feat: Properties of super commute
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jstoobysmith
parent
f7e669910c
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c18b4850e5
3 changed files with 357 additions and 13 deletions
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@ -96,8 +96,7 @@ lemma superCommuteF_ofStateList_ofState (φs : List 𝓕.States) (φ : 𝓕.Stat
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simp
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lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) :
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[anPartF φ, crPartF φ']ₛca =
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anPartF φ * crPartF φ' -
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[anPartF φ, crPartF φ']ₛca = anPartF φ * crPartF φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * anPartF φ := by
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match φ, φ' with
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| States.inAsymp φ, _ =>
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@ -125,10 +124,8 @@ lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) :
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simp [crAnStatistics, ← ofCrAnList_append]
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lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.States) :
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[crPartF φ, anPartF φ']ₛca =
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crPartF φ * anPartF φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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anPartF φ' * crPartF φ := by
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[crPartF φ, anPartF φ']ₛca = crPartF φ * anPartF φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * crPartF φ := by
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match φ, φ' with
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| States.outAsymp φ, _ =>
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simp only [crPartF_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
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@ -154,10 +151,8 @@ lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.States) :
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simp [crAnStatistics, ← ofCrAnList_append]
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lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.States) :
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[crPartF φ, crPartF φ']ₛca =
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crPartF φ * crPartF φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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crPartF φ' * crPartF φ := by
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[crPartF φ, crPartF φ']ₛca = crPartF φ * crPartF φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * crPartF φ := by
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match φ, φ' with
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| States.outAsymp φ, _ =>
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simp only [crPartF_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
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@ -183,9 +178,7 @@ lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.States) :
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lemma superCommuteF_anPartF_anPartF (φ φ' : 𝓕.States) :
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[anPartF φ, anPartF φ']ₛca =
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anPartF φ * anPartF φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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anPartF φ' * anPartF φ := by
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anPartF φ * anPartF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * anPartF φ := by
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match φ, φ' with
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| States.inAsymp φ, _ =>
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simp
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@ -362,6 +355,7 @@ lemma superCommuteF_ofCrAnState_ofCrAnState_symm (φ φ' : 𝓕.CrAnStates) :
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## Splitting the super commutor on lists into sums.
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-/
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lemma superCommuteF_ofCrAnList_ofCrAnList_cons (φ : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
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[ofCrAnList φs, ofCrAnList (φ :: φs')]ₛca =
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[ofCrAnList φs, ofCrAnState φ]ₛca * ofCrAnList φs' +
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@ -483,6 +477,7 @@ lemma summerCommute_jacobi_ofCrAnList (φs1 φs2 φs3 : List 𝓕.CrAnStates) :
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simp only [h1, h2, h3, mul_self, map_one, one_smul, fermionic_exchangeSign_fermionic, neg_smul,
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neg_sub]
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abel
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/-!
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## Interaction with grading.
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@ -451,15 +451,57 @@ def ofCrAnFieldOpList (φs : List 𝓕.CrAnStates) : 𝓕.FieldOpAlgebra := ι (
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lemma ofCrAnFieldOpList_eq_ι_ofCrAnList (φs : List 𝓕.CrAnStates) :
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ofCrAnFieldOpList φs = ι (ofCrAnList φs) := rfl
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lemma ofCrAnFieldOpList_append (φs ψs : List 𝓕.CrAnStates) :
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ofCrAnFieldOpList (φs ++ ψs) = ofCrAnFieldOpList φs * ofCrAnFieldOpList ψs := by
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simp only [ofCrAnFieldOpList]
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rw [ofCrAnList_append]
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simp
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lemma ofCrAnFieldOpList_singleton (φ : 𝓕.CrAnStates) :
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ofCrAnFieldOpList [φ] = ofCrAnFieldOp φ := by
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simp only [ofCrAnFieldOpList, ofCrAnFieldOp, ofCrAnList_singleton]
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/-- The annihilation part of a state. -/
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def anPart (φ : 𝓕.States) : 𝓕.FieldOpAlgebra := ι (anPartF φ)
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lemma anPart_eq_ι_anPartF (φ : 𝓕.States) : anPart φ = ι (anPartF φ) := rfl
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@[simp]
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lemma anPart_negAsymp (φ : 𝓕.IncomingAsymptotic) :
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anPart (States.inAsymp φ) = 0 := by
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simp [anPart, anPartF]
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@[simp]
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lemma anPart_position (φ : 𝓕.PositionStates) :
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anPart (States.position φ) =
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ofCrAnFieldOp ⟨States.position φ, CreateAnnihilate.annihilate⟩ := by
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simp [anPart, ofCrAnFieldOp]
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@[simp]
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lemma anPart_posAsymp (φ : 𝓕.OutgoingAsymptotic) :
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anPart (States.outAsymp φ) = ofCrAnFieldOp ⟨States.outAsymp φ, ()⟩ := by
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simp [anPart, ofCrAnFieldOp]
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/-- The creation part of a state. -/
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def crPart (φ : 𝓕.States) : 𝓕.FieldOpAlgebra := ι (crPartF φ)
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lemma crPart_eq_ι_crPartF (φ : 𝓕.States) : crPart φ = ι (crPartF φ) := rfl
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@[simp]
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lemma crPart_negAsymp (φ : 𝓕.IncomingAsymptotic) :
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crPart (States.inAsymp φ) = ofCrAnFieldOp ⟨States.inAsymp φ, ()⟩ := by
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simp [crPart, ofCrAnFieldOp]
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@[simp]
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lemma crPart_position (φ : 𝓕.PositionStates) :
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crPart (States.position φ) =
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ofCrAnFieldOp ⟨States.position φ, CreateAnnihilate.create⟩ := by
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simp [crPart, ofCrAnFieldOp]
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@[simp]
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lemma crPart_posAsymp (φ : 𝓕.OutgoingAsymptotic) :
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crPart (States.outAsymp φ) = 0 := by
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simp [crPart]
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end FieldOpAlgebra
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end FieldSpecification
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@ -117,5 +117,312 @@ scoped[FieldSpecification.FieldOpAlgebra] notation "[" a "," b "]ₛ" => superCo
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lemma superCommute_eq_ι_superCommuteF (a b : 𝓕.CrAnAlgebra) :
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[ι a, ι b]ₛ = ι [a, b]ₛca := rfl
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/-!
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## Properties of `superCommute`.
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-/
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/-!
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## Properties from the definition of FieldOpAlgebra
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-/
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lemma superCommute_create_create {φ φ' : 𝓕.CrAnStates}
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(h : 𝓕 |>ᶜ φ = .create) (h' : 𝓕 |>ᶜ φ' = .create) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = 0 := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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rw [superCommute_eq_ι_superCommuteF, ι_superCommuteF_of_create_create _ _ h h']
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lemma superCommute_annihilate_annihilate {φ φ' : 𝓕.CrAnStates}
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(h : 𝓕 |>ᶜ φ = .annihilate) (h' : 𝓕 |>ᶜ φ' = .annihilate) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = 0 := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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rw [superCommute_eq_ι_superCommuteF, ι_superCommuteF_of_annihilate_annihilate _ _ h h']
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lemma superCommute_diff_statistic {φ φ' : 𝓕.CrAnStates} (h : (𝓕 |>ₛ φ) ≠ 𝓕 |>ₛ φ') :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = 0 := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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rw [superCommute_eq_ι_superCommuteF, ι_superCommuteF_of_diff_statistic h]
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp_mem_center (φ φ' : 𝓕.CrAnStates) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ ∈ Subalgebra.center ℂ (FieldOpAlgebra 𝓕) := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp, superCommute_eq_ι_superCommuteF]
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exact ι_superCommuteF_ofCrAnState_ofCrAnState_mem_center φ φ'
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/-!
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### `superCommute` on different constructors.
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-/
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lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList (φs φs' : List 𝓕.CrAnStates) :
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[ofCrAnFieldOpList φs, ofCrAnFieldOpList φs']ₛ =
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ofCrAnFieldOpList (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnFieldOpList (φs' ++ φs) := by
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rw [ofCrAnFieldOpList_eq_ι_ofCrAnList, ofCrAnFieldOpList_eq_ι_ofCrAnList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnList_ofCrAnList]
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rfl
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lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp (φ φ' : 𝓕.CrAnStates) :
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[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ = ofCrAnFieldOp φ * ofCrAnFieldOp φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofCrAnFieldOp φ' * ofCrAnFieldOp φ := by
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rw [ofCrAnFieldOp, ofCrAnFieldOp]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnState_ofCrAnState]
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rfl
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lemma superCommute_ofCrAnFieldOpList_ofFieldOpList (φcas : List 𝓕.CrAnStates)
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(φs : List 𝓕.States) :
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[ofCrAnFieldOpList φcas, ofFieldOpList φs]ₛ = ofCrAnFieldOpList φcas * ofFieldOpList φs -
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𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofFieldOpList φs * ofCrAnFieldOpList φcas := by
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rw [ofCrAnFieldOpList, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnList_ofStatesList]
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rfl
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lemma superCommute_ofFieldOpList_ofFieldOpList (φs φs' : List 𝓕.States) :
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[ofFieldOpList φs, ofFieldOpList φs']ₛ = ofFieldOpList φs * ofFieldOpList φs' -
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOpList φs := by
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rw [ofFieldOpList, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofStateList_ofStatesList]
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rfl
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lemma superCommute_ofFieldOp_ofFieldOpList (φ : 𝓕.States) (φs : List 𝓕.States) :
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[ofFieldOp φ, ofFieldOpList φs]ₛ = ofFieldOp φ * ofFieldOpList φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * ofFieldOp φ := by
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rw [ofFieldOp, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofState_ofStatesList]
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rfl
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lemma superCommute_ofFieldOpList_ofFieldOp (φs : List 𝓕.States) (φ : 𝓕.States) :
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[ofFieldOpList φs, ofFieldOp φ]ₛ = ofFieldOpList φs * ofFieldOp φ -
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOp φ * ofFieldOpList φs := by
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rw [ofFieldOpList, ofFieldOp]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofStateList_ofState]
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rfl
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lemma superCommute_anPart_crPart (φ φ' : 𝓕.States) :
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[anPart φ, crPart φ']ₛ = anPart φ * crPart φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ := by
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rw [anPart, crPart]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_crPartF]
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rfl
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lemma superCommute_crPart_anPart (φ φ' : 𝓕.States) :
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[crPart φ, anPart φ']ₛ = crPart φ * anPart φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * crPart φ := by
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rw [anPart, crPart]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_anPartF]
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rfl
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@[simp]
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lemma superCommute_crPart_crPart (φ φ' : 𝓕.States) : [crPart φ, crPart φ']ₛ = 0 := by
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match φ, φ' with
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| States.outAsymp φ, _ =>
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simp
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| _, States.outAsymp φ =>
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simp
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| States.position φ, States.position φ' =>
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simp only [crPart_position]
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apply superCommute_create_create
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· rfl
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· rfl
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| States.position φ, States.inAsymp φ' =>
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simp only [crPart_position, crPart_negAsymp]
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apply superCommute_create_create
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· rfl
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· rfl
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| States.inAsymp φ, States.inAsymp φ' =>
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simp only [crPart_negAsymp]
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apply superCommute_create_create
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· rfl
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· rfl
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| States.inAsymp φ, States.position φ' =>
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simp only [crPart_negAsymp, crPart_position]
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apply superCommute_create_create
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· rfl
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· rfl
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@[simp]
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lemma superCommute_anPart_anPart (φ φ' : 𝓕.States) : [anPart φ, anPart φ']ₛ = 0 := by
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match φ, φ' with
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| States.inAsymp φ, _ =>
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simp
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| _, States.inAsymp φ =>
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simp
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| States.position φ, States.position φ' =>
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simp only [anPart_position]
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apply superCommute_annihilate_annihilate
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· rfl
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· rfl
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| States.position φ, States.outAsymp φ' =>
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simp only [anPart_position, anPart_posAsymp]
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apply superCommute_annihilate_annihilate
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· rfl
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· rfl
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| States.outAsymp φ, States.outAsymp φ' =>
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simp only [anPart_posAsymp]
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apply superCommute_annihilate_annihilate
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· rfl
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· rfl
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| States.outAsymp φ, States.position φ' =>
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simp only [anPart_posAsymp, anPart_position]
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apply superCommute_annihilate_annihilate
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· rfl
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· rfl
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lemma superCommute_crPart_ofFieldOpList (φ : 𝓕.States) (φs : List 𝓕.States) :
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[crPart φ, ofFieldOpList φs]ₛ = crPart φ * ofFieldOpList φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * crPart φ := by
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rw [crPart, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_ofStateList]
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rfl
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lemma superCommute_anPart_ofFieldOpList (φ : 𝓕.States) (φs : List 𝓕.States) :
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[anPart φ, ofFieldOpList φs]ₛ = anPart φ * ofFieldOpList φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * anPart φ := by
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rw [anPart, ofFieldOpList]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_ofStateList]
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rfl
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lemma superCommute_crPart_ofFieldOp (φ φ' : 𝓕.States) :
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[crPart φ, ofFieldOp φ']ₛ = crPart φ * ofFieldOp φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * crPart φ := by
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rw [crPart, ofFieldOp]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_ofState]
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rfl
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lemma superCommute_anPart_ofFieldOp (φ φ' : 𝓕.States) :
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[anPart φ, ofFieldOp φ']ₛ = anPart φ * ofFieldOp φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * anPart φ := by
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rw [anPart, ofFieldOp]
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rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_ofState]
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rfl
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/-!
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## Mul equal superCommute
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Lemmas which rewrite a multiplication of two elements of the algebra as their commuted
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multiplication with a sign plus the super commutor.
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-/
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lemma ofCrAnFieldOpList_mul_ofCrAnFieldOpList_eq_superCommute (φs φs' : List 𝓕.CrAnStates) :
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ofCrAnFieldOpList φs * ofCrAnFieldOpList φs' =
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnFieldOpList φs' * ofCrAnFieldOpList φs
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+ [ofCrAnFieldOpList φs, ofCrAnFieldOpList φs']ₛ := by
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rw [superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList]
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simp [ofCrAnFieldOpList_append]
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lemma ofCrAnFieldOp_mul_ofCrAnFieldOpList_eq_superCommute (φ : 𝓕.CrAnStates)
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(φs' : List 𝓕.CrAnStates) : ofCrAnFieldOp φ * ofCrAnFieldOpList φs' =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofCrAnFieldOpList φs' * ofCrAnFieldOp φ
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+ [ofCrAnFieldOp φ, ofCrAnFieldOpList φs']ₛ := by
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rw [← ofCrAnFieldOpList_singleton, ofCrAnFieldOpList_mul_ofCrAnFieldOpList_eq_superCommute]
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simp
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||||
|
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lemma ofFieldOpList_mul_ofFieldOpList_eq_superCommute (φs φs' : List 𝓕.States) :
|
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ofFieldOpList φs * ofFieldOpList φs' =
|
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOpList φs
|
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+ [ofFieldOpList φs, ofFieldOpList φs']ₛ := by
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rw [superCommute_ofFieldOpList_ofFieldOpList]
|
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simp
|
||||
|
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lemma ofFieldOp_mul_ofFieldOpList_eq_superCommute (φ : 𝓕.States) (φs' : List 𝓕.States) :
|
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ofFieldOp φ * ofFieldOpList φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOp φ
|
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+ [ofFieldOp φ, ofFieldOpList φs']ₛ := by
|
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rw [superCommute_ofFieldOp_ofFieldOpList]
|
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simp
|
||||
|
||||
lemma ofFieldOpList_mul_ofFieldOp_eq_superCommute (φs : List 𝓕.States) (φ : 𝓕.States) :
|
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ofFieldOpList φs * ofFieldOp φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOp φ * ofFieldOpList φs
|
||||
+ [ofFieldOpList φs, ofFieldOp φ]ₛ := by
|
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rw [superCommute_ofFieldOpList_ofFieldOp]
|
||||
simp
|
||||
|
||||
lemma ofCrAnFieldOpList_mul_ofFieldOpList_eq_superCommute (φs : List 𝓕.CrAnStates)
|
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(φs' : List 𝓕.States) : ofCrAnFieldOpList φs * ofFieldOpList φs' =
|
||||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofCrAnFieldOpList φs
|
||||
+ [ofCrAnFieldOpList φs, ofFieldOpList φs']ₛ := by
|
||||
rw [superCommute_ofCrAnFieldOpList_ofFieldOpList]
|
||||
simp
|
||||
|
||||
lemma crPart_mul_anPart_eq_superCommute (φ φ' : 𝓕.States) :
|
||||
crPart φ * anPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * crPart φ
|
||||
+ [crPart φ, anPart φ']ₛ := by
|
||||
rw [superCommute_crPart_anPart]
|
||||
simp
|
||||
|
||||
lemma anPart_mul_crPart_eq_superCommute (φ φ' : 𝓕.States) :
|
||||
anPart φ * crPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ
|
||||
+ [anPart φ, crPart φ']ₛ := by
|
||||
rw [superCommute_anPart_crPart]
|
||||
simp
|
||||
|
||||
lemma crPart_mul_crPart_swap (φ φ' : 𝓕.States) :
|
||||
crPart φ * crPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * crPart φ := by
|
||||
trans 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * crPart φ + [crPart φ, crPart φ']ₛ
|
||||
· rw [crPart, crPart, superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_crPartF]
|
||||
simp
|
||||
· simp
|
||||
|
||||
lemma anPart_mul_anPart_swap (φ φ' : 𝓕.States) :
|
||||
anPart φ * anPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * anPart φ := by
|
||||
trans 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * anPart φ + [anPart φ, anPart φ']ₛ
|
||||
· rw [anPart, anPart, superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_anPartF]
|
||||
simp
|
||||
· simp
|
||||
|
||||
/-!
|
||||
|
||||
## Symmetry of the super commutor.
|
||||
|
||||
-/
|
||||
|
||||
lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList_symm (φs φs' : List 𝓕.CrAnStates) :
|
||||
[ofCrAnFieldOpList φs, ofCrAnFieldOpList φs']ₛ =
|
||||
(- 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs')) • [ofCrAnFieldOpList φs', ofCrAnFieldOpList φs]ₛ := by
|
||||
rw [ofCrAnFieldOpList, ofCrAnFieldOpList, superCommute_eq_ι_superCommuteF,
|
||||
superCommuteF_ofCrAnList_ofCrAnList_symm]
|
||||
rfl
|
||||
|
||||
lemma superCommute_ofCrAnFieldOp_ofCrAnFieldOp_symm (φ φ' : 𝓕.CrAnStates) :
|
||||
[ofCrAnFieldOp φ, ofCrAnFieldOp φ']ₛ =
|
||||
(- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • [ofCrAnFieldOp φ', ofCrAnFieldOp φ]ₛ := by
|
||||
rw [ofCrAnFieldOp, ofCrAnFieldOp, superCommute_eq_ι_superCommuteF,
|
||||
superCommuteF_ofCrAnState_ofCrAnState_symm]
|
||||
rfl
|
||||
|
||||
/-!
|
||||
|
||||
## splitting the super commute into sums
|
||||
|
||||
-/
|
||||
|
||||
lemma superCommute_ofCrAnFieldOpList_ofCrAnFieldOpList_eq_sum (φs φs' : List 𝓕.CrAnStates) :
|
||||
[ofCrAnFieldOpList φs, ofCrAnFieldOpList φs']ₛ =
|
||||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
|
||||
ofCrAnFieldOpList (φs'.take n) * [ofCrAnFieldOpList φs, ofCrAnFieldOp (φs'.get n)]ₛ *
|
||||
ofCrAnFieldOpList (φs'.drop (n + 1)) := by
|
||||
conv_lhs =>
|
||||
rw [ofCrAnFieldOpList, ofCrAnFieldOpList, superCommute_eq_ι_superCommuteF,
|
||||
superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
|
||||
rw [map_sum]
|
||||
rfl
|
||||
|
||||
lemma superCommute_ofCrAnFieldOpList_ofFieldOpList_eq_sum (φs : List 𝓕.CrAnStates)
|
||||
(φs' : List 𝓕.States) : [ofCrAnFieldOpList φs, ofFieldOpList φs']ₛ =
|
||||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
|
||||
ofFieldOpList (φs'.take n) * [ofCrAnFieldOpList φs, ofFieldOp (φs'.get n)]ₛ *
|
||||
ofFieldOpList (φs'.drop (n + 1)) := by
|
||||
conv_lhs =>
|
||||
rw [ofCrAnFieldOpList, ofFieldOpList, superCommute_eq_ι_superCommuteF,
|
||||
superCommuteF_ofCrAnList_ofStateList_eq_sum]
|
||||
rw [map_sum]
|
||||
rfl
|
||||
|
||||
|
||||
end FieldOpAlgebra
|
||||
end FieldSpecification
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue