feat: Property of time-order w.r.t. superCommute
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8 changed files with 781 additions and 8 deletions
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/-
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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.Algebras.CrAnAlgebra.TimeOrder
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import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.Basic
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/-!
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# Time Ordering on Field operator algebra
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-/
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namespace FieldSpecification
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open CrAnAlgebra
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open HepLean.List
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open FieldStatistic
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namespace FieldOpAlgebra
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variable {𝓕 : FieldSpecification}
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lemma ι_timeOrder_superCommute_time {φ ψ : 𝓕.CrAnStates}
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(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.CrAnAlgebra) :
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ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
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ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b)) := by
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let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
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ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b))
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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) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * ofCrAnList φs) =
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ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a* 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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conv_lhs =>
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [mul_sub, sub_mul, ← ofCrAnList_append]
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rw [timeOrder_ofCrAnList, timeOrder_ofCrAnList]
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have h1 : crAnTimeOrderSign (φs' ++ φ :: ψ :: φs) = crAnTimeOrderSign (φs' ++ ψ :: φ :: φs) := by
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trans crAnTimeOrderSign (φs' ++ [φ, ψ] ++ φs)
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simp
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rw [crAnTimeOrderSign]
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have hp : List.Perm [φ,ψ] [ψ,φ] := by exact List.Perm.swap ψ φ []
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rw [Wick.koszulSign_perm_eq _ _ φ _ _ _ _ _ hp]
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simp
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rfl
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simp_all
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rw [h1]
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simp
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [φ, ψ] φs
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(by simp_all)
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rw [crAnTimeOrderList, show φs' ++ φ :: ψ :: φs = φs' ++ [φ, ψ] ++ φs by simp, h1]
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have h2 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [ψ, φ] φs
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(by simp_all)
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rw [crAnTimeOrderList, show φs' ++ ψ :: φ :: φs = φs' ++ [ψ, φ] ++ φs by simp, h2]
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repeat rw [ofCrAnList_append]
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rw [smul_smul, mul_comm, ← smul_smul, ← smul_sub]
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rw [map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul]
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rw [← mul_smul_comm]
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rw [mul_assoc, mul_assoc, mul_assoc ,mul_assoc ,mul_assoc ,mul_assoc]
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rw [← mul_sub, ← mul_sub, mul_smul_comm, mul_smul_comm, ← smul_mul_assoc,
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← smul_mul_assoc]
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rw [← sub_mul]
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have h1 : (ι (ofCrAnList [φ, ψ]) - (exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnList [ψ, φ])) =
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ι [ofCrAnState φ, ofCrAnState ψ]ₛca := by
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rw [superCommute_ofCrAnState_ofCrAnState]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_append]
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simp only [instCommGroup.eq_1, List.singleton_append, Algebra.smul_mul_assoc, map_sub,
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map_smul]
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rw [← ofCrAnList_append]
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simp
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rw [h1]
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have hc : ι ((superCommute (ofCrAnState φ)) (ofCrAnState ψ)) ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
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apply ι_superCommute_ofCrAnState_ofCrAnState_mem_center
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rw [Subalgebra.mem_center_iff] at hc
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repeat rw [← mul_assoc]
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rw [hc]
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repeat rw [mul_assoc]
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rw [smul_mul_assoc]
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rw [← map_mul, ← map_mul, ← map_mul, ← map_mul]
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rw [← ofCrAnList_append, ← ofCrAnList_append, ← ofCrAnList_append, ← ofCrAnList_append]
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have h1 := insertionSort_of_takeWhile_filter 𝓕.crAnTimeOrderRel φ φs' φs
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simp at h1 ⊢
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rw [← h1]
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rw [← crAnTimeOrderList]
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by_cases hq : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)
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· rw [ι_superCommute_of_diff_statistic hq]
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simp
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· rw [crAnTimeOrderSign, Wick.koszulSign_eq_rel_eq_stat _ _, ← crAnTimeOrderSign]
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rw [timeOrder_ofCrAnList]
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simp
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exact hψφ
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exact hφψ
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simpa using hq
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· simp [pa]
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· intro x y hx hy hpx hpy
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simp_all [pa,mul_add, add_mul]
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· intro x hx hpx
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simp_all [pa, hpx]
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· simp [pb]
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· intro x y hx hy hpx hpy
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simp_all [pb,mul_add, add_mul]
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· intro x hx hpx
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simp_all [pb, hpx]
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/-!
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## Defining normal order for `FiedOpAlgebra`.
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-/
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lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
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(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓣ᶠ(a) = 0 := by
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at h
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let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓣ᶠ(a) = 0
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change p a h
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apply AddSubgroup.closure_induction
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· intro x hx
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obtain ⟨a, ha, b, hb, rfl⟩ := Set.mem_mul.mp hx
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obtain ⟨a, ha, c, hc, rfl⟩ := ha
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simp only [p]
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simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq] at hc
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match hc with
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| Or.inl hc =>
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obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
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sorry
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| Or.inr (Or.inl hc) =>
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obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
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sorry
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| Or.inr (Or.inr (Or.inl hc)) =>
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obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
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sorry
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| Or.inr (Or.inr (Or.inr hc)) =>
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obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
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sorry
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· simp [p]
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· intro x y hx hy
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simp only [map_add, p]
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intro h1 h2
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simp [h1, h2]
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· intro x hx
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simp [p]
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lemma ι_timeOrder_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
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ι 𝓣ᶠ(a) = ι 𝓣ᶠ(b) := by
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rw [equiv_iff_sub_mem_ideal] at h
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rw [LinearMap.sub_mem_ker_iff.mp]
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simp only [LinearMap.mem_ker, ← map_sub]
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exact ι_timeOrder_zero_of_mem_ideal (a - b) h
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/-- Normal ordering on `FieldOpAlgebra`. -/
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noncomputable def timeOrder : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
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toFun := Quotient.lift (ι.toLinearMap ∘ₗ CrAnAlgebra.timeOrder) ι_timeOrder_eq_of_equiv
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map_add' x y := by
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obtain ⟨x, hx⟩ := ι_surjective x
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obtain ⟨y, hy⟩ := ι_surjective y
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subst hx hy
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rw [← map_add, ι_apply, ι_apply, ι_apply]
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rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
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simp
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map_smul' c y := by
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obtain ⟨y, hy⟩ := ι_surjective y
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subst hy
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rw [← map_smul, ι_apply, ι_apply]
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simp
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end FieldOpAlgebra
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end FieldSpecification
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