890 lines
38 KiB
Text
890 lines
38 KiB
Text
/-
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Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.FieldOpFreeAlgebra.Basic
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import HepLean.PerturbationTheory.FieldOpFreeAlgebra.Grading
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/-!
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# Super Commute
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-/
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namespace FieldSpecification
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variable {𝓕 : FieldSpecification}
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namespace FieldOpFreeAlgebra
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/-!
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## The super commutator on the FieldOpFreeAlgebra.
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-/
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open FieldStatistic
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/-- For a field specification `𝓕`, the super commutator `superCommuteF` is defined as the linear
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map `𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra`
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which on the lists `φs` and `φs'` of `𝓕.CrAnFieldOp` gives
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`superCommuteF φs φs' = φs * φs' - 𝓢(φs, φs') • φs' * φs`.
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The notation `[a, b]ₛF` can be used for `superCommuteF a b`. -/
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noncomputable def superCommuteF : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ]
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𝓕.FieldOpFreeAlgebra :=
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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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@[inherit_doc superCommuteF]
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scoped[FieldSpecification.FieldOpFreeAlgebra] notation "[" φs "," φs' "]ₛF" => superCommuteF φs φs'
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/-!
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## The super commutator of different types of elements
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-/
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lemma superCommuteF_ofCrAnListF_ofCrAnListF (φs φs' : List 𝓕.CrAnFieldOp) :
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[ofCrAnListF φs, ofCrAnListF φs']ₛF =
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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_ofCrAnOpF_ofCrAnOpF (φ φ' : 𝓕.CrAnFieldOp) :
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[ofCrAnOpF φ, ofCrAnOpF φ']ₛF =
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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 [ofCrAnListF_append]
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rw [FieldStatistic.ofList_singleton, FieldStatistic.ofList_singleton, smul_mul_assoc]
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lemma superCommuteF_ofCrAnListF_ofFieldOpFsList (φcas : List 𝓕.CrAnFieldOp) (φs : List 𝓕.FieldOp) :
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[ofCrAnListF φcas, ofFieldOpListF φs]ₛF = 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_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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lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
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[ofFieldOpListF φ, ofFieldOpListF φs]ₛF = 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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simp only [map_sum, LinearMap.coeFn_sum, Finset.sum_apply, instCommGroup.eq_1,
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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_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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lemma superCommuteF_ofFieldOpF_ofFieldOpFsList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
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[ofFieldOpF φ, ofFieldOpListF φs]ₛF = 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,
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ofFieldOpListF_singleton]
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simp
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lemma superCommuteF_ofFieldOpListF_ofFieldOpF (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
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[ofFieldOpListF φs, ofFieldOpF φ]ₛF = 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,
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ofFieldOpListF_singleton]
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simp
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lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.FieldOp) :
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[anPartF φ, crPartF φ']ₛF = 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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| FieldOp.inAsymp φ, _ =>
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simp
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| _, FieldOp.outAsymp φ =>
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simp only [crPartF_posAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
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sub_self]
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| FieldOp.position φ, FieldOp.position φ' =>
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simp only [anPartF_position, crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| FieldOp.outAsymp φ, FieldOp.position φ' =>
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simp only [anPartF_posAsymp, crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| FieldOp.position φ, FieldOp.inAsymp φ' =>
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simp only [anPartF_position, crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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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, crAnFieldOpToFieldOp_prod, ←
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ofCrAnListF_append]
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| FieldOp.outAsymp φ, FieldOp.inAsymp φ' =>
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simp only [anPartF_posAsymp, crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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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.FieldOp) :
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[crPartF φ, anPartF φ']ₛF = 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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| FieldOp.outAsymp φ, _ =>
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simp only [crPartF_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
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mul_zero, sub_self]
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| _, FieldOp.inAsymp φ =>
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simp only [anPartF_negAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
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sub_self]
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| FieldOp.position φ, FieldOp.position φ' =>
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simp only [crPartF_position, anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| FieldOp.position φ, FieldOp.outAsymp φ' =>
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simp only [crPartF_position, anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| FieldOp.inAsymp φ, FieldOp.position φ' =>
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simp only [crPartF_negAsymp, anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| FieldOp.inAsymp φ, FieldOp.outAsymp φ' =>
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simp only [crPartF_negAsymp, anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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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.FieldOp) :
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[crPartF φ, crPartF φ']ₛF = 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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| FieldOp.outAsymp φ, _ =>
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simp only [crPartF_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
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mul_zero, sub_self]
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| _, FieldOp.outAsymp φ =>
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simp only [crPartF_posAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
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sub_self]
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| FieldOp.position φ, FieldOp.position φ' =>
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simp only [crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| FieldOp.position φ, FieldOp.inAsymp φ' =>
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simp only [crPartF_position, crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| FieldOp.inAsymp φ, FieldOp.position φ' =>
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simp only [crPartF_negAsymp, crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| FieldOp.inAsymp φ, FieldOp.inAsymp φ' =>
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simp only [crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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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.FieldOp) :
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[anPartF φ, anPartF φ']ₛF =
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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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| FieldOp.inAsymp φ, _ =>
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simp
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| _, FieldOp.inAsymp φ =>
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simp
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| FieldOp.position φ, FieldOp.position φ' =>
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simp only [anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| FieldOp.position φ, FieldOp.outAsymp φ' =>
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simp only [anPartF_position, anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| FieldOp.outAsymp φ, FieldOp.position φ' =>
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simp only [anPartF_posAsymp, anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [crAnStatistics, ← ofCrAnListF_append]
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| FieldOp.outAsymp φ, FieldOp.outAsymp φ' =>
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simp only [anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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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.FieldOp) (φs : List 𝓕.FieldOp) :
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[crPartF φ, ofFieldOpListF φs]ₛF =
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crPartF φ * ofFieldOpListF φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs *
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crPartF φ := by
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match φ with
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| FieldOp.inAsymp φ =>
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simp only [crPartF_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
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simp [crAnStatistics]
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| FieldOp.position φ =>
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simp only [crPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
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simp [crAnStatistics]
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| FieldOp.outAsymp φ =>
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simp
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lemma superCommuteF_anPartF_ofFieldOpListF (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
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[anPartF φ, ofFieldOpListF φs]ₛF =
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anPartF φ * ofFieldOpListF φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) •
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ofFieldOpListF φs * anPartF φ := by
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match φ with
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| FieldOp.inAsymp φ =>
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simp
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| FieldOp.position φ =>
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simp only [anPartF_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofFieldOpFsList]
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simp [crAnStatistics]
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| FieldOp.outAsymp φ =>
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simp only [anPartF_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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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.FieldOp) :
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[crPartF φ, ofFieldOpF φ']ₛF =
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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_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
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[anPartF φ, ofFieldOpF φ']ₛF =
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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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/-!
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## Mul equal superCommuteF
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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 commutator.
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-/
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lemma ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF (φs φs' : List 𝓕.CrAnFieldOp) :
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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']ₛF := by
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rw [superCommuteF_ofCrAnListF_ofCrAnListF]
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simp [ofCrAnListF_append]
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lemma ofCrAnOpF_mul_ofCrAnListF_eq_superCommuteF (φ : 𝓕.CrAnFieldOp) (φs' : List 𝓕.CrAnFieldOp) :
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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']ₛF := 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.FieldOp) :
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ofFieldOpListF φs * ofFieldOpListF φs' =
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpListF φs
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+ [ofFieldOpListF φs, ofFieldOpListF φs']ₛF := by
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rw [superCommuteF_ofFieldOpListF_ofFieldOpFsList]
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simp
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lemma ofFieldOpF_mul_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.FieldOp) (φs' : List 𝓕.FieldOp) :
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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']ₛF := by
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rw [superCommuteF_ofFieldOpF_ofFieldOpFsList]
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simp
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lemma ofFieldOpListF_mul_ofFieldOpF_eq_superCommuteF (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
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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]ₛF := 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.FieldOp) :
|
||
crPartF φ * anPartF φ' =
|
||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * crPartF φ +
|
||
[crPartF φ, anPartF φ']ₛF := by
|
||
rw [superCommuteF_crPartF_anPartF]
|
||
simp
|
||
|
||
lemma anPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
|
||
anPartF φ * crPartF φ' =
|
||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
|
||
crPartF φ' * anPartF φ +
|
||
[anPartF φ, crPartF φ']ₛF := by
|
||
rw [superCommuteF_anPartF_crPartF]
|
||
simp
|
||
|
||
lemma crPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
|
||
crPartF φ * crPartF φ' =
|
||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * crPartF φ +
|
||
[crPartF φ, crPartF φ']ₛF := by
|
||
rw [superCommuteF_crPartF_crPartF]
|
||
simp
|
||
|
||
lemma anPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
|
||
anPartF φ * anPartF φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * anPartF φ +
|
||
[anPartF φ, anPartF φ']ₛF := by
|
||
rw [superCommuteF_anPartF_anPartF]
|
||
simp
|
||
|
||
lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnFieldOp)
|
||
(φs' : List 𝓕.FieldOp) : ofCrAnListF φs * ofFieldOpListF φs' =
|
||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofCrAnListF φs
|
||
+ [ofCrAnListF φs, ofFieldOpListF φs']ₛF := by
|
||
rw [superCommuteF_ofCrAnListF_ofFieldOpFsList]
|
||
simp
|
||
|
||
/-!
|
||
|
||
## Symmetry of the super commutator.
|
||
|
||
-/
|
||
|
||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_symm (φs φs' : List 𝓕.CrAnFieldOp) :
|
||
[ofCrAnListF φs, ofCrAnListF φs']ₛF =
|
||
(- 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs')) • [ofCrAnListF φs', ofCrAnListF φs]ₛF := 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 =>
|
||
rhs
|
||
rw [exchangeSign_symm, exchangeSign_mul_self]
|
||
simp only [one_smul]
|
||
abel
|
||
|
||
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_symm (φ φ' : 𝓕.CrAnFieldOp) :
|
||
[ofCrAnOpF φ, ofCrAnOpF φ']ₛF =
|
||
(- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • [ofCrAnOpF φ', ofCrAnOpF φ]ₛF := 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]
|
||
conv_rhs =>
|
||
rhs
|
||
rw [exchangeSign_symm, exchangeSign_mul_self]
|
||
simp only [one_smul]
|
||
abel
|
||
|
||
/-!
|
||
|
||
## Splitting the super commutator on lists into sums.
|
||
|
||
-/
|
||
|
||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (φ : 𝓕.CrAnFieldOp) (φs φs' : List 𝓕.CrAnFieldOp) :
|
||
[ofCrAnListF φs, ofCrAnListF (φ :: φs')]ₛF =
|
||
[ofCrAnListF φs, ofCrAnOpF φ]ₛF * ofCrAnListF φs' +
|
||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ)
|
||
• ofCrAnOpF φ * [ofCrAnListF φs, ofCrAnListF φs']ₛF := by
|
||
rw [superCommuteF_ofCrAnListF_ofCrAnListF]
|
||
conv_rhs =>
|
||
lhs
|
||
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF, sub_mul,
|
||
← ofCrAnListF_append]
|
||
rhs
|
||
rw [FieldStatistic.ofList_singleton, ofCrAnListF_append, ofCrAnListF_singleton, smul_mul_assoc,
|
||
mul_assoc, ← ofCrAnListF_append]
|
||
conv_rhs =>
|
||
rhs
|
||
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 [← ofCrAnListF_cons, smul_smul, FieldStatistic.ofList_cons_eq_mul]
|
||
simp only [instCommGroup, map_mul, mul_comm]
|
||
|
||
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_cons (φ : 𝓕.FieldOp) (φs : List 𝓕.CrAnFieldOp)
|
||
(φs' : List 𝓕.FieldOp) : [ofCrAnListF φs, ofFieldOpListF (φ :: φs')]ₛF =
|
||
[ofCrAnListF φs, ofFieldOpF φ]ₛF * ofFieldOpListF φs' +
|
||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * [ofCrAnListF φs, ofFieldOpListF φs']ₛF := by
|
||
rw [superCommuteF_ofCrAnListF_ofFieldOpFsList]
|
||
conv_rhs =>
|
||
lhs
|
||
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_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]
|
||
simp [mul_comm]
|
||
|
||
/--
|
||
For a field specification `𝓕`, and two lists `φs = φ₀…φₙ` and `φs'` of `𝓕.CrAnFieldOp`
|
||
the following super commutation relation holds:
|
||
|
||
`[φs', φ₀…φₙ]ₛF = ∑ i, 𝓢(φs', φ₀…φᵢ₋₁) • φ₀…φᵢ₋₁ * [φs', φᵢ]ₛF * φᵢ₊₁ … φₙ`
|
||
|
||
The proof of this relation is via induction on the length of `φs`.
|
||
-/
|
||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum (φs : List 𝓕.CrAnFieldOp) :
|
||
(φs' : List 𝓕.CrAnFieldOp) → [ofCrAnListF φs, ofCrAnListF φs']ₛF =
|
||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
|
||
ofCrAnListF (φs'.take n) * [ofCrAnListF φs, ofCrAnOpF (φs'.get n)]ₛF *
|
||
ofCrAnListF (φs'.drop (n + 1))
|
||
| [] => by
|
||
simp [← ofCrAnListF_nil, superCommuteF_ofCrAnListF_ofCrAnListF]
|
||
| φ :: φs' => by
|
||
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, ofCrAnListF_cons, mul_assoc,
|
||
FieldStatistic.ofList_cons_eq_mul, mul_comm]
|
||
|
||
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnFieldOp) :
|
||
(φs' : List 𝓕.FieldOp) →
|
||
[ofCrAnListF φs, ofFieldOpListF φs']ₛF =
|
||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
|
||
ofFieldOpListF (φs'.take n) * [ofCrAnListF φs, ofFieldOpF (φs'.get n)]ₛF *
|
||
ofFieldOpListF (φs'.drop (n + 1))
|
||
| [] => by
|
||
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_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_ofCrAnListF (φs1 φs2 φs3 : List 𝓕.CrAnFieldOp) :
|
||
[ofCrAnListF φs1, [ofCrAnListF φs2, ofCrAnListF φs3]ₛF]ₛF =
|
||
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs3) •
|
||
(- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3) • [ofCrAnListF φs3, [ofCrAnListF φs1, ofCrAnListF φs2]ₛF]ₛF -
|
||
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs2) • [ofCrAnListF φs2, [ofCrAnListF φs3, ofCrAnListF φs1]ₛF]ₛF) := by
|
||
repeat rw [superCommuteF_ofCrAnListF_ofCrAnListF]
|
||
simp only [instCommGroup, map_sub, map_smul, neg_smul]
|
||
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 <;>
|
||
by_cases h3 : (𝓕 |>ₛ φs3) = bosonic
|
||
· simp only [h1, h2, h3, mul_self, bosonic_exchangeSign, one_smul, exchangeSign_bosonic, neg_sub]
|
||
abel
|
||
· simp only [h1, h2, bosonic_exchangeSign, one_smul, mul_bosonic, mul_self, map_one,
|
||
exchangeSign_bosonic, neg_sub]
|
||
abel
|
||
· simp only [h1, h3, mul_bosonic, bosonic_exchangeSign, one_smul, exchangeSign_bosonic, neg_sub,
|
||
mul_self, map_one]
|
||
abel
|
||
· simp only [neq_bosonic_iff_eq_fermionic] at h1 h2 h3
|
||
simp only [h1, h2, h3, mul_self, bosonic_exchangeSign, one_smul,
|
||
fermionic_exchangeSign_fermionic, neg_smul, neg_sub, bosonic_mul_fermionic, sub_neg_eq_add,
|
||
mul_bosonic, smul_add, exchangeSign_bosonic]
|
||
abel
|
||
· simp only [neq_bosonic_iff_eq_fermionic] at h1 h2 h3
|
||
simp only [h1, h2, h3, mul_self, map_one, one_smul, exchangeSign_bosonic, mul_bosonic,
|
||
bosonic_exchangeSign, bosonic_mul_fermionic, neg_sub]
|
||
abel
|
||
· simp only [neq_bosonic_iff_eq_fermionic] at h1 h2 h3
|
||
simp only [h1, h2, h3, bosonic_mul_fermionic, fermionic_exchangeSign_fermionic, neg_smul,
|
||
one_smul, sub_neg_eq_add, bosonic_exchangeSign, mul_bosonic, smul_add, exchangeSign_bosonic,
|
||
neg_sub, mul_self]
|
||
abel
|
||
· simp only [neq_bosonic_iff_eq_fermionic] at h1 h2 h3
|
||
simp only [h1, h2, h3, mul_bosonic, fermionic_exchangeSign_fermionic, neg_smul, one_smul,
|
||
sub_neg_eq_add, exchangeSign_bosonic, bosonic_mul_fermionic, smul_add, mul_self,
|
||
bosonic_exchangeSign, neg_sub]
|
||
abel
|
||
· simp only [neq_bosonic_iff_eq_fermionic] at h1 h2 h3
|
||
simp only [h1, h2, h3, mul_self, map_one, one_smul, fermionic_exchangeSign_fermionic, neg_smul,
|
||
neg_sub]
|
||
abel
|
||
|
||
/-!
|
||
|
||
## Interaction with grading.
|
||
|
||
-/
|
||
|
||
lemma superCommuteF_grade {a b : 𝓕.FieldOpFreeAlgebra} {f1 f2 : FieldStatistic}
|
||
(ha : a ∈ statisticSubmodule f1) (hb : b ∈ statisticSubmodule f2) :
|
||
[a, b]ₛF ∈ statisticSubmodule (f1 + f2) := by
|
||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
|
||
[a, a2]ₛF ∈ statisticSubmodule (f1 + f2)
|
||
change p b hb
|
||
apply Submodule.span_induction (p := p)
|
||
· intro x hx
|
||
obtain ⟨φs, rfl, hφs⟩ := hx
|
||
simp only [add_eq_mul, instCommGroup, p]
|
||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule f1) : Prop :=
|
||
[a2, ofCrAnListF φs]ₛF ∈ 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_ofCrAnListF_ofCrAnListF]
|
||
apply Submodule.sub_mem _
|
||
· apply ofCrAnListF_mem_statisticSubmodule_of
|
||
rw [ofList_append_eq_mul, hφs, hφs']
|
||
· apply Submodule.smul_mem
|
||
apply ofCrAnListF_mem_statisticSubmodule_of
|
||
rw [ofList_append_eq_mul, hφs, hφs']
|
||
rw [mul_comm]
|
||
· simp [p]
|
||
· intro x y hx hy hp1 hp2
|
||
simp only [add_eq_mul, instCommGroup, map_add, LinearMap.add_apply, p]
|
||
exact Submodule.add_mem _ hp1 hp2
|
||
· intro c x hx hp1
|
||
simp only [add_eq_mul, instCommGroup, map_smul, LinearMap.smul_apply, p]
|
||
exact Submodule.smul_mem _ c hp1
|
||
· exact ha
|
||
· simp [p]
|
||
· intro x y hx hy hp1 hp2
|
||
simp only [add_eq_mul, instCommGroup, map_add, p]
|
||
exact Submodule.add_mem _ hp1 hp2
|
||
· intro c x hx hp1
|
||
simp only [add_eq_mul, instCommGroup, map_smul, p]
|
||
exact Submodule.smul_mem _ c hp1
|
||
· exact hb
|
||
|
||
lemma superCommuteF_bosonic_bosonic {a b : 𝓕.FieldOpFreeAlgebra}
|
||
(ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule bosonic) :
|
||
[a, b]ₛF = a * b - b * a := by
|
||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
|
||
[a, a2]ₛF = a * a2 - a2 * a
|
||
change p b hb
|
||
apply Submodule.span_induction (p := p)
|
||
· intro x hx
|
||
obtain ⟨φs, rfl, hφs⟩ := hx
|
||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
|
||
[a2, ofCrAnListF φs]ₛF = 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_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]
|
||
abel
|
||
· intro c x hx hp1
|
||
simp_all [p, smul_sub]
|
||
· exact ha
|
||
· simp [p]
|
||
· intro x y hx hy hp1 hp2
|
||
simp_all only [p, map_add, mul_add, add_mul]
|
||
abel
|
||
· intro c x hx hp1
|
||
simp_all [p, smul_sub]
|
||
· exact hb
|
||
|
||
lemma superCommuteF_bosonic_fermionic {a b : 𝓕.FieldOpFreeAlgebra}
|
||
(ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule fermionic) :
|
||
[a, b]ₛF = a * b - b * a := by
|
||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
|
||
[a, a2]ₛF = a * a2 - a2 * a
|
||
change p b hb
|
||
apply Submodule.span_induction (p := p)
|
||
· intro x hx
|
||
obtain ⟨φs, rfl, hφs⟩ := hx
|
||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
|
||
[a2, ofCrAnListF φs]ₛF = 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_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]
|
||
abel
|
||
· intro c x hx hp1
|
||
simp_all [p, smul_sub]
|
||
· exact ha
|
||
· simp [p]
|
||
· intro x y hx hy hp1 hp2
|
||
simp_all only [p, map_add, mul_add, add_mul]
|
||
abel
|
||
· intro c x hx hp1
|
||
simp_all [p, smul_sub]
|
||
· exact hb
|
||
|
||
lemma superCommuteF_fermionic_bonsonic {a b : 𝓕.FieldOpFreeAlgebra}
|
||
(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule bosonic) :
|
||
[a, b]ₛF = a * b - b * a := by
|
||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
|
||
[a, a2]ₛF = a * a2 - a2 * a
|
||
change p b hb
|
||
apply Submodule.span_induction (p := p)
|
||
· intro x hx
|
||
obtain ⟨φs, rfl, hφs⟩ := hx
|
||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
|
||
[a2, ofCrAnListF φs]ₛF = 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_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]
|
||
abel
|
||
· intro c x hx hp1
|
||
simp_all [p, smul_sub]
|
||
· exact ha
|
||
· simp [p]
|
||
· intro x y hx hy hp1 hp2
|
||
simp_all only [map_add, mul_add, add_mul, p]
|
||
abel
|
||
· intro c x hx hp1
|
||
simp_all [p, smul_sub]
|
||
· exact hb
|
||
|
||
lemma superCommuteF_bonsonic {a b : 𝓕.FieldOpFreeAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
|
||
[a, b]ₛF = a * b - b * a := by
|
||
rw [← bosonicProjF_add_fermionicProjF a]
|
||
simp only [map_add, LinearMap.add_apply]
|
||
rw [superCommuteF_bosonic_bosonic (by simp) hb, superCommuteF_fermionic_bonsonic (by simp) hb]
|
||
simp only [add_mul, mul_add]
|
||
abel
|
||
|
||
lemma bosonic_superCommuteF {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
|
||
[a, b]ₛF = a * b - b * a := by
|
||
rw [← bosonicProjF_add_fermionicProjF b]
|
||
simp only [map_add, LinearMap.add_apply]
|
||
rw [superCommuteF_bosonic_bosonic ha (by simp), superCommuteF_bosonic_fermionic ha (by simp)]
|
||
simp only [add_mul, mul_add]
|
||
abel
|
||
|
||
lemma superCommuteF_bonsonic_symm {a b : 𝓕.FieldOpFreeAlgebra}
|
||
(hb : b ∈ statisticSubmodule bosonic) :
|
||
[a, b]ₛF = - [b, a]ₛF := by
|
||
rw [bosonic_superCommuteF hb, superCommuteF_bonsonic hb]
|
||
simp
|
||
|
||
lemma bonsonic_superCommuteF_symm {a b : 𝓕.FieldOpFreeAlgebra}
|
||
(ha : a ∈ statisticSubmodule bosonic) :
|
||
[a, b]ₛF = - [b, a]ₛF := by
|
||
rw [bosonic_superCommuteF ha, superCommuteF_bonsonic ha]
|
||
simp
|
||
|
||
lemma superCommuteF_fermionic_fermionic {a b : 𝓕.FieldOpFreeAlgebra}
|
||
(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
|
||
[a, b]ₛF = a * b + b * a := by
|
||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
|
||
[a, a2]ₛF = a * a2 + a2 * a
|
||
change p b hb
|
||
apply Submodule.span_induction (p := p)
|
||
· intro x hx
|
||
obtain ⟨φs, rfl, hφs⟩ := hx
|
||
let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
|
||
[a2, ofCrAnListF φs]ₛF = 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_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]
|
||
abel
|
||
· intro c x hx hp1
|
||
simp_all [p, smul_sub]
|
||
· exact ha
|
||
· simp [p]
|
||
· intro x y hx hy hp1 hp2
|
||
simp_all only [map_add, mul_add, add_mul, p]
|
||
abel
|
||
· intro c x hx hp1
|
||
simp_all [p, smul_sub]
|
||
· exact hb
|
||
|
||
lemma superCommuteF_fermionic_fermionic_symm {a b : 𝓕.FieldOpFreeAlgebra}
|
||
(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
|
||
[a, b]ₛF = [b, a]ₛF := by
|
||
rw [superCommuteF_fermionic_fermionic ha hb]
|
||
rw [superCommuteF_fermionic_fermionic hb ha]
|
||
abel
|
||
|
||
lemma superCommuteF_expand_bosonicProjF_fermionicProjF (a b : 𝓕.FieldOpFreeAlgebra) :
|
||
[a, b]ₛF = bosonicProjF a * bosonicProjF b - bosonicProjF b * bosonicProjF a +
|
||
bosonicProjF a * fermionicProjF b - fermionicProjF b * bosonicProjF a +
|
||
fermionicProjF a * bosonicProjF b - bosonicProjF b * fermionicProjF a +
|
||
fermionicProjF a * fermionicProjF b + fermionicProjF b * fermionicProjF a := by
|
||
conv_lhs => rw [← bosonicProjF_add_fermionicProjF a, ← bosonicProjF_add_fermionicProjF b]
|
||
simp only [map_add, LinearMap.add_apply]
|
||
rw [superCommuteF_bonsonic (by simp),
|
||
superCommuteF_fermionic_bonsonic (by simp) (by simp),
|
||
superCommuteF_bosonic_fermionic (by simp) (by simp),
|
||
superCommuteF_fermionic_fermionic (by simp) (by simp)]
|
||
abel
|
||
|
||
lemma superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnFieldOp) :
|
||
[ofCrAnListF φs, ofCrAnListF φs']ₛF ∈ statisticSubmodule bosonic ∨
|
||
[ofCrAnListF φs, ofCrAnListF φs']ₛF ∈ statisticSubmodule fermionic := by
|
||
by_cases h1 : (𝓕 |>ₛ φs) = bosonic <;> by_cases h2 : (𝓕 |>ₛ φs') = bosonic
|
||
· left
|
||
have h : bosonic = bosonic + bosonic := by
|
||
simp only [add_eq_mul, instCommGroup, mul_self]
|
||
rfl
|
||
rw [h]
|
||
apply superCommuteF_grade
|
||
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 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 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 ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h1)
|
||
apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h2)
|
||
|
||
lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic (φ φ' : 𝓕.CrAnFieldOp) :
|
||
[ofCrAnOpF φ, ofCrAnOpF φ']ₛF ∈ statisticSubmodule bosonic ∨
|
||
[ofCrAnOpF φ, ofCrAnOpF φ']ₛF ∈ statisticSubmodule fermionic := by
|
||
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
|
||
exact superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ] [φ']
|
||
|
||
lemma superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic (φ1 φ2 φ3 : 𝓕.CrAnFieldOp) :
|
||
[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF ∈ statisticSubmodule bosonic ∨
|
||
[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF ∈ 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]
|
||
rfl
|
||
rw [h]
|
||
apply superCommuteF_grade h1 hs
|
||
· right
|
||
have h : fermionic = fermionic + bosonic := by
|
||
simp only [add_eq_mul, instCommGroup, mul_self]
|
||
rfl
|
||
rw [h]
|
||
apply superCommuteF_grade h1 hs
|
||
· right
|
||
have h : fermionic = bosonic + fermionic := by
|
||
simp only [add_eq_mul, instCommGroup, mul_self]
|
||
rfl
|
||
rw [h]
|
||
apply superCommuteF_grade h1 hs
|
||
· left
|
||
have h : bosonic = fermionic + fermionic := by
|
||
simp only [add_eq_mul, instCommGroup, mul_self]
|
||
rfl
|
||
rw [h]
|
||
apply superCommuteF_grade h1 hs
|
||
|
||
lemma superCommuteF_bosonic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnFieldOp)
|
||
(ha : a ∈ statisticSubmodule bosonic) :
|
||
[a, ofCrAnListF φs]ₛF = ∑ (n : Fin φs.length),
|
||
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛF *
|
||
ofCrAnListF (φs.drop (n + 1)) := by
|
||
let p (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ statisticSubmodule bosonic) : Prop :=
|
||
[a, ofCrAnListF φs]ₛF = ∑ (n : Fin φs.length),
|
||
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛF *
|
||
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_ofCrAnListF_ofCrAnListF_eq_sum]
|
||
congr
|
||
funext n
|
||
simp [hφs]
|
||
· simp [p]
|
||
· intro x y hx hy hpx hpy
|
||
simp_all only [List.get_eq_getElem, map_add, LinearMap.add_apply, p]
|
||
rw [← Finset.sum_add_distrib]
|
||
congr
|
||
funext n
|
||
simp [mul_add, add_mul]
|
||
· intro c x hx hpx
|
||
simp_all [p, Finset.smul_sum]
|
||
· exact ha
|
||
|
||
lemma superCommuteF_fermionic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra)
|
||
(φs : List 𝓕.CrAnFieldOp) (ha : a ∈ statisticSubmodule fermionic) :
|
||
[a, ofCrAnListF φs]ₛF = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
|
||
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛF *
|
||
ofCrAnListF (φs.drop (n + 1)) := by
|
||
let p (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ statisticSubmodule fermionic) : Prop :=
|
||
[a, ofCrAnListF φs]ₛF = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
|
||
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛF *
|
||
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_ofCrAnListF_ofCrAnListF_eq_sum]
|
||
congr
|
||
funext n
|
||
simp [hφs]
|
||
· simp [p]
|
||
· intro x y hx hy hpx hpy
|
||
simp_all only [p, instCommGroup, List.get_eq_getElem, Algebra.smul_mul_assoc, map_add,
|
||
LinearMap.add_apply]
|
||
rw [← Finset.sum_add_distrib]
|
||
congr
|
||
funext n
|
||
simp [mul_add, add_mul]
|
||
· intro c x hx hpx
|
||
simp_all only [p, instCommGroup, List.get_eq_getElem, Algebra.smul_mul_assoc, map_smul,
|
||
LinearMap.smul_apply, Finset.smul_sum, Algebra.mul_smul_comm]
|
||
congr
|
||
funext x
|
||
simp [smul_smul, mul_comm]
|
||
· exact ha
|
||
|
||
lemma statistic_neq_of_superCommuteF_fermionic {φs φs' : List 𝓕.CrAnFieldOp}
|
||
(h : [ofCrAnListF φs, ofCrAnListF φs']ₛF ∈ statisticSubmodule fermionic) :
|
||
(𝓕 |>ₛ φs) ≠ (𝓕 |>ₛ φs') ∨ [ofCrAnListF φs, ofCrAnListF φs']ₛF = 0 := by
|
||
by_cases h0 : [ofCrAnListF φs, ofCrAnListF φs']ₛF = 0
|
||
· simp [h0]
|
||
simp only [ne_eq, h0, or_false]
|
||
by_contra hn
|
||
refine h0 (eq_zero_of_bosonic_and_fermionic ?_ h)
|
||
by_cases hc : (𝓕 |>ₛ φs) = bosonic
|
||
· have h1 : bosonic = bosonic + bosonic := by
|
||
simp only [add_eq_mul, instCommGroup, mul_self]
|
||
rfl
|
||
rw [h1]
|
||
apply superCommuteF_grade
|
||
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 ofCrAnListF_mem_statisticSubmodule_of _ _
|
||
simpa using hc
|
||
apply ofCrAnListF_mem_statisticSubmodule_of _ _
|
||
rw [← hn]
|
||
simpa using hc
|
||
|
||
end FieldOpFreeAlgebra
|
||
|
||
end FieldSpecification
|