361 lines
15 KiB
Text
361 lines
15 KiB
Text
/-
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Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.FieldSpecification.TimeOrder
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import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.SuperCommute
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import HepLean.PerturbationTheory.Koszul.KoszulSign
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/-!
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# Time Ordering in the CrAnAlgebra
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-/
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namespace FieldSpecification
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variable {𝓕 : FieldSpecification}
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open FieldStatistic
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namespace CrAnAlgebra
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noncomputable section
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open HepLean.List
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/-!
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## Time order
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-/
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/-- Time ordering for the `CrAnAlgebra`. -/
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def timeOrder : CrAnAlgebra 𝓕 →ₗ[ℂ] CrAnAlgebra 𝓕 :=
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Basis.constr ofCrAnListBasis ℂ fun φs =>
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crAnTimeOrderSign φs • ofCrAnList (crAnTimeOrderList φs)
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@[inherit_doc timeOrder]
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scoped[FieldSpecification.CrAnAlgebra] notation "𝓣ᶠ(" a ")" => timeOrder a
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lemma timeOrder_ofCrAnList (φs : List 𝓕.CrAnStates) :
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𝓣ᶠ(ofCrAnList φs) = crAnTimeOrderSign φs • ofCrAnList (crAnTimeOrderList φs) := by
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rw [← ofListBasis_eq_ofList]
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simp only [timeOrder, Basis.constr_basis]
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lemma timeOrder_timeOrder_mid (a b c : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c) := by
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let pc (c : 𝓕.CrAnAlgebra) (hc : c ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := 𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c)
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change pc c (Basis.mem_span _ c)
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apply Submodule.span_induction
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· intro x hx
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obtain ⟨φs, rfl⟩ := hx
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simp [pc]
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let pb (b : 𝓕.CrAnAlgebra) (hb : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := 𝓣ᶠ(a * b * ofCrAnList φs) = 𝓣ᶠ(a * 𝓣ᶠ(b) * ofCrAnList φs)
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change pb b (Basis.mem_span _ b)
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apply Submodule.span_induction
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· intro x hx
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obtain ⟨φs', rfl⟩ := hx
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simp [pb]
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let pa (a : 𝓕.CrAnAlgebra) (ha : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := 𝓣ᶠ(a * ofCrAnList φs' * ofCrAnList φs) = 𝓣ᶠ(a * 𝓣ᶠ(ofCrAnList φs') * ofCrAnList φs)
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change pa a (Basis.mem_span _ a)
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apply Submodule.span_induction
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· intro x hx
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obtain ⟨φs'', rfl⟩ := hx
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simp [pa]
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rw [timeOrder_ofCrAnList]
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simp only [← ofCrAnList_append, Algebra.mul_smul_comm,
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Algebra.smul_mul_assoc, map_smul]
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rw [timeOrder_ofCrAnList, timeOrder_ofCrAnList, smul_smul]
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congr 1
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· simp only [crAnTimeOrderSign, crAnTimeOrderList]
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rw [Wick.koszulSign_of_append_eq_insertionSort, mul_comm]
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· congr 1
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simp only [crAnTimeOrderList]
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rw [insertionSort_append_insertionSort_append]
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· simp [pa]
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· intro x y hx hy h1 h2
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simp_all [pa, add_mul]
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· intro x hx h
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simp_all [pa]
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· simp [pb]
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· intro x y hx hy h1 h2
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simp_all [pb, mul_add, add_mul]
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· intro x hx h
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simp_all [pb]
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· simp [pc]
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· intro x y hx hy h1 h2
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simp_all [pc, mul_add]
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· intro x hx h hp
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simp_all [pc]
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lemma timeOrder_timeOrder_right (a b : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(a * 𝓣ᶠ(b)) := by
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trans 𝓣ᶠ(a * b * 1)
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· simp
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· rw [timeOrder_timeOrder_mid]
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simp
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lemma timeOrder_timeOrder_left (a b : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(𝓣ᶠ(a) * b) := by
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trans 𝓣ᶠ(1 * a * b)
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· simp
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· rw [timeOrder_timeOrder_mid]
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simp
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lemma timeOrder_ofStateList (φs : List 𝓕.States) :
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𝓣ᶠ(ofStateList φs) = timeOrderSign φs • ofStateList (timeOrderList φs) := by
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conv_lhs =>
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rw [ofStateList_sum, map_sum]
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enter [2, x]
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rw [timeOrder_ofCrAnList]
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simp only [crAnTimeOrderSign_crAnSection]
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rw [← Finset.smul_sum]
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congr
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rw [ofStateList_sum, sum_crAnSections_timeOrder]
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rfl
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lemma timeOrder_ofStateList_nil : timeOrder (𝓕 := 𝓕) (ofStateList []) = 1 := by
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rw [timeOrder_ofStateList]
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simp [timeOrderSign, Wick.koszulSign, timeOrderList]
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@[simp]
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lemma timeOrder_ofStateList_singleton (φ : 𝓕.States) : 𝓣ᶠ(ofStateList [φ]) = ofStateList [φ] := by
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simp [timeOrder_ofStateList, timeOrderSign, timeOrderList]
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lemma timeOrder_ofState_ofState_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
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𝓣ᶠ(ofState φ * ofState ψ) = ofState φ * ofState ψ := by
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rw [← ofStateList_singleton, ← ofStateList_singleton, ← ofStateList_append, timeOrder_ofStateList]
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simp only [List.singleton_append]
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rw [timeOrderSign_pair_ordered h, timeOrderList_pair_ordered h]
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simp
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lemma timeOrder_ofState_ofState_not_ordered {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
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𝓣ᶠ(ofState φ * ofState ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • ofState ψ * ofState φ := by
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rw [← ofStateList_singleton, ← ofStateList_singleton,
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← ofStateList_append, timeOrder_ofStateList]
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simp only [List.singleton_append, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [timeOrderSign_pair_not_ordered h, timeOrderList_pair_not_ordered h]
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simp [← ofStateList_append]
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lemma timeOrder_ofState_ofState_not_ordered_eq_timeOrder {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
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𝓣ᶠ(ofState φ * ofState ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓣ᶠ(ofState ψ * ofState φ) := by
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rw [timeOrder_ofState_ofState_not_ordered h]
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rw [timeOrder_ofState_ofState_ordered]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc]
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have hx := IsTotal.total (r := timeOrderRel) ψ φ
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simp_all
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) :
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𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca) = 0 := by
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rw [superCommute_ofCrAnState_ofCrAnState]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton,
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← ofCrAnList_append, ← ofCrAnList_append, timeOrder_ofCrAnList, timeOrder_ofCrAnList]
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simp only [List.singleton_append]
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rw [crAnTimeOrderSign_pair_not_ordered h, crAnTimeOrderList_pair_not_ordered h]
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rw [sub_eq_zero, smul_smul]
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have h1 := IsTotal.total (r := crAnTimeOrderRel) φ ψ
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congr
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· rw [crAnTimeOrderSign_pair_ordered, exchangeSign_symm]
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simp only [instCommGroup.eq_1, mul_one]
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simp_all
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· rw [crAnTimeOrderList_pair_ordered]
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simp_all
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
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𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca) = 0 := by
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rw [timeOrder_timeOrder_right,
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timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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simp
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
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𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca * a) = 0 := by
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rw [timeOrder_timeOrder_left,
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timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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simp
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_mid
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a b : 𝓕.CrAnAlgebra) :
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𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
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rw [timeOrder_timeOrder_mid,
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timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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simp
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lemma timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel
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{φ1 φ2 : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.CrAnAlgebra) :
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𝓣ᶠ([a, [ofCrAnState φ1, ofCrAnState φ2]ₛca]ₛca) = 0 := by
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rw [← bosonicProj_add_fermionicProj a]
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simp
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rw [bosonic_superCommute (Submodule.coe_mem (bosonicProj a))]
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simp
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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simp
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rcases superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h' | h'
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· rw [superCommute_bonsonic h']
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simp [ofCrAnList_singleton]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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simp
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· rw [superCommute_fermionic_fermionic (Submodule.coe_mem (fermionicProj a)) h']
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simp [ofCrAnList_singleton]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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simp
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lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel
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{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ1 φ2)
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(h13 : ¬ crAnTimeOrderRel φ1 φ3) :
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𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rw [summerCommute_jacobi_ofCrAnList]
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simp [ofCrAnList_singleton]
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right
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h12]
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rw [superCommute_ofCrAnState_ofCrAnState_symm φ3]
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simp
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h13]
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simp
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lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel'
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{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ2 φ1)
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(h13 : ¬ crAnTimeOrderRel φ3 φ1) :
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𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rw [summerCommute_jacobi_ofCrAnList]
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simp [ofCrAnList_singleton]
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right
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rw [superCommute_ofCrAnState_ofCrAnState_symm φ1]
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simp
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h12]
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simp
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h13]
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simp
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lemma timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel
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(φ1 φ2 φ3 : 𝓕.CrAnStates) (h : ¬
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(crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
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crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
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crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2)) :
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𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
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simp at h
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by_cases h23 : ¬ crAnTimeOrderRel φ2 φ3
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· simp_all
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h23]
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simp_all
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by_cases h32 : ¬ crAnTimeOrderRel φ3 φ2
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· simp_all
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rw [superCommute_ofCrAnState_ofCrAnState_symm]
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simp
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h32]
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simp
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simp_all
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by_cases h12 : ¬ crAnTimeOrderRel φ1 φ2
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· have h13 : ¬ crAnTimeOrderRel φ1 φ3 := by
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intro h13
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apply h12
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exact IsTrans.trans φ1 φ3 φ2 h13 h32
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rw [timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel h12 h13]
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simp_all
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have h13 : crAnTimeOrderRel φ1 φ3 := IsTrans.trans φ1 φ2 φ3 h12 h23
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simp_all
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by_cases h21 : ¬ crAnTimeOrderRel φ2 φ1
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· simp_all
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have h31 : ¬ crAnTimeOrderRel φ3 φ1 := by
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intro h31
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apply h21
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exact IsTrans.trans φ2 φ3 φ1 h23 h31
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rw [timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel' h21 h31]
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simp_all
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refine False.elim (h ?_)
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exact IsTrans.trans φ3 φ2 φ1 h32 h21
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_eq_time
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{φ ψ : 𝓕.CrAnStates} (h1 : crAnTimeOrderRel φ ψ) (h2 : crAnTimeOrderRel ψ φ) :
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𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca) = [ofCrAnState φ, ofCrAnState ψ]ₛca := by
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rw [superCommute_ofCrAnState_ofCrAnState]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton,
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← ofCrAnList_append, ← ofCrAnList_append, timeOrder_ofCrAnList, timeOrder_ofCrAnList]
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simp only [List.singleton_append]
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rw [crAnTimeOrderSign_pair_ordered h1, crAnTimeOrderList_pair_ordered h1,
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crAnTimeOrderSign_pair_ordered h2, crAnTimeOrderList_pair_ordered h2]
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simp
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/-!
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## Interaction with maxTimeField
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-/
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/-- In the state algebra time, ordering obeys `T(φ₀φ₁…φₙ) = s * φᵢ * T(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)`
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where `φᵢ` is the state
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which has maximum time and `s` is the exchange sign of `φᵢ` and `φ₀φ₁…φᵢ₋₁`. -/
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lemma timeOrder_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States) :
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𝓣ᶠ(ofStateList (φ :: φs)) =
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𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ (φ :: φs).take (maxTimeFieldPos φ φs)) •
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ofState (maxTimeField φ φs) * 𝓣ᶠ(ofStateList (eraseMaxTimeField φ φs)) := by
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rw [timeOrder_ofStateList, timeOrderList_eq_maxTimeField_timeOrderList]
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rw [ofStateList_cons, timeOrder_ofStateList]
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simp only [instCommGroup.eq_1, Algebra.mul_smul_comm, Algebra.smul_mul_assoc, smul_smul]
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congr
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rw [timerOrderSign_of_eraseMaxTimeField, mul_assoc]
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simp
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/-- In the state algebra time, ordering obeys `T(φ₀φ₁…φₙ) = s * φᵢ * T(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)`
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where `φᵢ` is the state
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which has maximum time and `s` is the exchange sign of `φᵢ` and `φ₀φ₁…φᵢ₋₁`.
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Here `s` is written using finite sets. -/
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lemma timeOrder_eq_maxTimeField_mul_finset (φ : 𝓕.States) (φs : List 𝓕.States) :
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𝓣ᶠ(ofStateList (φ :: φs)) = 𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ ⟨(eraseMaxTimeField φ φs).get,
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(Finset.filter (fun x =>
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(maxTimeFieldPosFin φ φs).succAbove x < maxTimeFieldPosFin φ φs) Finset.univ)⟩) •
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ofState (maxTimeField φ φs) * 𝓣ᶠ(ofStateList (eraseMaxTimeField φ φs)) := by
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rw [timeOrder_eq_maxTimeField_mul]
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congr 3
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apply FieldStatistic.ofList_perm
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nth_rewrite 1 [← List.finRange_map_get (φ :: φs)]
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simp only [List.length_cons, eraseMaxTimeField, insertionSortDropMinPos]
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rw [eraseIdx_get, ← List.map_take, ← List.map_map]
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refine List.Perm.map (φ :: φs).get ?_
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apply (List.perm_ext_iff_of_nodup _ _).mpr
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· intro i
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simp only [List.length_cons, maxTimeFieldPos, mem_take_finrange, Fin.val_fin_lt, List.mem_map,
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Finset.mem_sort, Finset.mem_filter, Finset.mem_univ, true_and, Function.comp_apply]
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refine Iff.intro (fun hi => ?_) (fun h => ?_)
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· have h2 := (maxTimeFieldPosFin φ φs).2
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simp only [eraseMaxTimeField, insertionSortDropMinPos, List.length_cons, Nat.succ_eq_add_one,
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maxTimeFieldPosFin, insertionSortMinPosFin] at h2
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use ⟨i, by omega⟩
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apply And.intro
|
||
· simp only [Fin.succAbove, List.length_cons, Fin.castSucc_mk, maxTimeFieldPosFin,
|
||
insertionSortMinPosFin, Nat.succ_eq_add_one, Fin.mk_lt_mk, Fin.val_fin_lt, Fin.succ_mk]
|
||
rw [Fin.lt_def]
|
||
split
|
||
· simp only [Fin.val_fin_lt]
|
||
omega
|
||
· omega
|
||
· simp only [Fin.succAbove, List.length_cons, Fin.castSucc_mk, Fin.succ_mk, Fin.ext_iff,
|
||
Fin.coe_cast]
|
||
split
|
||
· simp
|
||
· simp_all [Fin.lt_def]
|
||
· obtain ⟨j, h1, h2⟩ := h
|
||
subst h2
|
||
simp only [Fin.lt_def, Fin.coe_cast]
|
||
exact h1
|
||
· exact List.Sublist.nodup (List.take_sublist _ _) <|
|
||
List.nodup_finRange (φs.length + 1)
|
||
· refine List.Nodup.map ?_ ?_
|
||
· refine Function.Injective.comp ?hf.hg Fin.succAbove_right_injective
|
||
exact Fin.cast_injective (eraseIdx_length (φ :: φs) (insertionSortMinPos timeOrderRel φ φs))
|
||
· exact Finset.sort_nodup (fun x1 x2 => x1 ≤ x2)
|
||
(Finset.filter (fun x => (maxTimeFieldPosFin φ φs).succAbove x < maxTimeFieldPosFin φ φs)
|
||
Finset.univ)
|
||
|
||
end
|
||
|
||
end CrAnAlgebra
|
||
|
||
end FieldSpecification
|