413 lines
18 KiB
Text
413 lines
18 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.NormalOrder
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import HepLean.PerturbationTheory.Koszul.KoszulSign
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/-!
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# Normal ordering of the operator algebra
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-/
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namespace FieldSpecification
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variable {𝓕 : FieldSpecification}
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namespace ProtoOperatorAlgebra
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variable {𝓞 : ProtoOperatorAlgebra 𝓕}
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open CrAnAlgebra
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open FieldStatistic
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/-!
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## Normal order of super-commutators.
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The main result of this section is
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`crAnF_normalOrderF_superCommuteF_eq_zero_mul`.
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-/
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lemma crAnF_normalOrderF_superCommuteF_ofCrAnList_create_create_ofCrAnList
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(φc φc' : 𝓕.CrAnStates) (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
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(hφc' : 𝓕 |>ᶜ φc' = CreateAnnihilate.create) (φs φs' : List 𝓕.CrAnStates) :
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𝓞.crAnF (𝓝ᶠ(ofCrAnList φs * [ofCrAnState φc, ofCrAnState φc']ₛca * ofCrAnList φs')) = 0 := by
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rw [normalOrderF_superCommuteF_ofCrAnList_create_create_ofCrAnList φc φc' hφc hφc' φs φs']
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rw [map_smul, map_mul, map_mul, map_mul, 𝓞.superCommuteF_create_create φc φc' hφc hφc']
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simp
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lemma crAnF_normalOrderF_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList
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(φa φa' : 𝓕.CrAnStates) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(hφa' : 𝓕 |>ᶜ φa' = CreateAnnihilate.annihilate) (φs φs' : List 𝓕.CrAnStates) :
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𝓞.crAnF (𝓝ᶠ(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * ofCrAnList φs')) = 0 := by
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rw [normalOrderF_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList φa φa' hφa hφa' φs φs']
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rw [map_smul, map_mul, map_mul, map_mul, 𝓞.superCommuteF_annihilate_annihilate φa φa' hφa hφa']
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simp
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lemma crAnF_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero
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(φa φa' : 𝓕.CrAnStates) (φs φs' : List 𝓕.CrAnStates) :
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𝓞.crAnF (normalOrderF
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(ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca * ofCrAnList φs')) = 0 := by
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rcases CreateAnnihilate.eq_create_or_annihilate (𝓕 |>ᶜ φa) with hφa | hφa
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<;> rcases CreateAnnihilate.eq_create_or_annihilate (𝓕 |>ᶜ φa') with hφa' | hφa'
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· rw [normalOrderF_superCommuteF_ofCrAnList_create_create_ofCrAnList φa φa' hφa hφa' φs φs']
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rw [map_smul, map_mul, map_mul, map_mul, 𝓞.superCommuteF_create_create φa φa' hφa hφa']
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simp
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· rw [normalOrderF_superCommuteF_create_annihilate φa φa' hφa hφa' (ofCrAnList φs)
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(ofCrAnList φs')]
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simp
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· rw [normalOrderF_superCommuteF_annihilate_create φa' φa hφa' hφa (ofCrAnList φs)
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(ofCrAnList φs')]
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simp
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· rw [normalOrderF_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList φa φa' hφa hφa' φs φs']
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rw [map_smul, map_mul, map_mul, map_mul, 𝓞.superCommuteF_annihilate_annihilate φa φa' hφa hφa']
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simp
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lemma crAnF_normalOrderF_superCommuteF_ofCrAnList_eq_zero
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(φa φa' : 𝓕.CrAnStates) (φs : List 𝓕.CrAnStates)
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(a : 𝓕.CrAnAlgebra) : 𝓞.crAnF (normalOrderF (ofCrAnList φs *
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[ofCrAnState φa, ofCrAnState φa']ₛca * a)) = 0 := by
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change (𝓞.crAnF.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap ((ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca))) a = 0
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have hf : 𝓞.crAnF.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap (ofCrAnList φs * [ofCrAnState φa, ofCrAnState φa']ₛca) = 0 := by
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apply ofCrAnListBasis.ext
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intro l
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simp only [ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
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AlgHom.toLinearMap_apply, LinearMap.zero_apply]
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exact crAnF_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero φa φa' φs l
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rw [hf]
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simp
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lemma crAnF_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul (φa φa' : 𝓕.CrAnStates)
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(a b : 𝓕.CrAnAlgebra) :
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𝓞.crAnF (normalOrderF (a * [ofCrAnState φa, ofCrAnState φa']ₛca * b)) = 0 := by
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rw [mul_assoc]
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change (𝓞.crAnF.toLinearMap ∘ₗ normalOrderF ∘ₗ mulLinearMap.flip
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([ofCrAnState φa, ofCrAnState φa']ₛca * b)) a = 0
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have hf : 𝓞.crAnF.toLinearMap ∘ₗ normalOrderF ∘ₗ mulLinearMap.flip
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([ofCrAnState φa, ofCrAnState φa']ₛca * b) = 0 := by
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apply ofCrAnListBasis.ext
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intro l
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simp only [mulLinearMap, ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
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LinearMap.flip_apply, LinearMap.coe_mk, AddHom.coe_mk, AlgHom.toLinearMap_apply,
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LinearMap.zero_apply]
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rw [← mul_assoc]
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exact crAnF_normalOrderF_superCommuteF_ofCrAnList_eq_zero φa φa' _ _
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rw [hf]
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simp
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lemma crAnF_normalOrderF_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul (φa : 𝓕.CrAnStates)
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(φs : List 𝓕.CrAnStates)
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(a b : 𝓕.CrAnAlgebra) :
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𝓞.crAnF (normalOrderF (a * [ofCrAnState φa, ofCrAnList φs]ₛca * b)) = 0 := by
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList_eq_sum]
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rw [Finset.mul_sum, Finset.sum_mul]
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rw [map_sum, map_sum]
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apply Fintype.sum_eq_zero
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intro n
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rw [← mul_assoc, ← mul_assoc]
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rw [mul_assoc _ _ b, ofCrAnList_singleton]
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rw [crAnF_normalOrderF_superCommuteF_ofCrAnState_eq_zero_mul]
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lemma crAnF_normalOrderF_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul (φa : 𝓕.CrAnStates)
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(φs : List 𝓕.CrAnStates)
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(a b : 𝓕.CrAnAlgebra) :
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𝓞.crAnF (normalOrderF (a * [ofCrAnList φs, ofCrAnState φa]ₛca * b)) = 0 := by
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rw [← ofCrAnList_singleton, superCommuteF_ofCrAnList_ofCrAnList_symm, ofCrAnList_singleton]
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simp only [FieldStatistic.instCommGroup.eq_1, FieldStatistic.ofList_singleton, mul_neg,
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Algebra.mul_smul_comm, neg_mul, Algebra.smul_mul_assoc, map_neg, map_smul]
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rw [crAnF_normalOrderF_superCommuteF_ofCrAnState_ofCrAnList_eq_zero_mul]
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simp
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lemma crAnF_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul
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(φs φs' : List 𝓕.CrAnStates)
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(a b : 𝓕.CrAnAlgebra) :
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𝓞.crAnF (normalOrderF (a * [ofCrAnList φs, ofCrAnList φs']ₛca * b)) = 0 := by
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rw [superCommuteF_ofCrAnList_ofCrAnList_eq_sum, Finset.mul_sum, Finset.sum_mul]
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rw [map_sum, map_sum]
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apply Fintype.sum_eq_zero
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intro n
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rw [← mul_assoc, ← mul_assoc]
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rw [mul_assoc _ _ b]
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rw [crAnF_normalOrderF_superCommuteF_ofCrAnList_ofCrAnState_eq_zero_mul]
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lemma crAnF_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul
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(φs : List 𝓕.CrAnStates)
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(a b c : 𝓕.CrAnAlgebra) :
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𝓞.crAnF (normalOrderF (a * [ofCrAnList φs, c]ₛca * b)) = 0 := by
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change (𝓞.crAnF.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnList φs)) c = 0
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have hf : (𝓞.crAnF.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnList φs)) = 0 := by
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apply ofCrAnListBasis.ext
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intro φs'
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simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
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LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
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LinearMap.zero_apply]
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rw [crAnF_normalOrderF_superCommuteF_ofCrAnList_ofCrAnList_eq_zero_mul]
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rw [hf]
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simp
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@[simp]
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lemma crAnF_normalOrderF_superCommuteF_eq_zero_mul
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(a b c d : 𝓕.CrAnAlgebra) : 𝓞.crAnF (normalOrderF (a * [d, c]ₛca * b)) = 0 := by
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change (𝓞.crAnF.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) d = 0
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have hf : (𝓞.crAnF.toLinearMap ∘ₗ normalOrderF ∘ₗ
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mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) = 0 := by
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apply ofCrAnListBasis.ext
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intro φs
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simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
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LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
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LinearMap.zero_apply]
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rw [crAnF_normalOrderF_superCommuteF_ofCrAnList_eq_zero_mul]
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rw [hf]
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simp
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@[simp]
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lemma crAnF_normalOrderF_superCommuteF_eq_zero_mul_right
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(b c d : 𝓕.CrAnAlgebra) : 𝓞.crAnF (normalOrderF ([d, c]ₛca * b)) = 0 := by
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rw [← crAnF_normalOrderF_superCommuteF_eq_zero_mul 1 b c d]
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simp
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@[simp]
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lemma crAnF_normalOrderF_superCommuteF_eq_zero_mul_left
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(a c d : 𝓕.CrAnAlgebra) : 𝓞.crAnF (normalOrderF (a * [d, c]ₛca)) = 0 := by
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rw [← crAnF_normalOrderF_superCommuteF_eq_zero_mul a 1 c d]
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simp
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@[simp]
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lemma crAnF_normalOrderF_superCommuteF_eq_zero
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(c d : 𝓕.CrAnAlgebra) : 𝓞.crAnF (normalOrderF [d, c]ₛca) = 0 := by
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rw [← crAnF_normalOrderF_superCommuteF_eq_zero_mul 1 1 c d]
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simp
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/-!
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## Swapping terms in a normal order.
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-/
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lemma crAnF_normalOrderF_ofState_ofState_swap (φ φ' : 𝓕.States) :
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𝓞.crAnF (normalOrderF (ofState φ * ofState φ')) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓞.crAnF (normalOrderF (ofState φ' * ofState φ)) := by
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rw [← ofStateList_singleton, ← ofStateList_singleton,
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ofStateList_mul_ofStateList_eq_superCommuteF]
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simp
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lemma crAnF_normalOrderF_ofCrAnState_ofCrAnList_swap (φ : 𝓕.CrAnStates)
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(φs : List 𝓕.CrAnStates) :
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𝓞.crAnF (normalOrderF (ofCrAnState φ * ofCrAnList φs)) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • 𝓞.crAnF (normalOrderF (ofCrAnList φs * ofCrAnState φ)) := by
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rw [← ofCrAnList_singleton, ofCrAnList_mul_ofCrAnList_eq_superCommuteF]
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simp
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lemma crAnF_normalOrderF_ofCrAnState_ofStatesList_swap (φ : 𝓕.CrAnStates)
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(φ' : List 𝓕.States) :
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𝓞.crAnF (normalOrderF (ofCrAnState φ * ofStateList φ')) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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𝓞.crAnF (normalOrderF (ofStateList φ' * ofCrAnState φ)) := by
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rw [← ofCrAnList_singleton, ofCrAnList_mul_ofStateList_eq_superCommuteF]
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simp
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lemma crAnF_normalOrderF_anPart_ofStatesList_swap (φ : 𝓕.States)
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(φ' : List 𝓕.States) :
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𝓞.crAnF (normalOrderF (anPart φ * ofStateList φ')) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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𝓞.crAnF (normalOrderF (ofStateList φ' * anPart φ)) := by
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match φ with
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| .inAsymp φ =>
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simp
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| .position φ =>
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simp only [anPart_position, instCommGroup.eq_1]
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rw [crAnF_normalOrderF_ofCrAnState_ofStatesList_swap]
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rfl
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| .outAsymp φ =>
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simp only [anPart_posAsymp, instCommGroup.eq_1]
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rw [crAnF_normalOrderF_ofCrAnState_ofStatesList_swap]
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rfl
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lemma crAnF_normalOrderF_ofStatesList_anPart_swap (φ : 𝓕.States) (φ' : List 𝓕.States) :
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𝓞.crAnF (normalOrderF (ofStateList φ' * anPart φ))
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= 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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𝓞.crAnF (normalOrderF (anPart φ * ofStateList φ')) := by
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rw [crAnF_normalOrderF_anPart_ofStatesList_swap]
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simp [smul_smul, FieldStatistic.exchangeSign_mul_self]
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lemma crAnF_normalOrderF_ofStatesList_mul_anPart_swap (φ : 𝓕.States)
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(φ' : List 𝓕.States) :
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𝓞.crAnF (normalOrderF (ofStateList φ') * anPart φ) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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𝓞.crAnF (normalOrderF (anPart φ * ofStateList φ')) := by
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rw [← normalOrderF_mul_anPart]
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rw [crAnF_normalOrderF_ofStatesList_anPart_swap]
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/-!
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## Super commutators with a normal ordered term as sums
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-/
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lemma crAnF_ofCrAnState_superCommuteF_normalOrderF_ofCrAnList_eq_sum (φ : 𝓕.CrAnStates)
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(φs : List 𝓕.CrAnStates) : 𝓞.crAnF ([ofCrAnState φ, normalOrderF (ofCrAnList φs)]ₛca) =
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∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
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𝓞.crAnF ([ofCrAnState φ, ofCrAnState φs[n]]ₛca)
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* 𝓞.crAnF (normalOrderF (ofCrAnList (φs.eraseIdx n))) := by
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rw [normalOrderF_ofCrAnList, map_smul, map_smul]
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rw [crAnF_superCommuteF_ofCrAnState_ofCrAnList_eq_sum, sum_normalOrderList_length]
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simp only [instCommGroup.eq_1, List.get_eq_getElem, normalOrderList_get_normalOrderEquiv,
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normalOrderList_eraseIdx_normalOrderEquiv, Algebra.smul_mul_assoc, map_sum, map_smul, map_mul,
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Finset.smul_sum, Fin.getElem_fin]
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congr
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funext n
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rw [ofCrAnList_eq_normalOrderF, map_smul, mul_smul_comm, smul_smul, smul_smul]
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by_cases hs : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[n])
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· congr
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erw [normalOrderSign_eraseIdx, ← hs]
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trans (normalOrderSign φs * normalOrderSign φs) *
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(𝓢(𝓕 |>ₛ (φs.get n), 𝓕 |>ₛ ((normalOrderList φs).take (normalOrderEquiv n))) *
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𝓢(𝓕 |>ₛ (φs.get n), 𝓕 |>ₛ ((normalOrderList φs).take (normalOrderEquiv n))))
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* 𝓢(𝓕 |>ₛ (φs.get n), 𝓕 |>ₛ (φs.take n))
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· ring_nf
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rw [hs]
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rfl
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· simp [hs]
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· erw [𝓞.superCommuteF_different_statistics _ _ hs]
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simp
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lemma crAnF_ofCrAnState_superCommuteF_normalOrderF_ofStateList_eq_sum (φ : 𝓕.CrAnStates)
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(φs : List 𝓕.States) : 𝓞.crAnF ([ofCrAnState φ, normalOrderF (ofStateList φs)]ₛca) =
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∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
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𝓞.crAnF ([ofCrAnState φ, ofState φs[n]]ₛca)
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* 𝓞.crAnF (normalOrderF (ofStateList (φs.eraseIdx n))) := by
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conv_lhs =>
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rw [ofStateList_sum, map_sum, map_sum, map_sum]
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enter [2, s]
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rw [crAnF_ofCrAnState_superCommuteF_normalOrderF_ofCrAnList_eq_sum,
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CrAnSection.sum_over_length]
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enter [2, n]
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rw [CrAnSection.take_statistics_eq_take_state_statistics, smul_mul_assoc]
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rw [Finset.sum_comm]
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refine Finset.sum_congr rfl (fun n _ => ?_)
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simp only [instCommGroup.eq_1, Fin.coe_cast, Fin.getElem_fin,
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CrAnSection.sum_eraseIdxEquiv n _ n.prop,
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CrAnSection.eraseIdxEquiv_symm_getElem,
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CrAnSection.eraseIdxEquiv_symm_eraseIdx, ← Finset.smul_sum, Algebra.smul_mul_assoc]
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conv_lhs =>
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enter [2, 2, n]
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rw [← Finset.mul_sum]
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rw [← Finset.sum_mul, ← map_sum, ← map_sum, ← ofState, ← map_sum, ← map_sum, ← ofStateList_sum]
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/--
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Within a proto-operator algebra we have that
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`[anPart φ, 𝓝ᶠ(φs)] = ∑ i, sᵢ • [anPart φ, φᵢ]ₛca * 𝓝ᶠ(φ₀…φᵢ₋₁φᵢ₊₁…φₙ)`
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where `sᵢ` is the exchange sign for `φ` and `φ₀…φᵢ₋₁`.
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The origin of this result is
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- `superCommuteF_ofCrAnList_ofCrAnList_eq_sum`
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-/
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lemma crAnF_anPart_superCommuteF_normalOrderF_ofStateList_eq_sum (φ : 𝓕.States) (φs : List 𝓕.States) :
|
||
𝓞.crAnF ([anPart φ, 𝓝ᶠ(φs)]ₛca) =
|
||
∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
|
||
𝓞.crAnF ([anPart φ, ofState φs[n]]ₛca) * 𝓞.crAnF 𝓝ᶠ(φs.eraseIdx n) := by
|
||
match φ with
|
||
| .inAsymp φ =>
|
||
simp
|
||
| .position φ =>
|
||
simp only [anPart_position, instCommGroup.eq_1, Fin.getElem_fin, Algebra.smul_mul_assoc]
|
||
rw [crAnF_ofCrAnState_superCommuteF_normalOrderF_ofStateList_eq_sum]
|
||
simp [crAnStatistics]
|
||
| .outAsymp φ =>
|
||
simp only [anPart_posAsymp, instCommGroup.eq_1, Fin.getElem_fin, Algebra.smul_mul_assoc]
|
||
rw [crAnF_ofCrAnState_superCommuteF_normalOrderF_ofStateList_eq_sum]
|
||
simp [crAnStatistics]
|
||
|
||
/-!
|
||
|
||
## Multiplying with normal ordered terms
|
||
|
||
-/
|
||
/--
|
||
Within a proto-operator algebra we have that
|
||
`anPart φ * 𝓝ᶠ(φ₀φ₁…φₙ) = 𝓝ᶠ((anPart φ)φ₀φ₁…φₙ) + [anpart φ, 𝓝ᶠ(φ₀φ₁…φₙ)]ₛca`.
|
||
-/
|
||
lemma crAnF_anPart_mul_normalOrderF_ofStatesList_eq_superCommuteF (φ : 𝓕.States)
|
||
(φ' : List 𝓕.States) :
|
||
𝓞.crAnF (anPart φ * normalOrderF (ofStateList φ')) =
|
||
𝓞.crAnF (normalOrderF (anPart φ * ofStateList φ')) +
|
||
𝓞.crAnF ([anPart φ, normalOrderF (ofStateList φ')]ₛca) := by
|
||
rw [anPart_mul_normalOrderF_ofStateList_eq_superCommuteF]
|
||
simp only [instCommGroup.eq_1, map_add, map_smul]
|
||
congr
|
||
rw [crAnF_normalOrderF_anPart_ofStatesList_swap]
|
||
|
||
/--
|
||
Within a proto-operator algebra we have that
|
||
`φ * 𝓝ᶠ(φ₀φ₁…φₙ) = 𝓝ᶠ(φφ₀φ₁…φₙ) + [anpart φ, 𝓝ᶠ(φ₀φ₁…φₙ)]ₛca`.
|
||
-/
|
||
lemma crAnF_ofState_mul_normalOrderF_ofStatesList_eq_superCommuteF (φ : 𝓕.States)
|
||
(φs : List 𝓕.States) : 𝓞.crAnF (ofState φ * 𝓝ᶠ(φs)) =
|
||
𝓞.crAnF (normalOrderF (ofState φ * ofStateList φs)) +
|
||
𝓞.crAnF ([anPart φ, 𝓝ᶠ(φs)]ₛca) := by
|
||
conv_lhs => rw [ofState_eq_crPart_add_anPart]
|
||
rw [add_mul, map_add, crAnF_anPart_mul_normalOrderF_ofStatesList_eq_superCommuteF, ← add_assoc,
|
||
← normalOrderF_crPart_mul, ← map_add]
|
||
conv_lhs =>
|
||
lhs
|
||
rw [← map_add, ← add_mul, ← ofState_eq_crPart_add_anPart]
|
||
|
||
/-- In the expansion of `ofState φ * normalOrderF (ofStateList φs)` the element
|
||
of `𝓞.A` associated with contracting `φ` with the (optional) `n`th element of `φs`. -/
|
||
noncomputable def contractStateAtIndex (φ : 𝓕.States) (φs : List 𝓕.States)
|
||
(n : Option (Fin φs.length)) : 𝓞.A :=
|
||
match n with
|
||
| none => 1
|
||
| some n => 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
|
||
𝓞.crAnF ([anPart φ, ofState φs[n]]ₛca)
|
||
|
||
/--
|
||
Within a proto-operator algebra,
|
||
`φ * N(φ₀φ₁…φₙ) = N(φφ₀φ₁…φₙ) + ∑ i, (sᵢ • [anPart φ, φᵢ]ₛ) * N(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)`,
|
||
where `sₙ` is the exchange sign for `φ` and `φ₀φ₁…φᵢ₋₁`.
|
||
-/
|
||
lemma crAnF_ofState_mul_normalOrderF_ofStatesList_eq_sum (φ : 𝓕.States)
|
||
(φs : List 𝓕.States) :
|
||
𝓞.crAnF (ofState φ * normalOrderF (ofStateList φs)) =
|
||
∑ n : Option (Fin φs.length),
|
||
contractStateAtIndex φ φs n *
|
||
𝓞.crAnF (normalOrderF (ofStateList (HepLean.List.optionEraseZ φs φ n))) := by
|
||
rw [crAnF_ofState_mul_normalOrderF_ofStatesList_eq_superCommuteF]
|
||
rw [crAnF_anPart_superCommuteF_normalOrderF_ofStateList_eq_sum]
|
||
simp only [instCommGroup.eq_1, Fin.getElem_fin, Algebra.smul_mul_assoc, contractStateAtIndex,
|
||
Fintype.sum_option, one_mul]
|
||
rfl
|
||
|
||
/-!
|
||
|
||
## Cons vs insertIdx for a normal ordered term.
|
||
|
||
-/
|
||
|
||
/--
|
||
Within a proto-operator algebra, `N(φφ₀φ₁…φₙ) = s • N(φ₀…φₖ₋₁φφₖ…φₙ)`, where
|
||
`s` is the exchange sign for `φ` and `φ₀…φₖ₋₁`.
|
||
-/
|
||
lemma crAnF_ofState_normalOrderF_insert (φ : 𝓕.States) (φs : List 𝓕.States)
|
||
(k : Fin φs.length.succ) :
|
||
𝓞.crAnF (normalOrderF (ofStateList (φ :: φs))) =
|
||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs.take k) • 𝓞.crAnF (normalOrderF (ofStateList (φs.insertIdx k φ))) := by
|
||
have hl : φs.insertIdx k φ = φs.take k ++ [φ] ++ φs.drop k := by
|
||
rw [HepLean.List.insertIdx_eq_take_drop]
|
||
simp
|
||
rw [hl]
|
||
rw [ofStateList_append, ofStateList_append]
|
||
rw [ofStateList_mul_ofStateList_eq_superCommuteF, add_mul]
|
||
simp only [instCommGroup.eq_1, Nat.succ_eq_add_one, ofList_singleton, Algebra.smul_mul_assoc,
|
||
map_add, map_smul, crAnF_normalOrderF_superCommuteF_eq_zero_mul_right, add_zero, smul_smul,
|
||
exchangeSign_mul_self_swap, one_smul]
|
||
rw [← ofStateList_append, ← ofStateList_append]
|
||
simp
|
||
|
||
end ProtoOperatorAlgebra
|
||
|
||
end FieldSpecification
|