacetime/LorentzOrthogonal.lean:57,219-220`. + +## Item 3 follow-up: approval granted, change applied + +- Re-checked `.git/lean-kit/approvals.json`: an entry for + `Physicslib4/Spacetime/LorentzCauchySchwarz.lean` (note "D7", expires + 2026-10-14T17:22:03+00:00) is now present. Applied the change. +- `Physicslib4/Spacetime/LorentzCauchySchwarz.lean`: dropped `_hv : B v v < 0` from + `reverse_cauchy_schwarz_of_lorentzianAt`'s signature (now `(hw : B w w < 0)` is the only + vector hypothesis). Updated the theorem's docstring and the module docstring to say only `w` + needs to be timelike, and to note this is stronger than the blueprint's `lmm:reverse-cauchy-schwarz` + (which states both `v` and `w` timelike). + `Spacetime.reverse_cauchy_schwarz` (the blueprint-tagged wrapper, same file) keeps both `hv`/`hw` + unchanged to match the blueprint's symmetric statement; its proof no longer needs `hv` (only used + to forward to the generalized lemma), so `hv` is renamed to `_hv` (binder rename only, not a + statement change) and its docstring notes the hypothesis on `v` is now unused-by-design. The + internal call site was updated to drop the obsolete `hv'` argument. +- Updated callers (grep-confirmed, no others found): + - `Physicslib4/Spacetime/LorentzCone.lean:73` (`reverse_triangle_of_lorentzianAt`'s proof): + dropped the `hv` argument from the call; `v` is still inferred, here from the `have hcs : ...` + type ascription. + - `Physicslib4/Spacetime/LorentzOrthogonal.lean:57` (`nonneg_of_orthogonal_timelike`'s proof): + dropped `ht`'s positional use for the (now-removed) first hypothesis; since the surrounding + `have hcs := ...` has no type ascription, `v` is no longer inferable from context, so added + `(v := t)` explicitly. + - `Physicslib4/Spacetime/LorentzOrthogonal.lean:219-220` (`bilin_neg_of_inner_t_neg`'s proof): + same pattern, both calls updated to `(v := t) hsymm hL hv` / `(v := t) hsymm hL hw`. + - No guard blocks encountered; all three call-site files are not themselves restricted beyond + the normal proof-editing rules (only `LorentzCauchySchwarz.lean`'s statement needed approval, + which was granted). +- Verified: `lake build Physicslib4.Spacetime.LorentzCauchySchwarz`, + `...LorentzCone`, `...LorentzOrthogonal` each build clean individually (first attempt at + `LorentzCone.lean` hit a stale-olean type mismatch from checking against the pre-edit compiled + `LorentzCauchySchwarz.olean`; rebuilding that file first resolved it). Whole-project + `lake build`: 3358/3358 jobs, succeeds, no warnings in any of the three edited files (confirmed + by grepping the build output for the three filenames: no hits). `lake lint`: "Linting passed for + Physicslib4" — in particular no `unusedArguments` warning fired on `Spacetime.reverse_cauchy_schwarz` + despite `_hv` now being unused there, since the binder is prefixed with `_`. diff --git a/notes/agents/stage5-progress.md b/notes/agents/stage5-progress.md new file mode 100644 index 0000000..0724a83 --- /dev/null +++ b/notes/agents/stage5-progress.md @@ -0,0 +1,16 @@ +# Stage 5 progress + +- 5a def:lorentz-transformation: Physicslib4.IsLorentz, Physicslib4.IsProper, Physicslib4.IsOrthochronous (leanDeclaration recorded; status not set, tool unavailable) +- 5a lmm:orthochronous-preserves-future-timelike: Physicslib4.isOrthochronous_pos_time_of_timelike (recorded; status not set) +- 5a lmm:lorentz-closure: Physicslib4.isLorentz_{refl,trans,symm}, Physicslib4.isProper_{refl,trans,symm}, Physicslib4.isOrthochronous_{refl,trans,symm} (recorded; status not set) +- 5a def:inhomogeneous-lorentz-group: Physicslib4.AQFT.HaagKastler.InhomogeneousLorentzGroup (recorded; status not set) +- 5a lmm:precedence-preserving-spacelike: Physicslib4.Spacetime.isSpacelikeRelated_congr, Physicslib4.Spacetime.isCompletelySpacelike_image (recorded; status not set) +- 5a def:lorentz-path: Physicslib4.lorentzPath (recorded; status not set) +- 5a lmm:lorentz-path-causal: Physicslib4.lorentzPath_mfderivWithin, lorentzPath_isCausal, lorentzPath_isFutureOriented, lorentzPath_isPastEndpoint, lorentzPath_isFutureEndpoint (recorded; status not set) +- 5a thrm:causal-precedence-lorentz-invariant: Physicslib4.causalSegmentPrecedes_smul, causallyPrecedes_smul, causallyPrecedes_smul_iff (recorded; status not set) +- 5a crllr:spacelike-lorentz-invariant: Physicslib4.isSpacelikeRelated_smul_iff, isCompletelySpacelike_smul_iff (recorded; status not set) +- 5a review fix: all nine set to proved by coordinator; moved def:lorentz-path, lmm:lorentz-path-causal, thrm:causal-precedence-lorentz-invariant, crllr:spacelike-lorentz-invariant to after lmm:isometry-preserves-geodesics (spacetime.tex 1618-1663); def:lorentz-path cites lmm:pushforward-path and defines the standard time orientation e_0; def:time-orientable added to the statement uses of the four moved nodes. Validate: only the 14 known conv: errors. +- 5b R06-1/R10-2: thrm:invariant-state-gns-unitary extended with IsInvariantState.exists_gns_unitaryRep (\lean edited, leanDeclaration recorded); new lmm:weak-continuity-implies-strong (Physicslib4.strongContinuous_of_weak) and thrm:invariant-state-gns-unitary-strongly-continuous (IsInvariantState.exists_gns_unitary_strongContinuous, IsInvariantState.exists_gns_unitaryRep_strongContinuous), haag-kastler-axioms.tex 1902, 1917 (leanDeclaration recorded; status not set) +- 5b R13-6 curved: new thrm:haag-kastler-net-in-curved-spacetime-nonempty (nonempty_haagKastlerNet_identityComponent, isometricCovariance_trivial_identityComponent, identityComponent_faithfulSMul) and new lmm:local-commutativity-mono-in-curved-spacetime (commute_of_spacelike_mono_geometric), leanDeclaration recorded, status not set; extended def:local-observable (exists_isLocalObservable), thrm:gns-unitary-stabilizer-in-curved-spacetime (exists_gns_unitaryRep_stabilizer), thrm:gns-unitary-stabilizer-strongly-continuous-in-curved-spacetime (exists_gns_unitaryRep_stabilizer_strongContinuous; proof cites lmm:weak-continuity-implies-strong), thrm:von-neumann-geometric-covariance-in-curved-spacetime (lieConj_image_localVonNeumannAlgebra). Validate: only the 14 known conv: errors. +- 5b R13-6: def:quasilocal-observable extended with Physicslib4.AQFT.HaagKastler.exists_isQuasilocalObservable; thrm:irreducible-factor extended with Physicslib4.GNS.isFactor_gnsVonNeumann_of_isIrreducible (\lean edited, leanDeclaration recorded). Validate: only the 14 known conv: errors. +- 5b R13-6 curved review fix: nonempty node states faithfulness for the identity component and its orientation-preserving subgroup (proof: faithfulness passes to subgroups) and that existence holds for every abstract interface spacetime (no existing node for HaagKastlerCurved.nonempty_haagKastlerNet); strongly continuous stabilizer node names the operator-covariance clause, proof uses def:cyclic-vector; lmm:local-commutativity-mono wording fixed. Validate: only the 14 known conv: errors. diff --git a/notes/agents/stage8-progress.md b/notes/agents/stage8-progress.md new file mode 100644 index 0000000..471b36e --- /dev/null +++ b/notes/agents/stage8-progress.md @@ -0,0 +1,8 @@ +# Stage 8 progress: home page refresh + +- Scripts: /tmp/hp/extract.py (HTML + tex + PDF data), /tmp/hp/apply.py (entries, insertion, old->new item renumbering in prose, summary pages/counts). +- All home_page/*.md: entries renumbered, Lean names refreshed from `\lean{}` (10 entries changed their "+N more"), 13 new entries inserted (9 in sec10-5, 2 in sec10-6, 2 in sec10-7); item numbers in prose remapped old->new by label; summary/heading pages and item counts recomputed. +- Pages: /tmp/hp/pages.py recomputed section ranges (TOC heading pages; printed page = PDF page - 2), layer and subsection pages from item heading pages in the PDF, in all section pages, index.md and guide.md. +- Prose: sec10-5 (146 items; sentences for Definitions 336, 339, Lemmas 337-338, 390, 437-440), sec10-6 (162 items; Lemma 604, Theorem 605), sec10-7 (51 items; Theorem 645, Lemma 647, §10.7.1 items 646-647), sec10-2 ("largest by proposition count", since §10.5 and §10.6 are now larger). +- index.md / guide.md: 345 pages, 675 declarations (153 def, 159 thm, 259 lem, 99 prop, 5 cor), 1,188 distinct Lean declarations, 522 provable / 482 with written proof, table rows and pages. +- Checks: 681 blueprint links, 0 bad anchors; check_temporal_wording.py OK. diff --git a/scripts/check_temporal_wording.py b/scripts/check_temporal_wording.py index 1a13771..4ff564a 100755 --- a/scripts/check_temporal_wording.py +++ b/scripts/check_temporal_wording.py @@ -3,7 +3,7 @@ Scans blueprint prose, Lean comments/docstrings, the home page and the README for wording that refers to earlier versions of the project. References to earlier places in the same -text ("as we previously proved") are allowed by the allowlist below. +text ("as we previously proved") are allowed by POSITIONAL below; a line containing "temporal-ok" is skipped. """ import re, sys, pathlib @@ -16,7 +16,17 @@ r"|\(new\)|\bnew (layer|block|subsection|node)\b|[Nn]ewest|\bblog\b" r"|left for later|for the time being|v4\.\d+\.\d+-rc\d|Mathlib \(at" r"|\bthe former Axiom|has been (removed|replaced|split|renamed)" + r"|\b[Nn]o longer\b|\b[Pp]reviously\b" + r"|\b(is|are) now (proved|proven|formali[sz]ed|available|supported|a theorem|in Mathlib)" + r"|\bnow (formali[sz]ed|proven|available|supported)\b" ) +# Positional references to earlier parts of the same text are allowed. +POSITIONAL = re.compile( + r"\b(we|We|as|As|which|that) (have )?previously\b" + r"|\bpreviously (proved|proven|established|found|defined|shown|stated|derived|computed)\b" +) +# A line may opt out of the check with this marker. +ALLOW = "temporal-ok" # Version statements are allowed in exactly these places. VERSION = re.compile(r"\bv4\.\d+\.\d+\b") VERSION_HOMES = {"blueprint/src/sections/sec10/introduction.tex", "home_page/index.md", "README.md"} @@ -31,7 +41,7 @@ def files(): bad = [] for f in files(): for n, line in enumerate(f.read_text(encoding="utf-8", errors="replace").splitlines(), 1): - if PATTERN.search(line): + if PATTERN.search(POSITIONAL.sub("", line)) and ALLOW not in line: bad.append(f"{f}:{n}: temporal wording: {line.strip()[:120]}") if VERSION.search(line) and str(f) not in VERSION_HOMES: bad.append(f"{f}:{n}: version string outside its single home: {line.strip()[:120]}")