444 lines
20 KiB
Text
444 lines
20 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.Algebras.CrAnAlgebra.Basic
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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 CrAnAlgebra
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/-!
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## The super commutor on the CrAnAlgebra.
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-/
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open FieldStatistic
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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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whilst for two fermionic operators this corresponds to the anti-commutator. -/
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noncomputable def superCommute : 𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra :=
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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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/-- 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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whilst for two fermionic operators this corresponds to the anti-commutator. -/
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scoped[FieldSpecification.CrAnAlgebra] notation "[" φs "," φs' "]ₛca" => superCommute φs φs'
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/-!
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## The super commutor of different types of elements
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-/
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lemma superCommute_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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rw [← ofListBasis_eq_ofList, ← ofListBasis_eq_ofList]
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simp only [superCommute, Basis.constr_basis]
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lemma superCommute_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 [superCommute_ofCrAnList_ofCrAnList, ofCrAnList_append]
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congr
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rw [ofCrAnList_append]
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rw [FieldStatistic.ofList_singleton, FieldStatistic.ofList_singleton, smul_mul_assoc]
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lemma superCommute_ofCrAnList_ofStatesList (φcas : List 𝓕.CrAnStates) (φs : List 𝓕.States) :
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[ofCrAnList φcas, ofStateList φs]ₛca = ofCrAnList φcas * ofStateList φs -
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𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofStateList φs * ofCrAnList φcas := by
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conv_lhs => rw [ofStateList_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 [superCommute_ofCrAnList_ofCrAnList, CrAnSection.statistics_eq_state_statistics,
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ofCrAnList_append, ofCrAnList_append]
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rw [Finset.sum_sub_distrib, ← Finset.mul_sum, ← Finset.smul_sum,
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← Finset.sum_mul, ← ofStateList_sum]
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simp
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lemma superCommute_ofStateList_ofStatesList (φ : List 𝓕.States) (φs : List 𝓕.States) :
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[ofStateList φ, ofStateList φs]ₛca = ofStateList φ * ofStateList φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofStateList φs * ofStateList φ := by
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conv_lhs => rw [ofStateList_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 [superCommute_ofCrAnList_ofStatesList]
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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, ← ofStateList_sum]
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lemma superCommute_ofState_ofStatesList (φ : 𝓕.States) (φs : List 𝓕.States) :
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[ofState φ, ofStateList φs]ₛca = ofState φ * ofStateList φs -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofStateList φs * ofState φ := by
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rw [← ofStateList_singleton, superCommute_ofStateList_ofStatesList, ofStateList_singleton]
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simp
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lemma superCommute_ofStateList_ofState (φs : List 𝓕.States) (φ : 𝓕.States) :
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[ofStateList φs, ofState φ]ₛca = ofStateList φs * ofState φ -
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * ofStateList φs := by
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rw [← ofStateList_singleton, superCommute_ofStateList_ofStatesList, ofStateList_singleton]
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simp
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lemma superCommute_anPart_crPart (φ φ' : 𝓕.States) :
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[anPart φ, crPart φ']ₛca =
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anPart φ * crPart φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ := by
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match φ, φ' with
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| States.inAsymp φ, _ =>
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simp
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| _, States.outAsymp φ =>
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simp only [crPart_posAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
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sub_self]
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| States.position φ, States.position φ' =>
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simp only [anPart_position, crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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| States.outAsymp φ, States.position φ' =>
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simp only [anPart_posAsymp, crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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| States.position φ, States.inAsymp φ' =>
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simp only [anPart_position, crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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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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| States.outAsymp φ, States.inAsymp φ' =>
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simp only [anPart_posAsymp, crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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lemma superCommute_crPart_anPart (φ φ' : 𝓕.States) :
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[crPart φ, anPart φ']ₛca =
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crPart φ * anPart φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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anPart φ' * crPart φ := by
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match φ, φ' with
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| States.outAsymp φ, _ =>
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simp only [crPart_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
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mul_zero, sub_self]
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| _, States.inAsymp φ =>
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simp only [anPart_negAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul,
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sub_self]
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| States.position φ, States.position φ' =>
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simp only [crPart_position, anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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| States.position φ, States.outAsymp φ' =>
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simp only [crPart_position, anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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| States.inAsymp φ, States.position φ' =>
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simp only [crPart_negAsymp, anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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| States.inAsymp φ, States.outAsymp φ' =>
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simp only [crPart_negAsymp, anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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lemma superCommute_crPart_crPart (φ φ' : 𝓕.States) :
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[crPart φ, crPart φ']ₛca =
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crPart φ * crPart φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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crPart φ' * crPart φ := by
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match φ, φ' with
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| States.outAsymp φ, _ =>
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simp only [crPart_posAsymp, map_zero, LinearMap.zero_apply, zero_mul, instCommGroup.eq_1,
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mul_zero, sub_self]
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| _, States.outAsymp φ =>
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simp only [crPart_posAsymp, map_zero, mul_zero, instCommGroup.eq_1, smul_zero, zero_mul, sub_self]
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| States.position φ, States.position φ' =>
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simp only [crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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| States.position φ, States.inAsymp φ' =>
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simp only [crPart_position, crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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| States.inAsymp φ, States.position φ' =>
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simp only [crPart_negAsymp, crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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| States.inAsymp φ, States.inAsymp φ' =>
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simp only [crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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lemma superCommute_anPart_anPart (φ φ' : 𝓕.States) :
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[anPart φ, anPart φ']ₛca =
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anPart φ * anPart φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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anPart φ' * anPart φ := by
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match φ, φ' with
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| States.inAsymp φ, _ =>
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simp
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| _, States.inAsymp φ =>
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simp
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| States.position φ, States.position φ' =>
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simp only [anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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| States.position φ, States.outAsymp φ' =>
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simp only [anPart_position, anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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| States.outAsymp φ, States.position φ' =>
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simp only [anPart_posAsymp, anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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| States.outAsymp φ, States.outAsymp φ' =>
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simp only [anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
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simp [crAnStatistics, ← ofCrAnList_append]
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lemma superCommute_crPart_ofStateList (φ : 𝓕.States) (φs : List 𝓕.States) :
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[crPart φ, ofStateList φs]ₛca =
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crPart φ * ofStateList φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofStateList φs *
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crPart φ := by
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match φ with
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| States.inAsymp φ =>
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simp only [crPart_negAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStatesList]
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simp [crAnStatistics]
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| States.position φ =>
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simp only [crPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStatesList]
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simp [crAnStatistics]
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| States.outAsymp φ =>
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simp
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lemma superCommute_anPart_ofStateList (φ : 𝓕.States) (φs : List 𝓕.States) :
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[anPart φ, ofStateList φs]ₛca =
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anPart φ * ofStateList φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) •
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ofStateList φs * anPart φ := by
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match φ with
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| States.inAsymp φ =>
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simp
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| States.position φ =>
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simp only [anPart_position, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStatesList]
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simp [crAnStatistics]
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| States.outAsymp φ =>
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simp only [anPart_posAsymp, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofStatesList]
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simp [crAnStatistics]
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lemma superCommute_crPart_ofState (φ φ' : 𝓕.States) :
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[crPart φ, ofState φ']ₛca =
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crPart φ * ofState φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofState φ' * crPart φ := by
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rw [← ofStateList_singleton, superCommute_crPart_ofStateList]
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simp
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lemma superCommute_anPart_ofState (φ φ' : 𝓕.States) :
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[anPart φ, ofState φ']ₛca =
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anPart φ * ofState φ' -
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofState φ' * anPart φ := by
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rw [← ofStateList_singleton, superCommute_anPart_ofStateList]
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simp
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/-!
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## Mul equal superCommute
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Lemmas which rewrite a multiplication of two elements of the algebra as their commuted
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multiplication with a sign plus the super commutor.
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-/
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lemma ofCrAnList_mul_ofCrAnList_eq_superCommute (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 [superCommute_ofCrAnList_ofCrAnList]
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simp [ofCrAnList_append]
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lemma ofCrAnState_mul_ofCrAnList_eq_superCommute (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_superCommute]
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simp
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lemma ofStateList_mul_ofStateList_eq_superCommute (φs φs' : List 𝓕.States) :
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ofStateList φs * ofStateList φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofStateList φs' * ofStateList φs
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+ [ofStateList φs, ofStateList φs']ₛca := by
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rw [superCommute_ofStateList_ofStatesList]
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simp
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lemma ofState_mul_ofStateList_eq_superCommute (φ : 𝓕.States) (φs' : List 𝓕.States) :
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ofState φ * ofStateList φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofStateList φs' * ofState φ
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+ [ofState φ, ofStateList φs']ₛca := by
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rw [superCommute_ofState_ofStatesList]
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simp
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lemma ofStateList_mul_ofState_eq_superCommute (φs : List 𝓕.States) (φ : 𝓕.States) :
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ofStateList φs * ofState φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * ofStateList φs
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+ [ofStateList φs, ofState φ]ₛca := by
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rw [superCommute_ofStateList_ofState]
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simp
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lemma crPart_mul_anPart_eq_superCommute (φ φ' : 𝓕.States) :
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crPart φ * anPart φ' =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * crPart φ +
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[crPart φ, anPart φ']ₛca := by
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rw [superCommute_crPart_anPart]
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simp
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lemma anPart_mul_crPart_eq_superCommute (φ φ' : 𝓕.States) :
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anPart φ * crPart φ' =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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crPart φ' * anPart φ +
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[anPart φ, crPart φ']ₛca := by
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rw [superCommute_anPart_crPart]
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simp
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lemma crPart_mul_crPart_eq_superCommute (φ φ' : 𝓕.States) :
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crPart φ * crPart φ' =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * crPart φ +
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[crPart φ, crPart φ']ₛca := by
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rw [superCommute_crPart_crPart]
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simp
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lemma anPart_mul_anPart_eq_superCommute (φ φ' : 𝓕.States) :
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anPart φ * anPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * anPart φ +
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[anPart φ, anPart φ']ₛca := by
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rw [superCommute_anPart_anPart]
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simp
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lemma ofCrAnList_mul_ofStateList_eq_superCommute (φs : List 𝓕.CrAnStates) (φs' : List 𝓕.States) :
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ofCrAnList φs * ofStateList φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofStateList φs' * ofCrAnList φs
|
||
+ [ofCrAnList φs, ofStateList φs']ₛca := by
|
||
rw [superCommute_ofCrAnList_ofStatesList]
|
||
simp
|
||
|
||
/-!
|
||
|
||
## Symmetry of the super commutor.
|
||
|
||
-/
|
||
|
||
lemma superCommute_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 [superCommute_ofCrAnList_ofCrAnList, superCommute_ofCrAnList_ofCrAnList, 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 superCommute_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 [superCommute_ofCrAnState_ofCrAnState, superCommute_ofCrAnState_ofCrAnState]
|
||
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 commutor on lists into sums.
|
||
|
||
-/
|
||
lemma superCommute_ofCrAnList_ofCrAnList_cons (φ : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
|
||
[ofCrAnList φs, ofCrAnList (φ :: φs')]ₛca =
|
||
[ofCrAnList φs, ofCrAnState φ]ₛca * ofCrAnList φs' +
|
||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ)
|
||
• ofCrAnState φ * [ofCrAnList φs, ofCrAnList φs']ₛca := by
|
||
rw [superCommute_ofCrAnList_ofCrAnList]
|
||
conv_rhs =>
|
||
lhs
|
||
rw [← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList, sub_mul, ← ofCrAnList_append]
|
||
rhs
|
||
rw [FieldStatistic.ofList_singleton, ofCrAnList_append, ofCrAnList_singleton, smul_mul_assoc,
|
||
mul_assoc, ← ofCrAnList_append]
|
||
conv_rhs =>
|
||
rhs
|
||
rw [superCommute_ofCrAnList_ofCrAnList, 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]
|
||
simp only [instCommGroup, map_mul, mul_comm]
|
||
|
||
lemma superCommute_ofCrAnList_ofStateList_cons (φ : 𝓕.States) (φs : List 𝓕.CrAnStates)
|
||
(φs' : List 𝓕.States) : [ofCrAnList φs, ofStateList (φ :: φs')]ₛca =
|
||
[ofCrAnList φs, ofState φ]ₛca * ofStateList φs' +
|
||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofState φ * [ofCrAnList φs, ofStateList φs']ₛca := by
|
||
rw [superCommute_ofCrAnList_ofStatesList]
|
||
conv_rhs =>
|
||
lhs
|
||
rw [← ofStateList_singleton, superCommute_ofCrAnList_ofStatesList, sub_mul, mul_assoc,
|
||
← ofStateList_append]
|
||
rhs
|
||
rw [FieldStatistic.ofList_singleton, ofStateList_singleton, smul_mul_assoc,
|
||
smul_mul_assoc, mul_assoc]
|
||
conv_rhs =>
|
||
rhs
|
||
rw [superCommute_ofCrAnList_ofStatesList, 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 [ofStateList_cons, mul_assoc, smul_smul, FieldStatistic.ofList_cons_eq_mul]
|
||
simp [mul_comm]
|
||
|
||
/--
|
||
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 superCommute_ofCrAnList_ofCrAnList_eq_sum (φs : List 𝓕.CrAnStates) :
|
||
(φs' : List 𝓕.CrAnStates) → [ofCrAnList φs, ofCrAnList 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))
|
||
| [] => by
|
||
simp [← ofCrAnList_nil, superCommute_ofCrAnList_ofCrAnList]
|
||
| φ :: φs' => by
|
||
rw [superCommute_ofCrAnList_ofCrAnList_cons, superCommute_ofCrAnList_ofCrAnList_eq_sum φs φs']
|
||
conv_rhs => erw [Fin.sum_univ_succ]
|
||
congr 1
|
||
· simp
|
||
· simp [Finset.mul_sum, smul_smul, ofCrAnList_cons, mul_assoc,
|
||
FieldStatistic.ofList_cons_eq_mul, mul_comm]
|
||
|
||
lemma superCommute_ofCrAnList_ofStateList_eq_sum (φs : List 𝓕.CrAnStates) : (φs' : List 𝓕.States) →
|
||
[ofCrAnList φs, ofStateList φs']ₛca =
|
||
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
|
||
ofStateList (φs'.take n) * [ofCrAnList φs, ofState (φs'.get n)]ₛca *
|
||
ofStateList (φs'.drop (n + 1))
|
||
| [] => by
|
||
simp only [superCommute_ofCrAnList_ofStatesList, 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 [superCommute_ofCrAnList_ofStateList_cons, superCommute_ofCrAnList_ofStateList_eq_sum φs φs']
|
||
conv_rhs => erw [Fin.sum_univ_succ]
|
||
congr 1
|
||
· simp
|
||
· simp [Finset.mul_sum, smul_smul, ofStateList_cons, mul_assoc,
|
||
FieldStatistic.ofList_cons_eq_mul, mul_comm]
|
||
|
||
end CrAnAlgebra
|
||
|
||
end FieldSpecification
|