refactor: ofState rename to ofFieldOpF
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jstoobysmith
parent
08260e709c
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93d06895c6
12 changed files with 85 additions and 85 deletions
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@ -21,7 +21,7 @@ The main structures defined in this module are:
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* `FieldOpFreeAlgebra` - The creation and annihilation algebra
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* `ofCrAnState` - Maps a creation/annihilation state to the algebra
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* `ofCrAnList` - Maps a list of creation/annihilation states to the algebra
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* `ofState` - Maps a state to a sum of creation and annihilation operators
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* `ofFieldOpF` - Maps a state to a sum of creation and annihilation operators
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* `crPartF` - The creation part of a state in the algebra
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* `anPartF` - The annihilation part of a state in the algebra
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* `superCommuteF` - The super commutator on the algebra
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@ -66,13 +66,13 @@ lemma ofCrAnList_singleton (φ : 𝓕.CrAnStates) :
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/-- Maps a state to the sum of creation and annihilation operators in
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creation and annihilation free-algebra. -/
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def ofState (φ : 𝓕.States) : FieldOpFreeAlgebra 𝓕 :=
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def ofFieldOpF (φ : 𝓕.States) : FieldOpFreeAlgebra 𝓕 :=
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∑ (i : 𝓕.statesToCrAnType φ), ofCrAnState ⟨φ, i⟩
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/-- Maps a list of states to the creation and annihilation free-algebra by taking
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the product of their sums of creation and annihlation operators.
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Roughly `[φ1, φ2]` gets sent to `(φ1ᶜ+ φ1ᵃ) * (φ2ᶜ+ φ2ᵃ)` etc. -/
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def ofFieldOpListF (φs : List 𝓕.States) : FieldOpFreeAlgebra 𝓕 := (List.map ofState φs).prod
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def ofFieldOpListF (φs : List 𝓕.States) : FieldOpFreeAlgebra 𝓕 := (List.map ofFieldOpF φs).prod
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/-- Coercion from `List 𝓕.States` to `FieldOpFreeAlgebra 𝓕` through `ofFieldOpListF`. -/
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instance : Coe (List 𝓕.States) (FieldOpFreeAlgebra 𝓕) := ⟨ofFieldOpListF⟩
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@ -81,10 +81,10 @@ instance : Coe (List 𝓕.States) (FieldOpFreeAlgebra 𝓕) := ⟨ofFieldOpListF
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lemma ofFieldOpListF_nil : ofFieldOpListF ([] : List 𝓕.States) = 1 := rfl
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lemma ofFieldOpListF_cons (φ : 𝓕.States) (φs : List 𝓕.States) :
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ofFieldOpListF (φ :: φs) = ofState φ * ofFieldOpListF φs := rfl
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ofFieldOpListF (φ :: φs) = ofFieldOpF φ * ofFieldOpListF φs := rfl
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lemma ofFieldOpListF_singleton (φ : 𝓕.States) :
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ofFieldOpListF [φ] = ofState φ := by simp [ofFieldOpListF]
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ofFieldOpListF [φ] = ofFieldOpF φ := by simp [ofFieldOpListF]
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lemma ofFieldOpListF_append (φs φs' : List 𝓕.States) :
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ofFieldOpListF (φs ++ φs') = ofFieldOpListF φs * ofFieldOpListF φs' := by
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@ -160,9 +160,9 @@ lemma anPartF_posAsymp (φ : 𝓕.OutgoingAsymptotic) :
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anPartF (States.outAsymp φ) = ofCrAnState ⟨States.outAsymp φ, ()⟩ := by
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simp [anPartF]
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lemma ofState_eq_crPartF_add_anPartF (φ : 𝓕.States) :
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ofState φ = crPartF φ + anPartF φ := by
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rw [ofState]
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lemma ofFieldOpF_eq_crPartF_add_anPartF (φ : 𝓕.States) :
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ofFieldOpF φ = crPartF φ + anPartF φ := by
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rw [ofFieldOpF]
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cases φ with
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| inAsymp φ => simp [statesToCrAnType]
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| position φ => simp [statesToCrAnType, CreateAnnihilate.sum_eq]
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@ -367,14 +367,14 @@ lemma normalOrderF_anPartF_mul_crPartF (φ φ' : 𝓕.States) :
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rw [← mul_assoc, normalOrderF_swap_anPartF_crPartF]
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simp
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lemma normalOrderF_ofState_mul_ofState (φ φ' : 𝓕.States) :
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𝓝ᶠ(ofState φ * ofState φ') =
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lemma normalOrderF_ofFieldOpF_mul_ofFieldOpF (φ φ' : 𝓕.States) :
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𝓝ᶠ(ofFieldOpF φ * ofFieldOpF φ') =
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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' * anPartF φ) +
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crPartF φ * anPartF φ' +
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anPartF φ * anPartF φ' := by
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rw [ofState_eq_crPartF_add_anPartF, ofState_eq_crPartF_add_anPartF, mul_add, add_mul, add_mul]
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rw [ofFieldOpF_eq_crPartF_add_anPartF, ofFieldOpF_eq_crPartF_add_anPartF, mul_add, add_mul, add_mul]
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simp only [map_add, normalOrderF_crPartF_mul_crPartF, normalOrderF_anPartF_mul_crPartF,
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instCommGroup.eq_1, normalOrderF_crPartF_mul_anPartF, normalOrderF_anPartF_mul_anPartF]
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abel
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@ -57,7 +57,7 @@ lemma superCommuteF_ofCrAnState_ofCrAnState (φ φ' : 𝓕.CrAnStates) :
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rw [ofCrAnList_append]
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rw [FieldStatistic.ofList_singleton, FieldStatistic.ofList_singleton, smul_mul_assoc]
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lemma superCommuteF_ofCrAnList_ofStatesList (φcas : List 𝓕.CrAnStates) (φs : List 𝓕.States) :
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lemma superCommuteF_ofCrAnList_ofFieldOpFsList (φcas : List 𝓕.CrAnStates) (φs : List 𝓕.States) :
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[ofCrAnList φcas, ofFieldOpListF φs]ₛca = ofCrAnList φcas * ofFieldOpListF φs -
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𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofCrAnList φcas := by
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conv_lhs => rw [ofFieldOpListF_sum]
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@ -70,7 +70,7 @@ lemma superCommuteF_ofCrAnList_ofStatesList (φcas : List 𝓕.CrAnStates) (φs
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← Finset.sum_mul, ← ofFieldOpListF_sum]
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simp
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lemma superCommuteF_ofFieldOpListF_ofStatesList (φ : List 𝓕.States) (φs : List 𝓕.States) :
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lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.States) (φs : List 𝓕.States) :
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[ofFieldOpListF φ, ofFieldOpListF φs]ₛca = ofFieldOpListF φ * ofFieldOpListF φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofFieldOpListF φ := by
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conv_lhs => rw [ofFieldOpListF_sum]
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@ -78,21 +78,21 @@ lemma superCommuteF_ofFieldOpListF_ofStatesList (φ : List 𝓕.States) (φs : L
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Algebra.smul_mul_assoc]
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conv_lhs =>
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enter [2, x]
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rw [superCommuteF_ofCrAnList_ofStatesList]
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rw [superCommuteF_ofCrAnList_ofFieldOpFsList]
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simp only [instCommGroup.eq_1, CrAnSection.statistics_eq_state_statistics,
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Algebra.smul_mul_assoc, Finset.sum_sub_distrib]
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rw [← Finset.sum_mul, ← Finset.smul_sum, ← Finset.mul_sum, ← ofFieldOpListF_sum]
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lemma superCommuteF_ofState_ofStatesList (φ : 𝓕.States) (φs : List 𝓕.States) :
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[ofState φ, ofFieldOpListF φs]ₛca = ofState φ * ofFieldOpListF φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofState φ := by
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rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofStatesList, ofFieldOpListF_singleton]
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lemma superCommuteF_ofFieldOpF_ofFieldOpFsList (φ : 𝓕.States) (φs : List 𝓕.States) :
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[ofFieldOpF φ, ofFieldOpListF φs]ₛca = ofFieldOpF φ * ofFieldOpListF φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofFieldOpF φ := by
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rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList, ofFieldOpListF_singleton]
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simp
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lemma superCommuteF_ofFieldOpListF_ofState (φs : List 𝓕.States) (φ : 𝓕.States) :
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[ofFieldOpListF φs, ofState φ]ₛca = ofFieldOpListF φs * ofState φ -
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * ofFieldOpListF φs := by
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rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofStatesList, ofFieldOpListF_singleton]
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lemma superCommuteF_ofFieldOpListF_ofFieldOpF (φs : List 𝓕.States) (φ : 𝓕.States) :
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[ofFieldOpListF φs, ofFieldOpF φ]ₛca = ofFieldOpListF φs * ofFieldOpF φ -
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs := by
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rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList, ofFieldOpListF_singleton]
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simp
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lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) :
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@ -209,11 +209,11 @@ lemma superCommuteF_crPartF_ofFieldOpListF (φ : 𝓕.States) (φs : List 𝓕.S
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match φ with
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| States.inAsymp φ =>
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simp only [crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStatesList]
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofFieldOpFsList]
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simp [crAnStatistics]
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| States.position φ =>
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simp only [crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStatesList]
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofFieldOpFsList]
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simp [crAnStatistics]
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| States.outAsymp φ =>
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simp
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@ -227,24 +227,24 @@ lemma superCommuteF_anPartF_ofFieldOpListF (φ : 𝓕.States) (φs : List 𝓕.S
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simp
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| States.position φ =>
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simp only [anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStatesList]
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofFieldOpFsList]
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simp [crAnStatistics]
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| States.outAsymp φ =>
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simp only [anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofStatesList]
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofFieldOpFsList]
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simp [crAnStatistics]
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lemma superCommuteF_crPartF_ofState (φ φ' : 𝓕.States) :
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[crPartF φ, ofState φ']ₛca =
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crPartF φ * ofState φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofState φ' * crPartF φ := by
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lemma superCommuteF_crPartF_ofFieldOpF (φ φ' : 𝓕.States) :
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[crPartF φ, ofFieldOpF φ']ₛca =
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crPartF φ * ofFieldOpF φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOpF φ' * crPartF φ := by
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rw [← ofFieldOpListF_singleton, superCommuteF_crPartF_ofFieldOpListF]
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simp
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lemma superCommuteF_anPartF_ofState (φ φ' : 𝓕.States) :
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[anPartF φ, ofState φ']ₛca =
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anPartF φ * ofState φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofState φ' * anPartF φ := by
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lemma superCommuteF_anPartF_ofFieldOpF (φ φ' : 𝓕.States) :
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[anPartF φ, ofFieldOpF φ']ₛca =
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anPartF φ * ofFieldOpF φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOpF φ' * anPartF φ := by
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rw [← ofFieldOpListF_singleton, superCommuteF_anPartF_ofFieldOpListF]
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simp
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@ -271,19 +271,19 @@ lemma ofCrAnState_mul_ofCrAnList_eq_superCommuteF (φ : 𝓕.CrAnStates) (φs' :
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lemma ofFieldOpListF_mul_ofFieldOpListF_eq_superCommuteF (φs φs' : List 𝓕.States) :
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ofFieldOpListF φs * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpListF φs
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+ [ofFieldOpListF φs, ofFieldOpListF φs']ₛca := by
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rw [superCommuteF_ofFieldOpListF_ofStatesList]
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rw [superCommuteF_ofFieldOpListF_ofFieldOpFsList]
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simp
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lemma ofState_mul_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.States) (φs' : List 𝓕.States) :
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ofState φ * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofState φ
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+ [ofState φ, ofFieldOpListF φs']ₛca := by
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rw [superCommuteF_ofState_ofStatesList]
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lemma ofFieldOpF_mul_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.States) (φs' : List 𝓕.States) :
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ofFieldOpF φ * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpF φ
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+ [ofFieldOpF φ, ofFieldOpListF φs']ₛca := by
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rw [superCommuteF_ofFieldOpF_ofFieldOpFsList]
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simp
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lemma ofFieldOpListF_mul_ofState_eq_superCommuteF (φs : List 𝓕.States) (φ : 𝓕.States) :
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ofFieldOpListF φs * ofState φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * ofFieldOpListF φs
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+ [ofFieldOpListF φs, ofState φ]ₛca := by
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rw [superCommuteF_ofFieldOpListF_ofState]
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lemma ofFieldOpListF_mul_ofFieldOpF_eq_superCommuteF (φs : List 𝓕.States) (φ : 𝓕.States) :
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ofFieldOpListF φs * ofFieldOpF φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs
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+ [ofFieldOpListF φs, ofFieldOpF φ]ₛca := by
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rw [superCommuteF_ofFieldOpListF_ofFieldOpF]
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simp
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lemma crPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
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@ -317,7 +317,7 @@ lemma anPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
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lemma ofCrAnList_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnStates) (φs' : List 𝓕.States) :
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ofCrAnList φs * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofCrAnList φs
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+ [ofCrAnList φs, ofFieldOpListF φs']ₛca := by
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rw [superCommuteF_ofCrAnList_ofStatesList]
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rw [superCommuteF_ofCrAnList_ofFieldOpFsList]
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simp
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/-!
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@ -379,19 +379,19 @@ lemma superCommuteF_ofCrAnList_ofCrAnList_cons (φ : 𝓕.CrAnStates) (φs φs'
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lemma superCommuteF_ofCrAnList_ofFieldOpListF_cons (φ : 𝓕.States) (φs : List 𝓕.CrAnStates)
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(φs' : List 𝓕.States) : [ofCrAnList φs, ofFieldOpListF (φ :: φs')]ₛca =
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[ofCrAnList φs, ofState φ]ₛca * ofFieldOpListF φs' +
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * [ofCrAnList φs, ofFieldOpListF φs']ₛca := by
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rw [superCommuteF_ofCrAnList_ofStatesList]
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[ofCrAnList φs, ofFieldOpF φ]ₛca * ofFieldOpListF φs' +
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * [ofCrAnList φs, ofFieldOpListF φs']ₛca := by
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rw [superCommuteF_ofCrAnList_ofFieldOpFsList]
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conv_rhs =>
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lhs
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rw [← ofFieldOpListF_singleton, superCommuteF_ofCrAnList_ofStatesList, sub_mul, mul_assoc,
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rw [← ofFieldOpListF_singleton, superCommuteF_ofCrAnList_ofFieldOpFsList, sub_mul, mul_assoc,
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← ofFieldOpListF_append]
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rhs
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rw [FieldStatistic.ofList_singleton, ofFieldOpListF_singleton, smul_mul_assoc,
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smul_mul_assoc, mul_assoc]
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conv_rhs =>
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rhs
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rw [superCommuteF_ofCrAnList_ofStatesList, mul_sub, smul_mul_assoc]
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rw [superCommuteF_ofCrAnList_ofFieldOpFsList, mul_sub, smul_mul_assoc]
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simp only [instCommGroup, Algebra.smul_mul_assoc, List.singleton_append, Algebra.mul_smul_comm,
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sub_add_sub_cancel, sub_right_inj]
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rw [ofFieldOpListF_cons, mul_assoc, smul_smul, FieldStatistic.ofList_cons_eq_mul]
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@ -420,10 +420,10 @@ lemma superCommuteF_ofCrAnList_ofCrAnList_eq_sum (φs : List 𝓕.CrAnStates) :
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lemma superCommuteF_ofCrAnList_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnStates) : (φs' : List 𝓕.States) →
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[ofCrAnList φs, ofFieldOpListF φs']ₛca =
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∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
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ofFieldOpListF (φs'.take n) * [ofCrAnList φs, ofState (φs'.get n)]ₛca *
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ofFieldOpListF (φs'.take n) * [ofCrAnList φs, ofFieldOpF (φs'.get n)]ₛca *
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ofFieldOpListF (φs'.drop (n + 1))
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| [] => by
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simp only [superCommuteF_ofCrAnList_ofStatesList, instCommGroup, ofList_empty,
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simp only [superCommuteF_ofCrAnList_ofFieldOpFsList, instCommGroup, ofList_empty,
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exchangeSign_bosonic, one_smul, List.length_nil, Finset.univ_eq_empty, List.take_nil,
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List.get_eq_getElem, List.drop_nil, Finset.sum_empty]
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simp
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@ -119,27 +119,27 @@ lemma timeOrderF_ofFieldOpListF_nil : timeOrderF (𝓕 := 𝓕) (ofFieldOpListF
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lemma timeOrderF_ofFieldOpListF_singleton (φ : 𝓕.States) : 𝓣ᶠ(ofFieldOpListF [φ]) = ofFieldOpListF [φ] := by
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simp [timeOrderF_ofFieldOpListF, timeOrderSign, timeOrderList]
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lemma timeOrderF_ofState_ofState_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
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𝓣ᶠ(ofState φ * ofState ψ) = ofState φ * ofState ψ := by
|
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lemma timeOrderF_ofFieldOpF_ofFieldOpF_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
|
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𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = ofFieldOpF φ * ofFieldOpF ψ := by
|
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rw [← ofFieldOpListF_singleton, ← ofFieldOpListF_singleton, ← ofFieldOpListF_append,
|
||||
timeOrderF_ofFieldOpListF]
|
||||
simp only [List.singleton_append]
|
||||
rw [timeOrderSign_pair_ordered h, timeOrderList_pair_ordered h]
|
||||
simp
|
||||
|
||||
lemma timeOrderF_ofState_ofState_not_ordered {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
|
||||
𝓣ᶠ(ofState φ * ofState ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • ofState ψ * ofState φ := by
|
||||
lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered {φ ψ : 𝓕.States} (h : ¬ timeOrderRel φ ψ) :
|
||||
𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • ofFieldOpF ψ * ofFieldOpF φ := by
|
||||
rw [← ofFieldOpListF_singleton, ← ofFieldOpListF_singleton,
|
||||
← ofFieldOpListF_append, timeOrderF_ofFieldOpListF]
|
||||
simp only [List.singleton_append, instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
rw [timeOrderSign_pair_not_ordered h, timeOrderList_pair_not_ordered h]
|
||||
simp [← ofFieldOpListF_append]
|
||||
|
||||
lemma timeOrderF_ofState_ofState_not_ordered_eq_timeOrderF {φ ψ : 𝓕.States}
|
||||
lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered_eq_timeOrderF {φ ψ : 𝓕.States}
|
||||
(h : ¬ timeOrderRel φ ψ) :
|
||||
𝓣ᶠ(ofState φ * ofState ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓣ᶠ(ofState ψ * ofState φ) := by
|
||||
rw [timeOrderF_ofState_ofState_not_ordered h]
|
||||
rw [timeOrderF_ofState_ofState_ordered]
|
||||
𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓣ᶠ(ofFieldOpF ψ * ofFieldOpF φ) := by
|
||||
rw [timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered h]
|
||||
rw [timeOrderF_ofFieldOpF_ofFieldOpF_ordered]
|
||||
simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc]
|
||||
have hx := IsTotal.total (r := timeOrderRel) ψ φ
|
||||
simp_all
|
||||
|
|
@ -300,7 +300,7 @@ lemma timeOrderF_superCommuteF_ofCrAnState_ofCrAnState_eq_time
|
|||
lemma timeOrderF_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States) :
|
||||
𝓣ᶠ(ofFieldOpListF (φ :: φs)) =
|
||||
𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ (φ :: φs).take (maxTimeFieldPos φ φs)) •
|
||||
ofState (maxTimeField φ φs) * 𝓣ᶠ(ofFieldOpListF (eraseMaxTimeField φ φs)) := by
|
||||
ofFieldOpF (maxTimeField φ φs) * 𝓣ᶠ(ofFieldOpListF (eraseMaxTimeField φ φs)) := by
|
||||
rw [timeOrderF_ofFieldOpListF, timeOrderList_eq_maxTimeField_timeOrderList]
|
||||
rw [ofFieldOpListF_cons, timeOrderF_ofFieldOpListF]
|
||||
simp only [instCommGroup.eq_1, Algebra.mul_smul_comm, Algebra.smul_mul_assoc, smul_smul]
|
||||
|
|
@ -316,7 +316,7 @@ lemma timeOrderF_eq_maxTimeField_mul_finset (φ : 𝓕.States) (φs : List 𝓕.
|
|||
𝓣ᶠ(ofFieldOpListF (φ :: φs)) = 𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ ⟨(eraseMaxTimeField φ φs).get,
|
||||
(Finset.filter (fun x =>
|
||||
(maxTimeFieldPosFin φ φs).succAbove x < maxTimeFieldPosFin φ φs) Finset.univ)⟩) •
|
||||
ofState (maxTimeField φ φs) * 𝓣ᶠ(ofFieldOpListF (eraseMaxTimeField φ φs)) := by
|
||||
ofFieldOpF (maxTimeField φ φs) * 𝓣ᶠ(ofFieldOpListF (eraseMaxTimeField φ φs)) := by
|
||||
rw [timeOrderF_eq_maxTimeField_mul]
|
||||
congr 3
|
||||
apply FieldStatistic.ofList_perm
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue