refactor: Remove ProtoOperatorAlgebra

HepLean/PerturbationTheory/WickContraction/TimeContract.lean

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Tooby-Smith
 -/
 import HepLean.PerturbationTheory.WickContraction.Sign
-import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.TimeContraction
+import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.TimeContraction
 /-!
 
 # Time contractions
@@ -17,16 +17,16 @@ variable {𝓕 : FieldSpecification}
 namespace WickContraction
 variable {n : ℕ} (c : WickContraction n)
 open HepLean.List
-
+open FieldOpAlgebra
 /-- Given a Wick contraction `φsΛ` associated with a list `φs`, the
   product of all time-contractions of pairs of contracted elements in `φs`,
   as a member of the center of `𝓞.A`. -/
-noncomputable def timeContract (𝓞 : 𝓕.ProtoOperatorAlgebra) {φs : List 𝓕.States}
+noncomputable def timeContract {φs : List 𝓕.States}
     (φsΛ : WickContraction φs.length) :
-    Subalgebra.center ℂ 𝓞.A :=
-  ∏ (a : φsΛ.1), ⟨𝓞.timeContract
+    Subalgebra.center ℂ 𝓕.FieldOpAlgebra :=
+  ∏ (a : φsΛ.1), ⟨FieldOpAlgebra.timeContract
     (φs.get (φsΛ.fstFieldOfContract a)) (φs.get (φsΛ.sndFieldOfContract a)),
-    𝓞.timeContract_mem_center _ _⟩
+    timeContract_mem_center _ _⟩
 
 /-- For `φsΛ` a Wick contraction for `φs`, the time contraction `(φsΛ ↩Λ φ i none).timeContract 𝓞`
   is equal to `φsΛ.timeContract 𝓞`.
@@ -34,10 +34,10 @@ noncomputable def timeContract (𝓞 : 𝓕.ProtoOperatorAlgebra) {φs : List
 This result follows from the fact that `timeContract` only depends on contracted pairs,
 and `(φsΛ ↩Λ φ i none)` has the 'same' contracted pairs as `φsΛ`. -/
 @[simp]
-lemma timeContract_insertAndContract_none (𝓞 : 𝓕.ProtoOperatorAlgebra)
+lemma timeContract_insertAndContract_none
     (φ : 𝓕.States) (φs : List 𝓕.States)
     (φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) :
-    (φsΛ ↩Λ φ i none).timeContract 𝓞 = φsΛ.timeContract 𝓞 := by
+    (φsΛ ↩Λ φ i none).timeContract = φsΛ.timeContract  := by
   rw [timeContract, insertAndContract_none_prod_contractions]
   congr
   ext a
@@ -51,13 +51,13 @@ lemma timeContract_insertAndContract_none (𝓞 : 𝓕.ProtoOperatorAlgebra)
 This follows from the fact that `(φsΛ ↩Λ φ i (some j))` has one more contracted pair than `φsΛ`,
 corresponding to `φ` contracted with `φⱼ`. The order depends on whether we insert `φ` before
 or after `φⱼ`. -/
-lemma timeConract_insertAndContract_some (𝓞 : 𝓕.ProtoOperatorAlgebra)
+lemma timeConract_insertAndContract_some
     (φ : 𝓕.States) (φs : List 𝓕.States)
     (φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
-    (φsΛ ↩Λ φ i (some j)).timeContract 𝓞 =
+    (φsΛ ↩Λ φ i (some j)).timeContract =
     (if i < i.succAbove j then
-      ⟨𝓞.timeContract φ φs[j.1], 𝓞.timeContract_mem_center _ _⟩
-    else ⟨𝓞.timeContract φs[j.1] φ, 𝓞.timeContract_mem_center _ _⟩) * φsΛ.timeContract 𝓞 := by
+      ⟨FieldOpAlgebra.timeContract φ φs[j.1], timeContract_mem_center _ _⟩
+    else ⟨FieldOpAlgebra.timeContract φs[j.1] φ, timeContract_mem_center _ _⟩) * φsΛ.timeContract := by
   rw [timeContract, insertAndContract_some_prod_contractions]
   congr 1
   · simp only [Nat.succ_eq_add_one, insertAndContract_fstFieldOfContract_some_incl, finCongr_apply,
@@ -72,27 +72,25 @@ lemma timeConract_insertAndContract_some (𝓞 : 𝓕.ProtoOperatorAlgebra)
 open FieldStatistic
 
 lemma timeConract_insertAndContract_some_eq_mul_contractStateAtIndex_lt
-    (𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
+    (φ : 𝓕.States) (φs : List 𝓕.States)
     (φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (k : φsΛ.uncontracted)
     (ht : 𝓕.timeOrderRel φ φs[k.1]) (hik : i < i.succAbove k) :
-    (φsΛ ↩Λ φ i (some k)).timeContract 𝓞 =
+    (φsΛ ↩Λ φ i (some k)).timeContract  =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (φsΛ.uncontracted.filter (fun x => x < k))⟩)
-    • (𝓞.contractStateAtIndex φ [φsΛ]ᵘᶜ ((uncontractedStatesEquiv φs φsΛ) (some k)) *
-      φsΛ.timeContract 𝓞) := by
+    • (contractStateAtIndex φ [φsΛ]ᵘᶜ ((uncontractedStatesEquiv φs φsΛ) (some k)) *
+      φsΛ.timeContract ) := by
   rw [timeConract_insertAndContract_some]
   simp only [Nat.succ_eq_add_one, Fin.getElem_fin, ite_mul, instCommGroup.eq_1,
-    ProtoOperatorAlgebra.contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
+    contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
     Equiv.coe_trans, Option.map_some', Function.comp_apply, finCongr_apply, Fin.coe_cast,
     List.getElem_map, uncontractedList_getElem_uncontractedIndexEquiv_symm, List.get_eq_getElem,
     Algebra.smul_mul_assoc, uncontractedListGet]
   · simp only [hik, ↓reduceIte, MulMemClass.coe_mul]
-    rw [𝓞.timeContract_of_timeOrderRel]
-    trans (1 : ℂ) • (𝓞.crAnF ((CrAnAlgebra.superCommuteF
-      (CrAnAlgebra.anPartF φ)) (CrAnAlgebra.ofState φs[k.1])) *
-      ↑(timeContract 𝓞 φsΛ))
+    rw [timeContract_of_timeOrderRel]
+    trans (1 : ℂ) • ((superCommute (anPart φ)) (ofFieldOp φs[k.1]) * ↑φsΛ.timeContract )
     · simp
     simp only [smul_smul]
-    congr
+    congr 1
     have h1 : ofList 𝓕.statesStatistic (List.take (↑(φsΛ.uncontractedIndexEquiv.symm k))
         (List.map φs.get φsΛ.uncontractedList))
         = (𝓕 |>ₛ ⟨φs.get, (Finset.filter (fun x => x < k) φsΛ.uncontracted)⟩) := by
@@ -107,21 +105,21 @@ lemma timeConract_insertAndContract_some_eq_mul_contractStateAtIndex_lt
     · exact ht
 
 lemma timeConract_insertAndContract_some_eq_mul_contractStateAtIndex_not_lt
-    (𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
+    (φ : 𝓕.States) (φs : List 𝓕.States)
     (φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (k : φsΛ.uncontracted)
     (ht : ¬ 𝓕.timeOrderRel φs[k.1] φ) (hik : ¬ i < i.succAbove k) :
-    (φsΛ ↩Λ φ i (some k)).timeContract 𝓞 =
+    (φsΛ ↩Λ φ i (some k)).timeContract =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (φsΛ.uncontracted.filter (fun x => x ≤ k))⟩)
-    • (𝓞.contractStateAtIndex φ [φsΛ]ᵘᶜ
-      ((uncontractedStatesEquiv φs φsΛ) (some k)) * φsΛ.timeContract 𝓞) := by
+    • (contractStateAtIndex φ [φsΛ]ᵘᶜ
+      ((uncontractedStatesEquiv φs φsΛ) (some k)) * φsΛ.timeContract) := by
   rw [timeConract_insertAndContract_some]
   simp only [Nat.succ_eq_add_one, Fin.getElem_fin, ite_mul, instCommGroup.eq_1,
-    ProtoOperatorAlgebra.contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
+    contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
     Equiv.coe_trans, Option.map_some', Function.comp_apply, finCongr_apply, Fin.coe_cast,
     List.getElem_map, uncontractedList_getElem_uncontractedIndexEquiv_symm, List.get_eq_getElem,
     Algebra.smul_mul_assoc, uncontractedListGet]
   simp only [hik, ↓reduceIte, MulMemClass.coe_mul]
-  rw [𝓞.timeContract_of_not_timeOrderRel, 𝓞.timeContract_of_timeOrderRel]
+  rw [timeContract_of_not_timeOrderRel, timeContract_of_timeOrderRel]
   simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, smul_smul]
   congr
   have h1 : ofList 𝓕.statesStatistic (List.take (↑(φsΛ.uncontractedIndexEquiv.symm k))
@@ -158,9 +156,9 @@ lemma timeConract_insertAndContract_some_eq_mul_contractStateAtIndex_not_lt
   simp_all only [Fin.getElem_fin, Nat.succ_eq_add_one, not_lt, false_or]
   exact ht
 
-lemma timeContract_of_not_gradingCompliant (𝓞 : 𝓕.ProtoOperatorAlgebra) (φs : List 𝓕.States)
+lemma timeContract_of_not_gradingCompliant (φs : List 𝓕.States)
     (φsΛ : WickContraction φs.length) (h : ¬ GradingCompliant φs φsΛ) :
-    φsΛ.timeContract 𝓞 = 0 := by
+    φsΛ.timeContract = 0 := by
   rw [timeContract]
   simp only [GradingCompliant, Fin.getElem_fin, Subtype.forall, not_forall] at h
   obtain ⟨a, ha⟩ := h
@@ -169,7 +167,7 @@ lemma timeContract_of_not_gradingCompliant (𝓞 : 𝓕.ProtoOperatorAlgebra) (
   simp only [Finset.univ_eq_attach, Finset.mem_attach]
   apply Subtype.eq
   simp only [List.get_eq_getElem, ZeroMemClass.coe_zero]
-  rw [ProtoOperatorAlgebra.timeContract_zero_of_diff_grade]
+  rw [timeContract_zero_of_diff_grade]
   simp [ha2]
 
 end WickContraction