refactor: move algebra files

HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/TimeContraction.lean

@@ -1,178 +0,0 @@
-/-
-Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Joseph Tooby-Smith
--/
-import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.NormalOrder
-import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.TimeOrder
-/-!
-
-# Time contractions
-
-We define the state algebra of a field structure to be the free algebra
-generated by the states.
-
--/
-
-namespace FieldSpecification
-variable {𝓕 : FieldSpecification}
-open FieldOpFreeAlgebra
-noncomputable section
-
-namespace FieldOpAlgebra
-
-open FieldStatistic
-
-/-- The time contraction of two FieldOp as an element of `𝓞.A` defined
-  as their time ordering in the state algebra minus their normal ordering in the
-  creation and annihlation algebra, both mapped to `𝓞.A`.. -/
-def timeContract (φ ψ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra :=
-    𝓣(ofFieldOp φ * ofFieldOp ψ) - 𝓝(ofFieldOp φ * ofFieldOp ψ)
-
-lemma timeContract_eq_smul (φ ψ : 𝓕.FieldOp) : timeContract φ ψ =
-    𝓣(ofFieldOp φ * ofFieldOp ψ) + (-1 : ℂ) • 𝓝(ofFieldOp φ * ofFieldOp ψ) := by rfl
-
-lemma timeContract_of_timeOrderRel (φ ψ : 𝓕.FieldOp) (h : timeOrderRel φ ψ) :
-    timeContract φ ψ = [anPart φ, ofFieldOp ψ]ₛ := by
-  conv_rhs =>
-    rw [ofFieldOp_eq_crPart_add_anPart]
-    rw [map_add, superCommute_anPart_anPart, superCommute_anPart_crPart]
-  simp only [timeContract, instCommGroup.eq_1, Algebra.smul_mul_assoc, add_zero]
-  rw [timeOrder_ofFieldOp_ofFieldOp_ordered h]
-  rw [normalOrder_ofFieldOp_mul_ofFieldOp]
-  simp only [instCommGroup.eq_1]
-  rw [ofFieldOp_eq_crPart_add_anPart, ofFieldOp_eq_crPart_add_anPart]
-  simp only [mul_add, add_mul]
-  abel_nf
-
-lemma timeContract_of_not_timeOrderRel (φ ψ : 𝓕.FieldOp) (h : ¬ timeOrderRel φ ψ) :
-    timeContract φ ψ = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • timeContract ψ φ := by
-  rw [timeContract_eq_smul]
-  simp only [Int.reduceNeg, one_smul, map_add]
-  rw [normalOrder_ofFieldOp_ofFieldOp_swap]
-  rw [timeOrder_ofFieldOp_ofFieldOp_not_ordered_eq_timeOrder h]
-  rw [timeContract_eq_smul]
-  simp only [instCommGroup.eq_1, map_smul, map_add, smul_add]
-  rw [smul_smul, smul_smul, mul_comm]
-
-lemma timeContract_of_not_timeOrderRel_expand (φ ψ : 𝓕.FieldOp) (h : ¬ timeOrderRel φ ψ) :
-    timeContract φ ψ = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • [anPart ψ, ofFieldOp φ]ₛ := by
-  rw [timeContract_of_not_timeOrderRel _ _ h]
-  rw [timeContract_of_timeOrderRel _ _ _]
-  have h1 := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ
-  simp_all
-
-lemma timeContract_mem_center (φ ψ : 𝓕.FieldOp) :
-    timeContract φ ψ ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
-  by_cases h : timeOrderRel φ ψ
-  · rw [timeContract_of_timeOrderRel _ _ h]
-    exact superCommute_anPart_ofFieldOp_mem_center φ ψ
-  · rw [timeContract_of_not_timeOrderRel _ _ h]
-    refine Subalgebra.smul_mem (Subalgebra.center ℂ _) ?_ 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ)
-    rw [timeContract_of_timeOrderRel]
-    exact superCommute_anPart_ofFieldOp_mem_center _ _
-    have h1 := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ
-    simp_all
-
-lemma timeContract_zero_of_diff_grade (φ ψ : 𝓕.FieldOp) (h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) :
-    timeContract φ ψ = 0 := by
-  by_cases h1 : timeOrderRel φ ψ
-  · rw [timeContract_of_timeOrderRel _ _ h1]
-    rw [superCommute_anPart_ofFieldOpF_diff_grade_zero]
-    exact h
-  · rw [timeContract_of_not_timeOrderRel _ _ h1]
-    rw [timeContract_of_timeOrderRel _ _ _]
-    rw [superCommute_anPart_ofFieldOpF_diff_grade_zero]
-    simp only [instCommGroup.eq_1, smul_zero]
-    exact h.symm
-    have ht := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ
-    simp_all
-
-lemma normalOrder_timeContract (φ ψ : 𝓕.FieldOp) :
-    𝓝(timeContract φ ψ) = 0 := by
-  by_cases h : timeOrderRel φ ψ
-  · rw [timeContract_of_timeOrderRel _ _ h]
-    simp
-  · rw [timeContract_of_not_timeOrderRel _ _ h]
-    simp only [instCommGroup.eq_1, map_smul, smul_eq_zero]
-    have h1 : timeOrderRel ψ φ := by
-      have ht : timeOrderRel φ ψ ∨ timeOrderRel ψ φ := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ
-      simp_all
-    rw [timeContract_of_timeOrderRel _ _ h1]
-    simp
-
-lemma timeOrder_timeContract_eq_time_mid {φ ψ : 𝓕.FieldOp}
-    (h1 : timeOrderRel φ ψ) (h2 : timeOrderRel ψ φ) (a b : 𝓕.FieldOpAlgebra) :
-    𝓣(a * timeContract φ ψ * b) = timeContract φ ψ * 𝓣(a * b) := by
-  rw [timeContract_of_timeOrderRel _ _ h1]
-  rw [ofFieldOp_eq_sum]
-  simp only [map_sum, Finset.mul_sum, Finset.sum_mul]
-  congr
-  funext x
-  match φ with
-  | .inAsymp φ =>
-    simp
-  | .position φ =>
-    simp only [anPart_position, instCommGroup.eq_1]
-    apply timeOrder_superCommute_eq_time_mid _ _
-    simp only [crAnTimeOrderRel, h1]
-    simp [crAnTimeOrderRel, h2]
-  | .outAsymp φ =>
-    simp only [anPart_posAsymp, instCommGroup.eq_1]
-    apply timeOrder_superCommute_eq_time_mid _ _
-    simp only [crAnTimeOrderRel, h1]
-    simp [crAnTimeOrderRel, h2]
-
-lemma timeOrder_timeContract_eq_time_left {φ ψ : 𝓕.FieldOp}
-    (h1 : timeOrderRel φ ψ) (h2 : timeOrderRel ψ φ) (b : 𝓕.FieldOpAlgebra) :
-    𝓣(timeContract φ ψ * b) = timeContract φ ψ * 𝓣(b) := by
-  trans 𝓣(1 * timeContract φ ψ * b)
-  simp only [one_mul]
-  rw [timeOrder_timeContract_eq_time_mid h1 h2]
-  simp
-
-lemma timeOrder_timeContract_neq_time {φ ψ : 𝓕.FieldOp}
-    (h1 : ¬ (timeOrderRel φ ψ ∧ timeOrderRel ψ φ)) :
-    𝓣(timeContract φ ψ) = 0 := by
-  by_cases h2 : timeOrderRel φ ψ
-  · simp_all only [true_and]
-    rw [timeContract_of_timeOrderRel _ _ h2]
-    simp only
-    rw [ofFieldOp_eq_sum]
-    simp only [map_sum]
-    apply Finset.sum_eq_zero
-    intro x hx
-    match φ with
-    | .inAsymp φ =>
-      simp
-    | .position φ =>
-      simp only [anPart_position, instCommGroup.eq_1]
-      apply timeOrder_superCommute_neq_time
-      simp_all [crAnTimeOrderRel]
-    | .outAsymp φ =>
-      simp only [anPart_posAsymp, instCommGroup.eq_1]
-      apply timeOrder_superCommute_neq_time
-      simp_all [crAnTimeOrderRel]
-  · rw [timeContract_of_not_timeOrderRel_expand _ _ h2]
-    simp only [instCommGroup.eq_1, map_smul, smul_eq_zero]
-    right
-    rw [ofFieldOp_eq_sum]
-    simp only [map_sum]
-    apply Finset.sum_eq_zero
-    intro x hx
-    match ψ with
-    | .inAsymp ψ =>
-      simp
-    | .position ψ =>
-      simp only [anPart_position, instCommGroup.eq_1]
-      apply timeOrder_superCommute_neq_time
-      simp_all [crAnTimeOrderRel]
-    | .outAsymp ψ =>
-      simp only [anPart_posAsymp, instCommGroup.eq_1]
-      apply timeOrder_superCommute_neq_time
-      simp_all [crAnTimeOrderRel]
-
-end FieldOpAlgebra
-
-end
-end FieldSpecification