Merge pull request #299 from HEPLean/FieldOpAlgebra

feat: Remove state algebra, add time order CrAnAlgebra

HepLean.lean

@@ -124,13 +124,12 @@ import HepLean.Meta.TransverseTactics
 import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.Basic
 import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.NormalOrder
 import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.SuperCommute
+import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.TimeOrder
 import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.Basic
 import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.NormalOrder
 import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.Basic
 import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.NormalOrder
 import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.TimeContraction
-import HepLean.PerturbationTheory.Algebras.StateAlgebra.Basic
-import HepLean.PerturbationTheory.Algebras.StateAlgebra.TimeOrder
 import HepLean.PerturbationTheory.CreateAnnihilate
 import HepLean.PerturbationTheory.FeynmanDiagrams.Basic
 import HepLean.PerturbationTheory.FeynmanDiagrams.Instances.ComplexScalar

HepLean/PerturbationTheory/Algebras/CrAnAlgebra/Basic.lean

@@ -3,7 +3,7 @@ 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.StateAlgebra.Basic
+import HepLean.PerturbationTheory.FieldSpecification.CrAnStates
 import HepLean.PerturbationTheory.FieldSpecification.CrAnSection
 /-!
 
@@ -43,8 +43,6 @@ abbrev CrAnAlgebra (𝓕 : FieldSpecification) : Type := FreeAlgebra ℂ 𝓕.Cr
 
 namespace CrAnAlgebra
 
-open StateAlgebra
-
 /-- Maps a creation and annihlation state to the creation and annihlation free-algebra. -/
 def ofCrAnState (φ : 𝓕.CrAnStates) : CrAnAlgebra 𝓕 :=
   FreeAlgebra.ι ℂ φ
@@ -71,15 +69,6 @@ lemma ofCrAnList_singleton (φ : 𝓕.CrAnStates) :
 def ofState (φ : 𝓕.States) : CrAnAlgebra 𝓕 :=
   ∑ (i : 𝓕.statesToCrAnType φ), ofCrAnState ⟨φ, i⟩
 
-/-- The algebra map from the state free-algebra to the creation and annihlation free-algebra. -/
-def ofStateAlgebra : StateAlgebra 𝓕 →ₐ[ℂ] CrAnAlgebra 𝓕 :=
-  FreeAlgebra.lift ℂ ofState
-
-@[simp]
-lemma ofStateAlgebra_ofState (φ : 𝓕.States) :
-    ofStateAlgebra (StateAlgebra.ofState φ) = ofState φ := by
-  simp [ofStateAlgebra, StateAlgebra.ofState]
-
 /-- Maps a list of states to the creation and annihilation free-algebra by taking
   the product of their sums of creation and annihlation operators.
   Roughly `[φ1, φ2]` gets sent to `(φ1ᶜ+ φ1ᵃ) * (φ2ᶜ+ φ2ᵃ)` etc. -/
@@ -102,14 +91,6 @@ lemma ofStateList_append (φs φs' : List 𝓕.States) :
   dsimp only [ofStateList]
   rw [List.map_append, List.prod_append]
 
-lemma ofStateAlgebra_ofList_eq_ofStateList : (φs : List 𝓕.States) →
-    ofStateAlgebra (ofList φs) = ofStateList φs
-  | [] => by simp [ofStateList]
-  | φ :: φs => by
-    rw [ofStateList_cons, StateAlgebra.ofList_cons, map_mul, ofStateAlgebra_ofState,
-      mul_eq_mul_left_iff]
-    exact .inl (ofStateAlgebra_ofList_eq_ofStateList φs)
-
 lemma ofStateList_sum (φs : List 𝓕.States) :
     ofStateList φs = ∑ (s : CrAnSection φs), ofCrAnList s.1 := by
   induction φs with
@@ -132,57 +113,55 @@ lemma ofStateList_sum (φs : List 𝓕.States) :
 /-- The algebra map taking an element of the free-state algbra to
   the part of it in the creation and annihlation free algebra
   spanned by creation operators. -/
-def crPart : 𝓕.StateAlgebra →ₐ[ℂ] 𝓕.CrAnAlgebra :=
-  FreeAlgebra.lift ℂ fun φ =>
+def crPart : 𝓕.States → 𝓕.CrAnAlgebra := fun φ =>
   match φ with
   | States.inAsymp φ => ofCrAnState ⟨States.inAsymp φ, ()⟩
   | States.position φ => ofCrAnState ⟨States.position φ, CreateAnnihilate.create⟩
   | States.outAsymp _ => 0
 
 @[simp]
 lemma crPart_negAsymp (φ : 𝓕.IncomingAsymptotic) :
-    crPart (StateAlgebra.ofState (States.inAsymp φ)) = ofCrAnState ⟨States.inAsymp φ, ()⟩ := by
-  simp [crPart, StateAlgebra.ofState]
+    crPart (States.inAsymp φ) = ofCrAnState ⟨States.inAsymp φ, ()⟩ := by
+  simp [crPart]
 
 @[simp]
 lemma crPart_position (φ : 𝓕.PositionStates) :
-    crPart (StateAlgebra.ofState (States.position φ)) =
+    crPart (States.position φ) =
     ofCrAnState ⟨States.position φ, CreateAnnihilate.create⟩ := by
-  simp [crPart, StateAlgebra.ofState]
+  simp [crPart]
 
 @[simp]
 lemma crPart_posAsymp (φ : 𝓕.OutgoingAsymptotic) :
-    crPart (StateAlgebra.ofState (States.outAsymp φ)) = 0 := by
-  simp [crPart, StateAlgebra.ofState]
+    crPart (States.outAsymp φ) = 0 := by
+  simp [crPart]
 
 /-- The algebra map taking an element of the free-state algbra to
   the part of it in the creation and annihilation free algebra
   spanned by annihilation operators. -/
-def anPart : 𝓕.StateAlgebra →ₐ[ℂ] 𝓕.CrAnAlgebra :=
-  FreeAlgebra.lift ℂ fun φ =>
+def anPart : 𝓕.States → 𝓕.CrAnAlgebra := fun φ =>
   match φ with
   | States.inAsymp _ => 0
   | States.position φ => ofCrAnState ⟨States.position φ, CreateAnnihilate.annihilate⟩
   | States.outAsymp φ => ofCrAnState ⟨States.outAsymp φ, ()⟩
 
 @[simp]
 lemma anPart_negAsymp (φ : 𝓕.IncomingAsymptotic) :
-    anPart (StateAlgebra.ofState (States.inAsymp φ)) = 0 := by
-  simp [anPart, StateAlgebra.ofState]
+    anPart (States.inAsymp φ) = 0 := by
+  simp [anPart]
 
 @[simp]
 lemma anPart_position (φ : 𝓕.PositionStates) :
-    anPart (StateAlgebra.ofState (States.position φ)) =
+    anPart (States.position φ) =
     ofCrAnState ⟨States.position φ, CreateAnnihilate.annihilate⟩ := by
-  simp [anPart, StateAlgebra.ofState]
+  simp [anPart]
 
 @[simp]
 lemma anPart_posAsymp (φ : 𝓕.OutgoingAsymptotic) :
-    anPart (StateAlgebra.ofState (States.outAsymp φ)) = ofCrAnState ⟨States.outAsymp φ, ()⟩ := by
-  simp [anPart, StateAlgebra.ofState]
+    anPart (States.outAsymp φ) = ofCrAnState ⟨States.outAsymp φ, ()⟩ := by
+  simp [anPart]
 
 lemma ofState_eq_crPart_add_anPart (φ : 𝓕.States) :
-    ofState φ = crPart (StateAlgebra.ofState φ) + anPart (StateAlgebra.ofState φ) := by
+    ofState φ = crPart φ + anPart φ := by
   rw [ofState]
   cases φ with
   | inAsymp φ => simp [statesToCrAnType]

HepLean/PerturbationTheory/Algebras/CrAnAlgebra/NormalOrder.lean

@@ -93,27 +93,27 @@ lemma normalOrder_mul_annihilate (φ : 𝓕.CrAnStates)
     normalOrder_ofCrAnList_append_annihilate φ hφ]
 
 lemma normalOrder_crPart_mul (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
-    𝓝(crPart (StateAlgebra.ofState φ) * a) =
-    crPart (StateAlgebra.ofState φ) * 𝓝(a) := by
+    𝓝(crPart φ * a) =
+    crPart φ * 𝓝(a) := by
   match φ with
   | .inAsymp φ =>
-    rw [crPart, StateAlgebra.ofState, FreeAlgebra.lift_ι_apply]
+    rw [crPart]
     exact normalOrder_create_mul ⟨States.inAsymp φ, ()⟩ rfl a
   | .position φ =>
-    rw [crPart, StateAlgebra.ofState, FreeAlgebra.lift_ι_apply]
+    rw [crPart]
     exact normalOrder_create_mul _ rfl _
   | .outAsymp φ => simp
 
 lemma normalOrder_mul_anPart (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
-    𝓝(a * anPart (StateAlgebra.ofState φ)) =
-    𝓝(a) * anPart (StateAlgebra.ofState φ) := by
+    𝓝(a * anPart φ) =
+    𝓝(a) * anPart φ := by
   match φ with
   | .inAsymp φ => simp
   | .position φ =>
-    rw [anPart, StateAlgebra.ofState, FreeAlgebra.lift_ι_apply]
+    rw [anPart]
     exact normalOrder_mul_annihilate _ rfl _
   | .outAsymp φ =>
-    rw [anPart, StateAlgebra.ofState, FreeAlgebra.lift_ι_apply]
+    rw [anPart]
     refine normalOrder_mul_annihilate _ rfl _
 
 /-!
@@ -182,9 +182,9 @@ lemma normalOrder_superCommute_annihilate_create (φc φa : 𝓕.CrAnStates)
   exact Or.inr (normalOrder_superCommute_create_annihilate φc φa hφc hφa ..)
 
 lemma normalOrder_swap_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
-    𝓝(a * (crPart (StateAlgebra.ofState φ)) * (anPart (StateAlgebra.ofState φ')) * b) =
+    𝓝(a * (crPart φ) * (anPart φ') * b) =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
-    𝓝(a * (anPart (StateAlgebra.ofState φ')) * (crPart (StateAlgebra.ofState φ)) * b) := by
+    𝓝(a * (anPart φ') * (crPart φ) * b) := by
   match φ, φ' with
   | _, .inAsymp φ' => simp
   | .outAsymp φ, _ => simp
@@ -218,14 +218,14 @@ Using the results from above.
 -/
 
 lemma normalOrder_swap_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
-    𝓝(a * (anPart (StateAlgebra.ofState φ)) * (crPart (StateAlgebra.ofState φ')) * b) =
-    𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓝(a * (crPart (StateAlgebra.ofState φ')) *
-      (anPart (StateAlgebra.ofState φ)) * b) := by
+    𝓝(a * (anPart φ) * (crPart φ') * b) =
+    𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓝(a * (crPart φ') *
+      (anPart φ) * b) := by
   simp [normalOrder_swap_crPart_anPart, smul_smul]
 
 lemma normalOrder_superCommute_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
     𝓝(a * superCommute
-      (crPart (StateAlgebra.ofState φ)) (anPart (StateAlgebra.ofState φ')) * b) = 0 := by
+      (crPart φ) (anPart φ') * b) = 0 := by
   match φ, φ' with
   | _, .inAsymp φ' => simp
   | .outAsymp φ', _ => simp
@@ -244,7 +244,7 @@ lemma normalOrder_superCommute_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnA
 
 lemma normalOrder_superCommute_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
     𝓝(a * superCommute
-    (anPart (StateAlgebra.ofState φ)) (crPart (StateAlgebra.ofState φ')) * b) = 0 := by
+    (anPart φ) (crPart φ') * b) = 0 := by
   match φ, φ' with
   | .inAsymp φ', _ => simp
   | _, .outAsymp φ' => simp
@@ -269,49 +269,49 @@ lemma normalOrder_superCommute_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnA
 
 @[simp]
 lemma normalOrder_crPart_mul_crPart (φ φ' : 𝓕.States) :
-    𝓝(crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ')) =
-    crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ') := by
+    𝓝(crPart φ * crPart φ') =
+    crPart φ * crPart φ' := by
   rw [normalOrder_crPart_mul]
-  conv_lhs => rw [← mul_one (crPart (StateAlgebra.ofState φ'))]
+  conv_lhs => rw [← mul_one (crPart φ')]
   rw [normalOrder_crPart_mul, normalOrder_one]
   simp
 
 @[simp]
 lemma normalOrder_anPart_mul_anPart (φ φ' : 𝓕.States) :
-    𝓝(anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ')) =
-    anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') := by
+    𝓝(anPart φ * anPart φ') =
+    anPart φ * anPart φ' := by
   rw [normalOrder_mul_anPart]
-  conv_lhs => rw [← one_mul (anPart (StateAlgebra.ofState φ))]
+  conv_lhs => rw [← one_mul (anPart φ)]
   rw [normalOrder_mul_anPart, normalOrder_one]
   simp
 
 @[simp]
 lemma normalOrder_crPart_mul_anPart (φ φ' : 𝓕.States) :
-    𝓝(crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ')) =
-    crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') := by
+    𝓝(crPart φ * anPart φ') =
+    crPart φ * anPart φ' := by
   rw [normalOrder_crPart_mul]
-  conv_lhs => rw [← one_mul (anPart (StateAlgebra.ofState φ'))]
+  conv_lhs => rw [← one_mul (anPart φ')]
   rw [normalOrder_mul_anPart, normalOrder_one]
   simp
 
 @[simp]
 lemma normalOrder_anPart_mul_crPart (φ φ' : 𝓕.States) :
-    𝓝(anPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ')) =
+    𝓝(anPart φ * crPart φ') =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
-    (crPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ)) := by
-  conv_lhs => rw [← one_mul (anPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ'))]
-  conv_lhs => rw [← mul_one (1 * (anPart (StateAlgebra.ofState φ) *
-    crPart (StateAlgebra.ofState φ')))]
+    (crPart φ' * anPart φ) := by
+  conv_lhs => rw [← one_mul (anPart φ * crPart φ')]
+  conv_lhs => rw [← mul_one (1 * (anPart φ *
+    crPart φ'))]
   rw [← mul_assoc, normalOrder_swap_anPart_crPart]
   simp
 
 lemma normalOrder_ofState_mul_ofState (φ φ' : 𝓕.States) :
     𝓝(ofState φ * ofState φ') =
-    crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ') +
+    crPart φ * crPart φ' +
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
-    (crPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ)) +
-    crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') +
-    anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') := by
+    (crPart φ' * anPart φ) +
+    crPart φ * anPart φ' +
+    anPart φ * anPart φ' := by
   rw [ofState_eq_crPart_add_anPart, ofState_eq_crPart_add_anPart, mul_add, add_mul, add_mul]
   simp only [map_add, normalOrder_crPart_mul_crPart, normalOrder_anPart_mul_crPart,
     instCommGroup.eq_1, normalOrder_crPart_mul_anPart, normalOrder_anPart_mul_anPart]
@@ -497,9 +497,9 @@ lemma ofCrAnState_mul_normalOrder_ofStateList_eq_superCommute (φ : 𝓕.CrAnSta
 
 lemma anPart_mul_normalOrder_ofStateList_eq_superCommute (φ : 𝓕.States)
     (φs' : List 𝓕.States) :
-    anPart (StateAlgebra.ofState φ) * 𝓝(ofStateList φs') =
-    𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝(ofStateList φs' * anPart (StateAlgebra.ofState φ))
-    + [anPart (StateAlgebra.ofState φ), 𝓝(ofStateList φs')]ₛca := by
+    anPart φ * 𝓝(ofStateList φs') =
+    𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝(ofStateList φs' * anPart φ)
+    + [anPart φ, 𝓝(ofStateList φs')]ₛca := by
   rw [normalOrder_mul_anPart]
   match φ with
   | .inAsymp φ => simp