refactor: Rename ofCrAnState and ofCrAnList
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jstoobysmith
parent
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11 changed files with 601 additions and 601 deletions
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@ -27,9 +27,9 @@ open FieldStatistic
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or a bosonic and fermionic operator this corresponds to the usual commutator
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whilst for two fermionic operators this corresponds to the anti-commutator. -/
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noncomputable def superCommuteF : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra :=
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Basis.constr ofCrAnListBasis ℂ fun φs =>
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Basis.constr ofCrAnListBasis ℂ fun φs' =>
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ofCrAnList (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnList (φs' ++ φs)
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Basis.constr ofCrAnListFBasis ℂ fun φs =>
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Basis.constr ofCrAnListFBasis ℂ fun φs' =>
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ofCrAnListF (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF (φs' ++ φs)
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/-- The super commutor on the creation and annihlation algebra. For two bosonic operators
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or a bosonic and fermionic operator this corresponds to the usual commutator
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@ -42,30 +42,30 @@ scoped[FieldSpecification.FieldOpFreeAlgebra] notation "[" φs "," φs' "]ₛca"
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-/
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lemma superCommuteF_ofCrAnList_ofCrAnList (φs φs' : List 𝓕.CrAnStates) :
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[ofCrAnList φs, ofCrAnList φs']ₛca =
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ofCrAnList (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnList (φs' ++ φs) := by
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lemma superCommuteF_ofCrAnListF_ofCrAnListF (φs φs' : List 𝓕.CrAnStates) :
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[ofCrAnListF φs, ofCrAnListF φs']ₛca =
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ofCrAnListF (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF (φs' ++ φs) := by
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rw [← ofListBasis_eq_ofList, ← ofListBasis_eq_ofList]
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simp only [superCommuteF, Basis.constr_basis]
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lemma superCommuteF_ofCrAnState_ofCrAnState (φ φ' : 𝓕.CrAnStates) :
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[ofCrAnState φ, ofCrAnState φ']ₛca =
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ofCrAnState φ * ofCrAnState φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofCrAnState φ' * ofCrAnState φ := by
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rw [superCommuteF_ofCrAnList_ofCrAnList, ofCrAnList_append]
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lemma superCommuteF_ofCrAnOpF_ofCrAnOpF (φ φ' : 𝓕.CrAnStates) :
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[ofCrAnOpF φ, ofCrAnOpF φ']ₛca =
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ofCrAnOpF φ * ofCrAnOpF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofCrAnOpF φ' * ofCrAnOpF φ := by
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
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rw [superCommuteF_ofCrAnListF_ofCrAnListF, ofCrAnListF_append]
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congr
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rw [ofCrAnList_append]
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rw [ofCrAnListF_append]
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rw [FieldStatistic.ofList_singleton, FieldStatistic.ofList_singleton, smul_mul_assoc]
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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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lemma superCommuteF_ofCrAnListF_ofFieldOpFsList (φcas : List 𝓕.CrAnStates) (φs : List 𝓕.States) :
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[ofCrAnListF φcas, ofFieldOpListF φs]ₛca = ofCrAnListF φcas * ofFieldOpListF φs -
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𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofCrAnListF φcas := by
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conv_lhs => rw [ofFieldOpListF_sum]
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rw [map_sum]
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conv_lhs =>
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enter [2, x]
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rw [superCommuteF_ofCrAnList_ofCrAnList, CrAnSection.statistics_eq_state_statistics,
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ofCrAnList_append, ofCrAnList_append]
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rw [superCommuteF_ofCrAnListF_ofCrAnListF, CrAnSection.statistics_eq_state_statistics,
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ofCrAnListF_append, ofCrAnListF_append]
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rw [Finset.sum_sub_distrib, ← Finset.mul_sum, ← Finset.smul_sum,
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← Finset.sum_mul, ← ofFieldOpListF_sum]
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simp
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@ -78,7 +78,7 @@ lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.States) (φs
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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_ofFieldOpFsList]
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rw [superCommuteF_ofCrAnListF_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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@ -106,22 +106,22 @@ lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.States) :
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sub_self]
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| States.position φ, States.position φ' =>
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simp only [anPartF_position, crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| States.outAsymp φ, States.position φ' =>
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simp only [anPartF_posAsymp, crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| States.position φ, States.inAsymp φ' =>
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simp only [anPartF_position, crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp only [List.singleton_append, instCommGroup.eq_1, crAnStatistics,
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FieldStatistic.ofList_singleton, Function.comp_apply, crAnStatesToStates_prod, ←
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ofCrAnList_append]
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ofCrAnListF_append]
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| States.outAsymp φ, States.inAsymp φ' =>
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simp only [anPartF_posAsymp, crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.States) :
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[crPartF φ, anPartF φ']ₛca = crPartF φ * anPartF φ' -
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@ -135,20 +135,20 @@ lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.States) :
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sub_self]
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| States.position φ, States.position φ' =>
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simp only [crPartF_position, anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| States.position φ, States.outAsymp φ' =>
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simp only [crPartF_position, anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| States.inAsymp φ, States.position φ' =>
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simp only [crPartF_negAsymp, anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| States.inAsymp φ, States.outAsymp φ' =>
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simp only [crPartF_negAsymp, anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.States) :
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[crPartF φ, crPartF φ']ₛca = crPartF φ * crPartF φ' -
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@ -162,20 +162,20 @@ lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.States) :
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sub_self]
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| States.position φ, 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, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| States.position φ, States.inAsymp φ' =>
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simp only [crPartF_position, crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| States.inAsymp φ, States.position φ' =>
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simp only [crPartF_negAsymp, crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| States.inAsymp φ, 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, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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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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@ -187,20 +187,20 @@ lemma superCommuteF_anPartF_anPartF (φ φ' : 𝓕.States) :
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simp
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| States.position φ, 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, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| States.position φ, States.outAsymp φ' =>
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simp only [anPartF_position, anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| States.outAsymp φ, States.position φ' =>
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simp only [anPartF_posAsymp, anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| States.outAsymp φ, 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, ← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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lemma superCommuteF_crPartF_ofFieldOpListF (φ : 𝓕.States) (φs : List 𝓕.States) :
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[crPartF φ, ofFieldOpListF φs]ₛca =
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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_ofFieldOpFsList]
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rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_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_ofFieldOpFsList]
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rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
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simp [crAnStatistics]
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| States.outAsymp φ =>
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simp
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@ -227,11 +227,11 @@ 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_ofFieldOpFsList]
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rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_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_ofFieldOpFsList]
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rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
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simp [crAnStatistics]
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lemma superCommuteF_crPartF_ofFieldOpF (φ φ' : 𝓕.States) :
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@ -256,16 +256,16 @@ Lemmas which rewrite a multiplication of two elements of the algebra as their co
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multiplication with a sign plus the super commutor.
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-/
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lemma ofCrAnList_mul_ofCrAnList_eq_superCommuteF (φs φs' : List 𝓕.CrAnStates) :
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ofCrAnList φs * ofCrAnList φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnList φs' * ofCrAnList φs
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+ [ofCrAnList φs, ofCrAnList φs']ₛca := by
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rw [superCommuteF_ofCrAnList_ofCrAnList]
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simp [ofCrAnList_append]
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lemma ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF (φs φs' : List 𝓕.CrAnStates) :
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ofCrAnListF φs * ofCrAnListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF φs' * ofCrAnListF φs
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+ [ofCrAnListF φs, ofCrAnListF φs']ₛca := by
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rw [superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [ofCrAnListF_append]
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lemma ofCrAnState_mul_ofCrAnList_eq_superCommuteF (φ : 𝓕.CrAnStates) (φs' : List 𝓕.CrAnStates) :
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ofCrAnState φ * ofCrAnList φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofCrAnList φs' * ofCrAnState φ
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+ [ofCrAnState φ, ofCrAnList φs']ₛca := by
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rw [← ofCrAnList_singleton, ofCrAnList_mul_ofCrAnList_eq_superCommuteF]
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lemma ofCrAnOpF_mul_ofCrAnListF_eq_superCommuteF (φ : 𝓕.CrAnStates) (φs' : List 𝓕.CrAnStates) :
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ofCrAnOpF φ * ofCrAnListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofCrAnListF φs' * ofCrAnOpF φ
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+ [ofCrAnOpF φ, ofCrAnListF φs']ₛca := by
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rw [← ofCrAnListF_singleton, ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF]
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simp
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lemma ofFieldOpListF_mul_ofFieldOpListF_eq_superCommuteF (φs φs' : List 𝓕.States) :
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@ -314,10 +314,10 @@ lemma anPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.States) :
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rw [superCommuteF_anPartF_anPartF]
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simp
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|
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lemma ofCrAnList_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnStates) (φs' : List 𝓕.States) :
|
||||
ofCrAnList φs * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofCrAnList φs
|
||||
+ [ofCrAnList φs, ofFieldOpListF φs']ₛca := by
|
||||
rw [superCommuteF_ofCrAnList_ofFieldOpFsList]
|
||||
lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnStates) (φs' : List 𝓕.States) :
|
||||
ofCrAnListF φs * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofCrAnListF φs
|
||||
+ [ofCrAnListF φs, ofFieldOpListF φs']ₛca := by
|
||||
rw [superCommuteF_ofCrAnListF_ofFieldOpFsList]
|
||||
simp
|
||||
|
||||
/-!
|
||||
|
|
@ -326,10 +326,10 @@ lemma ofCrAnList_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnStates
|
|||
|
||||
-/
|
||||
|
||||
lemma superCommuteF_ofCrAnList_ofCrAnList_symm (φs φs' : List 𝓕.CrAnStates) :
|
||||
[ofCrAnList φs, ofCrAnList φs']ₛca =
|
||||
(- 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs')) • [ofCrAnList φs', ofCrAnList φs]ₛca := by
|
||||
rw [superCommuteF_ofCrAnList_ofCrAnList, superCommuteF_ofCrAnList_ofCrAnList, smul_sub]
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_symm (φs φs' : List 𝓕.CrAnStates) :
|
||||
[ofCrAnListF φs, ofCrAnListF φs']ₛca =
|
||||
(- 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs')) • [ofCrAnListF φs', ofCrAnListF φs]ₛca := by
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF, superCommuteF_ofCrAnListF_ofCrAnListF, smul_sub]
|
||||
simp only [instCommGroup.eq_1, neg_smul, sub_neg_eq_add]
|
||||
rw [smul_smul]
|
||||
conv_rhs =>
|
||||
|
|
@ -338,10 +338,10 @@ lemma superCommuteF_ofCrAnList_ofCrAnList_symm (φs φs' : List 𝓕.CrAnStates)
|
|||
simp only [one_smul]
|
||||
abel
|
||||
|
||||
lemma superCommuteF_ofCrAnState_ofCrAnState_symm (φ φ' : 𝓕.CrAnStates) :
|
||||
[ofCrAnState φ, ofCrAnState φ']ₛca =
|
||||
(- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • [ofCrAnState φ', ofCrAnState φ]ₛca := by
|
||||
rw [superCommuteF_ofCrAnState_ofCrAnState, superCommuteF_ofCrAnState_ofCrAnState]
|
||||
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_symm (φ φ' : 𝓕.CrAnStates) :
|
||||
[ofCrAnOpF φ, ofCrAnOpF φ']ₛca =
|
||||
(- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • [ofCrAnOpF φ', ofCrAnOpF φ]ₛca := by
|
||||
rw [superCommuteF_ofCrAnOpF_ofCrAnOpF, superCommuteF_ofCrAnOpF_ofCrAnOpF]
|
||||
rw [smul_sub]
|
||||
simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, neg_smul, sub_neg_eq_add]
|
||||
rw [smul_smul]
|
||||
|
|
@ -357,41 +357,41 @@ lemma superCommuteF_ofCrAnState_ofCrAnState_symm (φ φ' : 𝓕.CrAnStates) :
|
|||
|
||||
-/
|
||||
|
||||
lemma superCommuteF_ofCrAnList_ofCrAnList_cons (φ : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
|
||||
[ofCrAnList φs, ofCrAnList (φ :: φs')]ₛca =
|
||||
[ofCrAnList φs, ofCrAnState φ]ₛca * ofCrAnList φs' +
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (φ : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
|
||||
[ofCrAnListF φs, ofCrAnListF (φ :: φs')]ₛca =
|
||||
[ofCrAnListF φs, ofCrAnOpF φ]ₛca * ofCrAnListF φs' +
|
||||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ)
|
||||
• ofCrAnState φ * [ofCrAnList φs, ofCrAnList φs']ₛca := by
|
||||
rw [superCommuteF_ofCrAnList_ofCrAnList]
|
||||
• ofCrAnOpF φ * [ofCrAnListF φs, ofCrAnListF φs']ₛca := by
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
conv_rhs =>
|
||||
lhs
|
||||
rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList, sub_mul, ← ofCrAnList_append]
|
||||
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF, sub_mul, ← ofCrAnListF_append]
|
||||
rhs
|
||||
rw [FieldStatistic.ofList_singleton, ofCrAnList_append, ofCrAnList_singleton, smul_mul_assoc,
|
||||
mul_assoc, ← ofCrAnList_append]
|
||||
rw [FieldStatistic.ofList_singleton, ofCrAnListF_append, ofCrAnListF_singleton, smul_mul_assoc,
|
||||
mul_assoc, ← ofCrAnListF_append]
|
||||
conv_rhs =>
|
||||
rhs
|
||||
rw [superCommuteF_ofCrAnList_ofCrAnList, mul_sub, smul_mul_assoc]
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF, mul_sub, smul_mul_assoc]
|
||||
simp only [instCommGroup.eq_1, List.cons_append, List.append_assoc, List.nil_append,
|
||||
Algebra.mul_smul_comm, Algebra.smul_mul_assoc, sub_add_sub_cancel, sub_right_inj]
|
||||
rw [← ofCrAnList_cons, smul_smul, FieldStatistic.ofList_cons_eq_mul]
|
||||
rw [← ofCrAnListF_cons, smul_smul, FieldStatistic.ofList_cons_eq_mul]
|
||||
simp only [instCommGroup, map_mul, mul_comm]
|
||||
|
||||
lemma superCommuteF_ofCrAnList_ofFieldOpListF_cons (φ : 𝓕.States) (φs : List 𝓕.CrAnStates)
|
||||
(φs' : List 𝓕.States) : [ofCrAnList φs, ofFieldOpListF (φ :: φs')]ₛca =
|
||||
[ofCrAnList φs, ofFieldOpF φ]ₛca * ofFieldOpListF φs' +
|
||||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * [ofCrAnList φs, ofFieldOpListF φs']ₛca := by
|
||||
rw [superCommuteF_ofCrAnList_ofFieldOpFsList]
|
||||
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_cons (φ : 𝓕.States) (φs : List 𝓕.CrAnStates)
|
||||
(φs' : List 𝓕.States) : [ofCrAnListF φs, ofFieldOpListF (φ :: φs')]ₛca =
|
||||
[ofCrAnListF φs, ofFieldOpF φ]ₛca * ofFieldOpListF φs' +
|
||||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * [ofCrAnListF φs, ofFieldOpListF φs']ₛca := by
|
||||
rw [superCommuteF_ofCrAnListF_ofFieldOpFsList]
|
||||
conv_rhs =>
|
||||
lhs
|
||||
rw [← ofFieldOpListF_singleton, superCommuteF_ofCrAnList_ofFieldOpFsList, sub_mul, mul_assoc,
|
||||
rw [← ofFieldOpListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList, sub_mul, mul_assoc,
|
||||
← ofFieldOpListF_append]
|
||||
rhs
|
||||
rw [FieldStatistic.ofList_singleton, ofFieldOpListF_singleton, smul_mul_assoc,
|
||||
smul_mul_assoc, mul_assoc]
|
||||
conv_rhs =>
|
||||
rhs
|
||||
rw [superCommuteF_ofCrAnList_ofFieldOpFsList, mul_sub, smul_mul_assoc]
|
||||
rw [superCommuteF_ofCrAnListF_ofFieldOpFsList, mul_sub, smul_mul_assoc]
|
||||
simp only [instCommGroup, Algebra.smul_mul_assoc, List.singleton_append, Algebra.mul_smul_comm,
|
||||
sub_add_sub_cancel, sub_right_inj]
|
||||
rw [ofFieldOpListF_cons, mul_assoc, smul_smul, FieldStatistic.ofList_cons_eq_mul]
|
||||
|
|
@ -402,48 +402,48 @@ Within the creation and annihilation algebra, we have that
|
|||
`[φᶜᵃs, φᶜᵃ₀ … φᶜᵃₙ]ₛca = ∑ i, sᵢ • φᶜᵃs₀ … φᶜᵃᵢ₋₁ * [φᶜᵃs, φᶜᵃᵢ]ₛca * φᶜᵃᵢ₊₁ … φᶜᵃₙ`
|
||||
where `sᵢ` is the exchange sign for `φᶜᵃs` and `φᶜᵃs₀ … φᶜᵃᵢ₋₁`.
|
||||
-/
|
||||
lemma superCommuteF_ofCrAnList_ofCrAnList_eq_sum (φs : List 𝓕.CrAnStates) :
|
||||
(φs' : List 𝓕.CrAnStates) → [ofCrAnList φs, ofCrAnList φs']ₛca =
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum (φs : List 𝓕.CrAnStates) :
|
||||
(φs' : List 𝓕.CrAnStates) → [ofCrAnListF φs, ofCrAnListF φs']ₛca =
|
||||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
|
||||
ofCrAnList (φs'.take n) * [ofCrAnList φs, ofCrAnState (φs'.get n)]ₛca *
|
||||
ofCrAnList (φs'.drop (n + 1))
|
||||
ofCrAnListF (φs'.take n) * [ofCrAnListF φs, ofCrAnOpF (φs'.get n)]ₛca *
|
||||
ofCrAnListF (φs'.drop (n + 1))
|
||||
| [] => by
|
||||
simp [← ofCrAnList_nil, superCommuteF_ofCrAnList_ofCrAnList]
|
||||
simp [← ofCrAnListF_nil, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
| φ :: φs' => by
|
||||
rw [superCommuteF_ofCrAnList_ofCrAnList_cons, superCommuteF_ofCrAnList_ofCrAnList_eq_sum φs φs']
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF_cons, superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum φs φs']
|
||||
conv_rhs => erw [Fin.sum_univ_succ]
|
||||
congr 1
|
||||
· simp
|
||||
· simp [Finset.mul_sum, smul_smul, ofCrAnList_cons, mul_assoc,
|
||||
· simp [Finset.mul_sum, smul_smul, ofCrAnListF_cons, mul_assoc,
|
||||
FieldStatistic.ofList_cons_eq_mul, mul_comm]
|
||||
|
||||
lemma superCommuteF_ofCrAnList_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnStates) : (φs' : List 𝓕.States) →
|
||||
[ofCrAnList φs, ofFieldOpListF φs']ₛca =
|
||||
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnStates) : (φs' : List 𝓕.States) →
|
||||
[ofCrAnListF φs, ofFieldOpListF φs']ₛca =
|
||||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
|
||||
ofFieldOpListF (φs'.take n) * [ofCrAnList φs, ofFieldOpF (φs'.get n)]ₛca *
|
||||
ofFieldOpListF (φs'.take n) * [ofCrAnListF φs, ofFieldOpF (φs'.get n)]ₛca *
|
||||
ofFieldOpListF (φs'.drop (n + 1))
|
||||
| [] => by
|
||||
simp only [superCommuteF_ofCrAnList_ofFieldOpFsList, instCommGroup, ofList_empty,
|
||||
simp only [superCommuteF_ofCrAnListF_ofFieldOpFsList, instCommGroup, ofList_empty,
|
||||
exchangeSign_bosonic, one_smul, List.length_nil, Finset.univ_eq_empty, List.take_nil,
|
||||
List.get_eq_getElem, List.drop_nil, Finset.sum_empty]
|
||||
simp
|
||||
| φ :: φs' => by
|
||||
rw [superCommuteF_ofCrAnList_ofFieldOpListF_cons,
|
||||
superCommuteF_ofCrAnList_ofFieldOpListF_eq_sum φs φs']
|
||||
rw [superCommuteF_ofCrAnListF_ofFieldOpListF_cons,
|
||||
superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum φs φs']
|
||||
conv_rhs => erw [Fin.sum_univ_succ]
|
||||
congr 1
|
||||
· simp
|
||||
· simp [Finset.mul_sum, smul_smul, ofFieldOpListF_cons, mul_assoc,
|
||||
FieldStatistic.ofList_cons_eq_mul, mul_comm]
|
||||
|
||||
lemma summerCommute_jacobi_ofCrAnList (φs1 φs2 φs3 : List 𝓕.CrAnStates) :
|
||||
[ofCrAnList φs1, [ofCrAnList φs2, ofCrAnList φs3]ₛca]ₛca =
|
||||
lemma summerCommute_jacobi_ofCrAnListF (φs1 φs2 φs3 : List 𝓕.CrAnStates) :
|
||||
[ofCrAnListF φs1, [ofCrAnListF φs2, ofCrAnListF φs3]ₛca]ₛca =
|
||||
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs3) •
|
||||
(- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3) • [ofCrAnList φs3, [ofCrAnList φs1, ofCrAnList φs2]ₛca]ₛca -
|
||||
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs2) • [ofCrAnList φs2, [ofCrAnList φs3, ofCrAnList φs1]ₛca]ₛca) := by
|
||||
repeat rw [superCommuteF_ofCrAnList_ofCrAnList]
|
||||
(- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3) • [ofCrAnListF φs3, [ofCrAnListF φs1, ofCrAnListF φs2]ₛca]ₛca -
|
||||
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs2) • [ofCrAnListF φs2, [ofCrAnListF φs3, ofCrAnListF φs1]ₛca]ₛca) := by
|
||||
repeat rw [superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp only [instCommGroup, map_sub, map_smul, neg_smul]
|
||||
repeat rw [superCommuteF_ofCrAnList_ofCrAnList]
|
||||
repeat rw [superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp only [instCommGroup.eq_1, ofList_append_eq_mul, List.append_assoc]
|
||||
by_cases h1 : (𝓕 |>ₛ φs1) = bosonic <;>
|
||||
by_cases h2 : (𝓕 |>ₛ φs2) = bosonic <;>
|
||||
|
|
@ -497,18 +497,18 @@ lemma superCommuteF_grade {a b : 𝓕.FieldOpFreeAlgebra} {f1 f2 : FieldStatisti
|
|||
obtain ⟨φs, rfl, hφs⟩ := hx
|
||||
simp only [add_eq_mul, instCommGroup, p]
|
||||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule f1) : Prop :=
|
||||
[a2, ofCrAnList φs]ₛca ∈ statisticSubmodule (f1 + f2)
|
||||
[a2, ofCrAnListF φs]ₛca ∈ statisticSubmodule (f1 + f2)
|
||||
change p a ha
|
||||
apply Submodule.span_induction (p := p)
|
||||
· intro x hx
|
||||
obtain ⟨φs', rfl, hφs'⟩ := hx
|
||||
simp only [add_eq_mul, instCommGroup, p]
|
||||
rw [superCommuteF_ofCrAnList_ofCrAnList]
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
apply Submodule.sub_mem _
|
||||
· apply ofCrAnList_mem_statisticSubmodule_of
|
||||
· apply ofCrAnListF_mem_statisticSubmodule_of
|
||||
rw [ofList_append_eq_mul, hφs, hφs']
|
||||
· apply Submodule.smul_mem
|
||||
apply ofCrAnList_mem_statisticSubmodule_of
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of
|
||||
rw [ofList_append_eq_mul, hφs, hφs']
|
||||
rw [mul_comm]
|
||||
· simp [p]
|
||||
|
|
@ -538,14 +538,14 @@ lemma superCommuteF_bosonic_bosonic {a b : 𝓕.FieldOpFreeAlgebra}
|
|||
· intro x hx
|
||||
obtain ⟨φs, rfl, hφs⟩ := hx
|
||||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
|
||||
[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
|
||||
[a2, ofCrAnListF φs]ₛca = a2 * ofCrAnListF φs - ofCrAnListF φs * a2
|
||||
change p a ha
|
||||
apply Submodule.span_induction (p := p)
|
||||
· intro x hx
|
||||
obtain ⟨φs', rfl, hφs'⟩ := hx
|
||||
simp only [p]
|
||||
rw [superCommuteF_ofCrAnList_ofCrAnList]
|
||||
simp [hφs, ofCrAnList_append]
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [hφs, ofCrAnListF_append]
|
||||
· simp [p]
|
||||
· intro x y hx hy hp1 hp2
|
||||
simp_all only [p, map_add, LinearMap.add_apply, add_mul, mul_add]
|
||||
|
|
@ -571,14 +571,14 @@ lemma superCommuteF_bosonic_fermionic {a b : 𝓕.FieldOpFreeAlgebra}
|
|||
· intro x hx
|
||||
obtain ⟨φs, rfl, hφs⟩ := hx
|
||||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
|
||||
[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
|
||||
[a2, ofCrAnListF φs]ₛca = a2 * ofCrAnListF φs - ofCrAnListF φs * a2
|
||||
change p a ha
|
||||
apply Submodule.span_induction (p := p)
|
||||
· intro x hx
|
||||
obtain ⟨φs', rfl, hφs'⟩ := hx
|
||||
simp only [p]
|
||||
rw [superCommuteF_ofCrAnList_ofCrAnList]
|
||||
simp [hφs, hφs', ofCrAnList_append]
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [hφs, hφs', ofCrAnListF_append]
|
||||
· simp [p]
|
||||
· intro x y hx hy hp1 hp2
|
||||
simp_all only [p, map_add, LinearMap.add_apply, add_mul, mul_add]
|
||||
|
|
@ -604,14 +604,14 @@ lemma superCommuteF_fermionic_bonsonic {a b : 𝓕.FieldOpFreeAlgebra}
|
|||
· intro x hx
|
||||
obtain ⟨φs, rfl, hφs⟩ := hx
|
||||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
|
||||
[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
|
||||
[a2, ofCrAnListF φs]ₛca = a2 * ofCrAnListF φs - ofCrAnListF φs * a2
|
||||
change p a ha
|
||||
apply Submodule.span_induction (p := p)
|
||||
· intro x hx
|
||||
obtain ⟨φs', rfl, hφs'⟩ := hx
|
||||
simp only [p]
|
||||
rw [superCommuteF_ofCrAnList_ofCrAnList]
|
||||
simp [hφs, hφs', ofCrAnList_append]
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [hφs, hφs', ofCrAnListF_append]
|
||||
· simp [p]
|
||||
· intro x y hx hy hp1 hp2
|
||||
simp_all only [p, map_add, LinearMap.add_apply, add_mul, mul_add]
|
||||
|
|
@ -663,14 +663,14 @@ lemma superCommuteF_fermionic_fermionic {a b : 𝓕.FieldOpFreeAlgebra}
|
|||
· intro x hx
|
||||
obtain ⟨φs, rfl, hφs⟩ := hx
|
||||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
|
||||
[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs + ofCrAnList φs * a2
|
||||
[a2, ofCrAnListF φs]ₛca = a2 * ofCrAnListF φs + ofCrAnListF φs * a2
|
||||
change p a ha
|
||||
apply Submodule.span_induction (p := p)
|
||||
· intro x hx
|
||||
obtain ⟨φs', rfl, hφs'⟩ := hx
|
||||
simp only [p]
|
||||
rw [superCommuteF_ofCrAnList_ofCrAnList]
|
||||
simp [hφs, hφs', ofCrAnList_append]
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF]
|
||||
simp [hφs, hφs', ofCrAnListF_append]
|
||||
· simp [p]
|
||||
· intro x y hx hy hp1 hp2
|
||||
simp_all only [p, map_add, LinearMap.add_apply, add_mul, mul_add]
|
||||
|
|
@ -706,9 +706,9 @@ lemma superCommuteF_expand_bosonicProj_fermionicProj (a b : 𝓕.FieldOpFreeAlge
|
|||
superCommuteF_fermionic_fermionic (by simp) (by simp)]
|
||||
abel
|
||||
|
||||
lemma superCommuteF_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnStates) :
|
||||
[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule bosonic ∨
|
||||
[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule fermionic := by
|
||||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnStates) :
|
||||
[ofCrAnListF φs, ofCrAnListF φs']ₛca ∈ statisticSubmodule bosonic ∨
|
||||
[ofCrAnListF φs, ofCrAnListF φs']ₛca ∈ statisticSubmodule fermionic := by
|
||||
by_cases h1 : (𝓕 |>ₛ φs) = bosonic <;> by_cases h2 : (𝓕 |>ₛ φs') = bosonic
|
||||
· left
|
||||
have h : bosonic = bosonic + bosonic := by
|
||||
|
|
@ -716,44 +716,44 @@ lemma superCommuteF_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List
|
|||
rfl
|
||||
rw [h]
|
||||
apply superCommuteF_grade
|
||||
apply ofCrAnList_mem_statisticSubmodule_of _ _ h1
|
||||
apply ofCrAnList_mem_statisticSubmodule_of _ _ h2
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _ h1
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _ h2
|
||||
· right
|
||||
have h : fermionic = bosonic + fermionic := by
|
||||
simp only [add_eq_mul, instCommGroup, mul_self]
|
||||
rfl
|
||||
rw [h]
|
||||
apply superCommuteF_grade
|
||||
apply ofCrAnList_mem_statisticSubmodule_of _ _ h1
|
||||
apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h2)
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _ h1
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h2)
|
||||
· right
|
||||
have h : fermionic = fermionic + bosonic := by
|
||||
simp only [add_eq_mul, instCommGroup, mul_self]
|
||||
rfl
|
||||
rw [h]
|
||||
apply superCommuteF_grade
|
||||
apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h1)
|
||||
apply ofCrAnList_mem_statisticSubmodule_of _ _ h2
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h1)
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _ h2
|
||||
· left
|
||||
have h : bosonic = fermionic + fermionic := by
|
||||
simp only [add_eq_mul, instCommGroup, mul_self]
|
||||
rfl
|
||||
rw [h]
|
||||
apply superCommuteF_grade
|
||||
apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h1)
|
||||
apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h2)
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h1)
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h2)
|
||||
|
||||
lemma superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_fermionic (φ φ' : 𝓕.CrAnStates) :
|
||||
[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule bosonic ∨
|
||||
[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule fermionic := by
|
||||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
|
||||
exact superCommuteF_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ] [φ']
|
||||
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic (φ φ' : 𝓕.CrAnStates) :
|
||||
[ofCrAnOpF φ, ofCrAnOpF φ']ₛca ∈ statisticSubmodule bosonic ∨
|
||||
[ofCrAnOpF φ, ofCrAnOpF φ']ₛca ∈ statisticSubmodule fermionic := by
|
||||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
|
||||
exact superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ] [φ']
|
||||
|
||||
lemma superCommuteF_superCommuteF_ofCrAnState_bosonic_or_fermionic (φ1 φ2 φ3 : 𝓕.CrAnStates) :
|
||||
[ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca ∈ statisticSubmodule bosonic ∨
|
||||
[ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca ∈ statisticSubmodule fermionic := by
|
||||
rcases superCommuteF_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ2 φ3 with hs | hs
|
||||
<;> rcases ofCrAnState_bosonic_or_fermionic φ1 with h1 | h1
|
||||
lemma superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic (φ1 φ2 φ3 : 𝓕.CrAnStates) :
|
||||
[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca ∈ statisticSubmodule bosonic ∨
|
||||
[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca ∈ statisticSubmodule fermionic := by
|
||||
rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φ2 φ3 with hs | hs
|
||||
<;> rcases ofCrAnOpF_bosonic_or_fermionic φ1 with h1 | h1
|
||||
· left
|
||||
have h : bosonic = bosonic + bosonic := by
|
||||
simp only [add_eq_mul, instCommGroup, mul_self]
|
||||
|
|
@ -779,21 +779,21 @@ lemma superCommuteF_superCommuteF_ofCrAnState_bosonic_or_fermionic (φ1 φ2 φ3
|
|||
rw [h]
|
||||
apply superCommuteF_grade h1 hs
|
||||
|
||||
lemma superCommuteF_bosonic_ofCrAnList_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnStates)
|
||||
lemma superCommuteF_bosonic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnStates)
|
||||
(ha : a ∈ statisticSubmodule bosonic) :
|
||||
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length),
|
||||
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
|
||||
ofCrAnList (φs.drop (n + 1)) := by
|
||||
[a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length),
|
||||
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
|
||||
ofCrAnListF (φs.drop (n + 1)) := by
|
||||
let p (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ statisticSubmodule bosonic) : Prop :=
|
||||
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length),
|
||||
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
|
||||
ofCrAnList (φs.drop (n + 1))
|
||||
[a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length),
|
||||
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
|
||||
ofCrAnListF (φs.drop (n + 1))
|
||||
change p a ha
|
||||
apply Submodule.span_induction (p := p)
|
||||
· intro a ha
|
||||
obtain ⟨φs, rfl, hφs⟩ := ha
|
||||
simp only [List.get_eq_getElem, p]
|
||||
rw [superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum]
|
||||
congr
|
||||
funext n
|
||||
simp [hφs]
|
||||
|
|
@ -808,21 +808,21 @@ lemma superCommuteF_bosonic_ofCrAnList_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs
|
|||
simp_all [p, Finset.smul_sum]
|
||||
· exact ha
|
||||
|
||||
lemma superCommuteF_fermionic_ofCrAnList_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnStates)
|
||||
lemma superCommuteF_fermionic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnStates)
|
||||
(ha : a ∈ statisticSubmodule fermionic) :
|
||||
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
|
||||
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
|
||||
ofCrAnList (φs.drop (n + 1)) := by
|
||||
[a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
|
||||
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
|
||||
ofCrAnListF (φs.drop (n + 1)) := by
|
||||
let p (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ statisticSubmodule fermionic) : Prop :=
|
||||
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
|
||||
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
|
||||
ofCrAnList (φs.drop (n + 1))
|
||||
[a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
|
||||
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
|
||||
ofCrAnListF (φs.drop (n + 1))
|
||||
change p a ha
|
||||
apply Submodule.span_induction (p := p)
|
||||
· intro a ha
|
||||
obtain ⟨φs, rfl, hφs⟩ := ha
|
||||
simp only [instCommGroup, List.get_eq_getElem, Algebra.smul_mul_assoc, p]
|
||||
rw [superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
|
||||
rw [superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum]
|
||||
congr
|
||||
funext n
|
||||
simp [hφs]
|
||||
|
|
@ -843,9 +843,9 @@ lemma superCommuteF_fermionic_ofCrAnList_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (
|
|||
· exact ha
|
||||
|
||||
lemma statistic_neq_of_superCommuteF_fermionic {φs φs' : List 𝓕.CrAnStates}
|
||||
(h : [ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule fermionic) :
|
||||
(𝓕 |>ₛ φs) ≠ (𝓕 |>ₛ φs') ∨ [ofCrAnList φs, ofCrAnList φs']ₛca = 0 := by
|
||||
by_cases h0 : [ofCrAnList φs, ofCrAnList φs']ₛca = 0
|
||||
(h : [ofCrAnListF φs, ofCrAnListF φs']ₛca ∈ statisticSubmodule fermionic) :
|
||||
(𝓕 |>ₛ φs) ≠ (𝓕 |>ₛ φs') ∨ [ofCrAnListF φs, ofCrAnListF φs']ₛca = 0 := by
|
||||
by_cases h0 : [ofCrAnListF φs, ofCrAnListF φs']ₛca = 0
|
||||
· simp [h0]
|
||||
simp only [ne_eq, h0, or_false]
|
||||
by_contra hn
|
||||
|
|
@ -856,17 +856,17 @@ lemma statistic_neq_of_superCommuteF_fermionic {φs φs' : List 𝓕.CrAnStates}
|
|||
rfl
|
||||
rw [h1]
|
||||
apply superCommuteF_grade
|
||||
apply ofCrAnList_mem_statisticSubmodule_of _ _ hc
|
||||
apply ofCrAnList_mem_statisticSubmodule_of _ _
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _ hc
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _
|
||||
rw [← hn, hc]
|
||||
· have h1 : bosonic = fermionic + fermionic := by
|
||||
simp only [add_eq_mul, instCommGroup, mul_self]
|
||||
rfl
|
||||
rw [h1]
|
||||
apply superCommuteF_grade
|
||||
apply ofCrAnList_mem_statisticSubmodule_of _ _
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _
|
||||
simpa using hc
|
||||
apply ofCrAnList_mem_statisticSubmodule_of _ _
|
||||
apply ofCrAnListF_mem_statisticSubmodule_of _ _
|
||||
rw [← hn]
|
||||
simpa using hc
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue