refactor: rename normalOrder for CrAnAlgebra
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jstoobysmith
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5 changed files with 249 additions and 249 deletions
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@ -32,24 +32,24 @@ noncomputable section
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a list of CrAnStates to the normal-ordered list of states multiplied by
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the sign corresponding to the number of fermionic-fermionic
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exchanges done in ordering. -/
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def normalOrder : CrAnAlgebra 𝓕 →ₗ[ℂ] CrAnAlgebra 𝓕 :=
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def normalOrderF : CrAnAlgebra 𝓕 →ₗ[ℂ] CrAnAlgebra 𝓕 :=
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Basis.constr ofCrAnListBasis ℂ fun φs =>
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normalOrderSign φs • ofCrAnList (normalOrderList φs)
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@[inherit_doc normalOrder]
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scoped[FieldSpecification.CrAnAlgebra] notation "𝓝ᶠ(" a ")" => normalOrder a
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@[inherit_doc normalOrderF]
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scoped[FieldSpecification.CrAnAlgebra] notation "𝓝ᶠ(" a ")" => normalOrderF a
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lemma normalOrder_ofCrAnList (φs : List 𝓕.CrAnStates) :
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lemma normalOrderF_ofCrAnList (φs : List 𝓕.CrAnStates) :
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𝓝ᶠ(ofCrAnList φs) = normalOrderSign φs • ofCrAnList (normalOrderList φs) := by
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rw [← ofListBasis_eq_ofList, normalOrder, Basis.constr_basis]
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rw [← ofListBasis_eq_ofList, normalOrderF, Basis.constr_basis]
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lemma ofCrAnList_eq_normalOrder (φs : List 𝓕.CrAnStates) :
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lemma ofCrAnList_eq_normalOrderF (φs : List 𝓕.CrAnStates) :
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ofCrAnList (normalOrderList φs) = normalOrderSign φs • 𝓝ᶠ(ofCrAnList φs) := by
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rw [normalOrder_ofCrAnList, normalOrderList, smul_smul, normalOrderSign, Wick.koszulSign_mul_self,
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rw [normalOrderF_ofCrAnList, normalOrderList, smul_smul, normalOrderSign, Wick.koszulSign_mul_self,
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one_smul]
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lemma normalOrder_one : normalOrder (𝓕 := 𝓕) 1 = 1 := by
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rw [← ofCrAnList_nil, normalOrder_ofCrAnList, normalOrderSign_nil, normalOrderList_nil,
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lemma normalOrderF_one : normalOrderF (𝓕 := 𝓕) 1 = 1 := by
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rw [← ofCrAnList_nil, normalOrderF_ofCrAnList, normalOrderSign_nil, normalOrderList_nil,
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ofCrAnList_nil, one_smul]
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/-!
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@ -58,130 +58,130 @@ lemma normalOrder_one : normalOrder (𝓕 := 𝓕) 1 = 1 := by
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-/
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lemma normalOrder_ofCrAnList_cons_create (φ : 𝓕.CrAnStates)
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lemma normalOrderF_ofCrAnList_cons_create (φ : 𝓕.CrAnStates)
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(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) (φs : List 𝓕.CrAnStates) :
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𝓝ᶠ(ofCrAnList (φ :: φs)) = ofCrAnState φ * 𝓝ᶠ(ofCrAnList φs) := by
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rw [normalOrder_ofCrAnList, normalOrderSign_cons_create φ hφ, normalOrderList_cons_create φ hφ φs]
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rw [ofCrAnList_cons, normalOrder_ofCrAnList, mul_smul_comm]
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rw [normalOrderF_ofCrAnList, normalOrderSign_cons_create φ hφ, normalOrderList_cons_create φ hφ φs]
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rw [ofCrAnList_cons, normalOrderF_ofCrAnList, mul_smul_comm]
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lemma normalOrder_create_mul (φ : 𝓕.CrAnStates)
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lemma normalOrderF_create_mul (φ : 𝓕.CrAnStates)
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(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) (a : CrAnAlgebra 𝓕) :
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𝓝ᶠ(ofCrAnState φ * a) = ofCrAnState φ * 𝓝ᶠ(a) := by
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change (normalOrder ∘ₗ mulLinearMap (ofCrAnState φ)) a =
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(mulLinearMap (ofCrAnState φ) ∘ₗ normalOrder) a
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change (normalOrderF ∘ₗ mulLinearMap (ofCrAnState φ)) a =
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(mulLinearMap (ofCrAnState φ) ∘ₗ normalOrderF) a
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refine LinearMap.congr_fun (ofCrAnListBasis.ext fun l ↦ ?_) a
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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]
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rw [← ofCrAnList_cons, normalOrder_ofCrAnList_cons_create φ hφ]
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rw [← ofCrAnList_cons, normalOrderF_ofCrAnList_cons_create φ hφ]
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lemma normalOrder_ofCrAnList_append_annihilate (φ : 𝓕.CrAnStates)
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lemma normalOrderF_ofCrAnList_append_annihilate (φ : 𝓕.CrAnStates)
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(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.annihilate) (φs : List 𝓕.CrAnStates) :
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𝓝ᶠ(ofCrAnList (φs ++ [φ])) = 𝓝ᶠ(ofCrAnList φs) * ofCrAnState φ := by
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rw [normalOrder_ofCrAnList, normalOrderSign_append_annihlate φ hφ φs,
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rw [normalOrderF_ofCrAnList, normalOrderSign_append_annihlate φ hφ φs,
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normalOrderList_append_annihilate φ hφ φs, ofCrAnList_append, ofCrAnList_singleton,
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normalOrder_ofCrAnList, smul_mul_assoc]
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normalOrderF_ofCrAnList, smul_mul_assoc]
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lemma normalOrder_mul_annihilate (φ : 𝓕.CrAnStates)
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lemma normalOrderF_mul_annihilate (φ : 𝓕.CrAnStates)
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(hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.annihilate)
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(a : CrAnAlgebra 𝓕) : 𝓝ᶠ(a * ofCrAnState φ) = 𝓝ᶠ(a) * ofCrAnState φ := by
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change (normalOrder ∘ₗ mulLinearMap.flip (ofCrAnState φ)) a =
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(mulLinearMap.flip (ofCrAnState φ) ∘ₗ normalOrder) a
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change (normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnState φ)) a =
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(mulLinearMap.flip (ofCrAnState φ) ∘ₗ normalOrderF) a
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refine LinearMap.congr_fun (ofCrAnListBasis.ext fun l ↦ ?_) a
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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]
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rw [← ofCrAnList_singleton, ← ofCrAnList_append, ofCrAnList_singleton,
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normalOrder_ofCrAnList_append_annihilate φ hφ]
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normalOrderF_ofCrAnList_append_annihilate φ hφ]
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lemma normalOrder_crPart_mul (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
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lemma normalOrderF_crPart_mul (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
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𝓝ᶠ(crPart φ * a) =
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crPart φ * 𝓝ᶠ(a) := by
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match φ with
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| .inAsymp φ =>
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rw [crPart]
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exact normalOrder_create_mul ⟨States.inAsymp φ, ()⟩ rfl a
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exact normalOrderF_create_mul ⟨States.inAsymp φ, ()⟩ rfl a
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| .position φ =>
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rw [crPart]
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exact normalOrder_create_mul _ rfl _
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exact normalOrderF_create_mul _ rfl _
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| .outAsymp φ => simp
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lemma normalOrder_mul_anPart (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
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lemma normalOrderF_mul_anPart (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
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𝓝ᶠ(a * anPart φ) =
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𝓝ᶠ(a) * anPart φ := by
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match φ with
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| .inAsymp φ => simp
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| .position φ =>
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rw [anPart]
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exact normalOrder_mul_annihilate _ rfl _
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exact normalOrderF_mul_annihilate _ rfl _
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| .outAsymp φ =>
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rw [anPart]
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refine normalOrder_mul_annihilate _ rfl _
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refine normalOrderF_mul_annihilate _ rfl _
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/-!
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## Normal ordering for an adjacent creation and annihliation state
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The main result of this section is `normalOrder_superCommuteF_annihilate_create`.
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The main result of this section is `normalOrderF_superCommuteF_annihilate_create`.
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-/
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lemma normalOrder_swap_create_annihlate_ofCrAnList_ofCrAnList (φc φa : 𝓕.CrAnStates)
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lemma normalOrderF_swap_create_annihlate_ofCrAnList_ofCrAnList (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(φs φs' : List 𝓕.CrAnStates) :
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𝓝ᶠ(ofCrAnList φs' * ofCrAnState φc * ofCrAnState φa * ofCrAnList φs) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
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𝓝ᶠ(ofCrAnList φs' * ofCrAnState φa * ofCrAnState φc * ofCrAnList φs) := by
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rw [mul_assoc, mul_assoc, ← ofCrAnList_cons, ← ofCrAnList_cons, ← ofCrAnList_append]
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rw [normalOrder_ofCrAnList, normalOrderSign_swap_create_annihlate φc φa hφc hφa]
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rw [normalOrderList_swap_create_annihlate φc φa hφc hφa, ← smul_smul, ← normalOrder_ofCrAnList]
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rw [normalOrderF_ofCrAnList, normalOrderSign_swap_create_annihlate φc φa hφc hφa]
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rw [normalOrderList_swap_create_annihlate φc φa hφc hφa, ← smul_smul, ← normalOrderF_ofCrAnList]
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rw [ofCrAnList_append, ofCrAnList_cons, ofCrAnList_cons]
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noncomm_ring
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lemma normalOrder_swap_create_annihlate_ofCrAnList (φc φa : 𝓕.CrAnStates)
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lemma normalOrderF_swap_create_annihlate_ofCrAnList (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(φs : List 𝓕.CrAnStates) (a : 𝓕.CrAnAlgebra) :
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𝓝ᶠ(ofCrAnList φs * ofCrAnState φc * ofCrAnState φa * a) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
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𝓝ᶠ(ofCrAnList φs * ofCrAnState φa * ofCrAnState φc * a) := by
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change (normalOrder ∘ₗ mulLinearMap (ofCrAnList φs * ofCrAnState φc * ofCrAnState φa)) a =
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(smulLinearMap _ ∘ₗ normalOrder ∘ₗ
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change (normalOrderF ∘ₗ mulLinearMap (ofCrAnList φs * ofCrAnState φc * ofCrAnState φa)) a =
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(smulLinearMap _ ∘ₗ normalOrderF ∘ₗ
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mulLinearMap (ofCrAnList φs * ofCrAnState φa * ofCrAnState φc)) a
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refine LinearMap.congr_fun (ofCrAnListBasis.ext fun l ↦ ?_) a
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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, instCommGroup.eq_1]
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rw [normalOrder_swap_create_annihlate_ofCrAnList_ofCrAnList φc φa hφc hφa]
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rw [normalOrderF_swap_create_annihlate_ofCrAnList_ofCrAnList φc φa hφc hφa]
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rfl
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lemma normalOrder_swap_create_annihlate (φc φa : 𝓕.CrAnStates)
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lemma normalOrderF_swap_create_annihlate (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(a b : 𝓕.CrAnAlgebra) :
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𝓝ᶠ(a * ofCrAnState φc * ofCrAnState φa * b) = 𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa) •
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𝓝ᶠ(a * ofCrAnState φa * ofCrAnState φc * b) := by
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rw [mul_assoc, mul_assoc, mul_assoc, mul_assoc]
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change (normalOrder ∘ₗ mulLinearMap.flip (ofCrAnState φc * (ofCrAnState φa * b))) a =
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change (normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnState φc * (ofCrAnState φa * b))) a =
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(smulLinearMap (𝓢(𝓕 |>ₛ φc, 𝓕 |>ₛ φa)) ∘ₗ
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normalOrder ∘ₗ mulLinearMap.flip (ofCrAnState φa * (ofCrAnState φc * b))) a
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normalOrderF ∘ₗ mulLinearMap.flip (ofCrAnState φa * (ofCrAnState φc * b))) a
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refine LinearMap.congr_fun (ofCrAnListBasis.ext fun 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, instCommGroup.eq_1, ← mul_assoc,
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normalOrder_swap_create_annihlate_ofCrAnList φc φa hφc hφa]
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normalOrderF_swap_create_annihlate_ofCrAnList φc φa hφc hφa]
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rfl
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lemma normalOrder_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnStates)
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lemma normalOrderF_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(a b : 𝓕.CrAnAlgebra) :
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𝓝ᶠ(a * [ofCrAnState φc, ofCrAnState φa]ₛca * b) = 0 := by
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simp only [superCommuteF_ofCrAnState_ofCrAnState, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [mul_sub, sub_mul, map_sub, ← smul_mul_assoc, ← mul_assoc, ← mul_assoc,
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normalOrder_swap_create_annihlate φc φa hφc hφa]
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normalOrderF_swap_create_annihlate φc φa hφc hφa]
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simp
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lemma normalOrder_superCommuteF_annihilate_create (φc φa : 𝓕.CrAnStates)
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lemma normalOrderF_superCommuteF_annihilate_create (φc φa : 𝓕.CrAnStates)
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(hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(a b : 𝓕.CrAnAlgebra) :
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𝓝ᶠ(a * [ofCrAnState φa, ofCrAnState φc]ₛca * b) = 0 := by
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rw [superCommuteF_ofCrAnState_ofCrAnState_symm]
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simp only [instCommGroup.eq_1, neg_smul, mul_neg, Algebra.mul_smul_comm, neg_mul,
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Algebra.smul_mul_assoc, map_neg, map_smul, neg_eq_zero, smul_eq_zero]
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exact Or.inr (normalOrder_superCommuteF_create_annihilate φc φa hφc hφa ..)
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exact Or.inr (normalOrderF_superCommuteF_create_annihilate φc φa hφc hφa ..)
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lemma normalOrder_swap_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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lemma normalOrderF_swap_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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𝓝ᶠ(a * (crPart φ) * (anPart φ') * b) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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𝓝ᶠ(a * (anPart φ') * (crPart φ) * b) := by
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@ -190,22 +190,22 @@ lemma normalOrder_swap_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra
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| .outAsymp φ, _ => simp
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| .position φ, .position φ' =>
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simp only [crPart_position, anPart_position, instCommGroup.eq_1]
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rw [normalOrder_swap_create_annihlate]
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rw [normalOrderF_swap_create_annihlate]
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simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnStatesToStates_prod]
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rfl; rfl
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| .inAsymp φ, .outAsymp φ' =>
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simp only [crPart_negAsymp, anPart_posAsymp, instCommGroup.eq_1]
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rw [normalOrder_swap_create_annihlate]
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rw [normalOrderF_swap_create_annihlate]
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simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnStatesToStates_prod]
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rfl; rfl
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| .inAsymp φ, .position φ' =>
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simp only [crPart_negAsymp, anPart_position, instCommGroup.eq_1]
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rw [normalOrder_swap_create_annihlate]
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rw [normalOrderF_swap_create_annihlate]
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simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnStatesToStates_prod]
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rfl; rfl
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| .position φ, .outAsymp φ' =>
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simp only [crPart_position, anPart_posAsymp, instCommGroup.eq_1]
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rw [normalOrder_swap_create_annihlate]
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rw [normalOrderF_swap_create_annihlate]
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simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnStatesToStates_prod]
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rfl; rfl
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@ -217,13 +217,13 @@ Using the results from above.
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-/
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lemma normalOrder_swap_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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lemma normalOrderF_swap_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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𝓝ᶠ(a * (anPart φ) * (crPart φ') * b) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓝ᶠ(a * (crPart φ') *
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(anPart φ) * b) := by
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simp [normalOrder_swap_crPart_anPart, smul_smul]
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simp [normalOrderF_swap_crPart_anPart, smul_smul]
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lemma normalOrder_superCommuteF_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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lemma normalOrderF_superCommuteF_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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𝓝ᶠ(a * superCommuteF
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(crPart φ) (anPart φ') * b) = 0 := by
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match φ, φ' with
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@ -231,18 +231,18 @@ lemma normalOrder_superCommuteF_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAn
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| .outAsymp φ', _ => simp
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| .position φ, .position φ' =>
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rw [crPart_position, anPart_position]
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exact normalOrder_superCommuteF_create_annihilate _ _ rfl rfl ..
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exact normalOrderF_superCommuteF_create_annihilate _ _ rfl rfl ..
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| .inAsymp φ, .outAsymp φ' =>
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rw [crPart_negAsymp, anPart_posAsymp]
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exact normalOrder_superCommuteF_create_annihilate _ _ rfl rfl ..
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exact normalOrderF_superCommuteF_create_annihilate _ _ rfl rfl ..
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| .inAsymp φ, .position φ' =>
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rw [crPart_negAsymp, anPart_position]
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exact normalOrder_superCommuteF_create_annihilate _ _ rfl rfl ..
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exact normalOrderF_superCommuteF_create_annihilate _ _ rfl rfl ..
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| .position φ, .outAsymp φ' =>
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rw [crPart_position, anPart_posAsymp]
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exact normalOrder_superCommuteF_create_annihilate _ _ rfl rfl ..
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exact normalOrderF_superCommuteF_create_annihilate _ _ rfl rfl ..
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lemma normalOrder_superCommuteF_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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||||
lemma normalOrderF_superCommuteF_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
|
||||
𝓝ᶠ(a * superCommuteF
|
||||
(anPart φ) (crPart φ') * b) = 0 := by
|
||||
match φ, φ' with
|
||||
|
|
@ -250,16 +250,16 @@ lemma normalOrder_superCommuteF_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAn
|
|||
| _, .outAsymp φ' => simp
|
||||
| .position φ, .position φ' =>
|
||||
rw [anPart_position, crPart_position]
|
||||
exact normalOrder_superCommuteF_annihilate_create _ _ rfl rfl ..
|
||||
exact normalOrderF_superCommuteF_annihilate_create _ _ rfl rfl ..
|
||||
| .outAsymp φ', .inAsymp φ =>
|
||||
simp only [anPart_posAsymp, crPart_negAsymp]
|
||||
exact normalOrder_superCommuteF_annihilate_create _ _ rfl rfl ..
|
||||
exact normalOrderF_superCommuteF_annihilate_create _ _ rfl rfl ..
|
||||
| .position φ', .inAsymp φ =>
|
||||
simp only [anPart_position, crPart_negAsymp]
|
||||
exact normalOrder_superCommuteF_annihilate_create _ _ rfl rfl ..
|
||||
exact normalOrderF_superCommuteF_annihilate_create _ _ rfl rfl ..
|
||||
| .outAsymp φ, .position φ' =>
|
||||
simp only [anPart_posAsymp, crPart_position]
|
||||
exact normalOrder_superCommuteF_annihilate_create _ _ rfl rfl ..
|
||||
exact normalOrderF_superCommuteF_annihilate_create _ _ rfl rfl ..
|
||||
|
||||
/-!
|
||||
|
||||
|
|
@ -268,44 +268,44 @@ lemma normalOrder_superCommuteF_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAn
|
|||
-/
|
||||
|
||||
@[simp]
|
||||
lemma normalOrder_crPart_mul_crPart (φ φ' : 𝓕.States) :
|
||||
lemma normalOrderF_crPart_mul_crPart (φ φ' : 𝓕.States) :
|
||||
𝓝ᶠ(crPart φ * crPart φ') =
|
||||
crPart φ * crPart φ' := by
|
||||
rw [normalOrder_crPart_mul]
|
||||
rw [normalOrderF_crPart_mul]
|
||||
conv_lhs => rw [← mul_one (crPart φ')]
|
||||
rw [normalOrder_crPart_mul, normalOrder_one]
|
||||
rw [normalOrderF_crPart_mul, normalOrderF_one]
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
lemma normalOrder_anPart_mul_anPart (φ φ' : 𝓕.States) :
|
||||
lemma normalOrderF_anPart_mul_anPart (φ φ' : 𝓕.States) :
|
||||
𝓝ᶠ(anPart φ * anPart φ') =
|
||||
anPart φ * anPart φ' := by
|
||||
rw [normalOrder_mul_anPart]
|
||||
rw [normalOrderF_mul_anPart]
|
||||
conv_lhs => rw [← one_mul (anPart φ)]
|
||||
rw [normalOrder_mul_anPart, normalOrder_one]
|
||||
rw [normalOrderF_mul_anPart, normalOrderF_one]
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
lemma normalOrder_crPart_mul_anPart (φ φ' : 𝓕.States) :
|
||||
lemma normalOrderF_crPart_mul_anPart (φ φ' : 𝓕.States) :
|
||||
𝓝ᶠ(crPart φ * anPart φ') =
|
||||
crPart φ * anPart φ' := by
|
||||
rw [normalOrder_crPart_mul]
|
||||
rw [normalOrderF_crPart_mul]
|
||||
conv_lhs => rw [← one_mul (anPart φ')]
|
||||
rw [normalOrder_mul_anPart, normalOrder_one]
|
||||
rw [normalOrderF_mul_anPart, normalOrderF_one]
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
lemma normalOrder_anPart_mul_crPart (φ φ' : 𝓕.States) :
|
||||
lemma normalOrderF_anPart_mul_crPart (φ φ' : 𝓕.States) :
|
||||
𝓝ᶠ(anPart φ * crPart φ') =
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
|
||||
(crPart φ' * anPart φ) := by
|
||||
conv_lhs => rw [← one_mul (anPart φ * crPart φ')]
|
||||
conv_lhs => rw [← mul_one (1 * (anPart φ *
|
||||
crPart φ'))]
|
||||
rw [← mul_assoc, normalOrder_swap_anPart_crPart]
|
||||
rw [← mul_assoc, normalOrderF_swap_anPart_crPart]
|
||||
simp
|
||||
|
||||
lemma normalOrder_ofState_mul_ofState (φ φ' : 𝓕.States) :
|
||||
lemma normalOrderF_ofState_mul_ofState (φ φ' : 𝓕.States) :
|
||||
𝓝ᶠ(ofState φ * ofState φ') =
|
||||
crPart φ * crPart φ' +
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
|
||||
|
|
@ -313,8 +313,8 @@ lemma normalOrder_ofState_mul_ofState (φ φ' : 𝓕.States) :
|
|||
crPart φ * anPart φ' +
|
||||
anPart φ * anPart φ' := by
|
||||
rw [ofState_eq_crPart_add_anPart, ofState_eq_crPart_add_anPart, mul_add, add_mul, add_mul]
|
||||
simp only [map_add, normalOrder_crPart_mul_crPart, normalOrder_anPart_mul_crPart,
|
||||
instCommGroup.eq_1, normalOrder_crPart_mul_anPart, normalOrder_anPart_mul_anPart]
|
||||
simp only [map_add, normalOrderF_crPart_mul_crPart, normalOrderF_anPart_mul_crPart,
|
||||
instCommGroup.eq_1, normalOrderF_crPart_mul_anPart, normalOrderF_anPart_mul_anPart]
|
||||
abel
|
||||
|
||||
/-!
|
||||
|
|
@ -325,7 +325,7 @@ lemma normalOrder_ofState_mul_ofState (φ φ' : 𝓕.States) :
|
|||
|
||||
TODO "Split the following two lemmas up into smaller parts."
|
||||
|
||||
lemma normalOrder_superCommuteF_ofCrAnList_create_create_ofCrAnList
|
||||
lemma normalOrderF_superCommuteF_ofCrAnList_create_create_ofCrAnList
|
||||
(φc φc' : 𝓕.CrAnStates) (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
|
||||
(hφc' : 𝓕 |>ᶜ φc' = CreateAnnihilate.create) (φs φs' : List 𝓕.CrAnStates) :
|
||||
(𝓝ᶠ(ofCrAnList φs * [ofCrAnState φc, ofCrAnState φc']ₛca * ofCrAnList φs')) =
|
||||
|
|
@ -339,7 +339,7 @@ lemma normalOrder_superCommuteF_ofCrAnList_create_create_ofCrAnList
|
|||
← ofCrAnList_append]
|
||||
conv_lhs =>
|
||||
lhs
|
||||
rw [normalOrder_ofCrAnList, normalOrderList_eq_createFilter_append_annihilateFilter]
|
||||
rw [normalOrderF_ofCrAnList, normalOrderList_eq_createFilter_append_annihilateFilter]
|
||||
rw [createFilter_append, createFilter_append, createFilter_append,
|
||||
createFilter_singleton_create _ hφc, createFilter_singleton_create _ hφc']
|
||||
rw [annihilateFilter_append, annihilateFilter_append, annihilateFilter_append,
|
||||
|
|
@ -358,7 +358,7 @@ lemma normalOrder_superCommuteF_ofCrAnList_create_create_ofCrAnList
|
|||
rhs
|
||||
rw [map_smul]
|
||||
rhs
|
||||
rw [normalOrder_ofCrAnList, normalOrderList_eq_createFilter_append_annihilateFilter]
|
||||
rw [normalOrderF_ofCrAnList, normalOrderList_eq_createFilter_append_annihilateFilter]
|
||||
rw [createFilter_append, createFilter_append, createFilter_append,
|
||||
createFilter_singleton_create _ hφc, createFilter_singleton_create _ hφc']
|
||||
rw [annihilateFilter_append, annihilateFilter_append, annihilateFilter_append,
|
||||
|
|
@ -384,7 +384,7 @@ lemma normalOrder_superCommuteF_ofCrAnList_create_create_ofCrAnList
|
|||
ofCrAnList_singleton]
|
||||
rw [ofCrAnList_append, ofCrAnList_singleton, ofCrAnList_singleton, smul_mul_assoc]
|
||||
|
||||
lemma normalOrder_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList
|
||||
lemma normalOrderF_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList
|
||||
(φa φa' : 𝓕.CrAnStates)
|
||||
(hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
|
||||
(hφa' : 𝓕 |>ᶜ φa' = CreateAnnihilate.annihilate)
|
||||
|
|
@ -401,7 +401,7 @@ lemma normalOrder_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList
|
|||
← ofCrAnList_append]
|
||||
conv_lhs =>
|
||||
lhs
|
||||
rw [normalOrder_ofCrAnList, normalOrderList_eq_createFilter_append_annihilateFilter]
|
||||
rw [normalOrderF_ofCrAnList, normalOrderList_eq_createFilter_append_annihilateFilter]
|
||||
rw [createFilter_append, createFilter_append, createFilter_append,
|
||||
createFilter_singleton_annihilate _ hφa, createFilter_singleton_annihilate _ hφa']
|
||||
rw [annihilateFilter_append, annihilateFilter_append, annihilateFilter_append,
|
||||
|
|
@ -421,7 +421,7 @@ lemma normalOrder_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList
|
|||
rhs
|
||||
rw [map_smul]
|
||||
rhs
|
||||
rw [normalOrder_ofCrAnList, normalOrderList_eq_createFilter_append_annihilateFilter]
|
||||
rw [normalOrderF_ofCrAnList, normalOrderList_eq_createFilter_append_annihilateFilter]
|
||||
rw [createFilter_append, createFilter_append, createFilter_append,
|
||||
createFilter_singleton_annihilate _ hφa, createFilter_singleton_annihilate _ hφa']
|
||||
rw [annihilateFilter_append, annihilateFilter_append, annihilateFilter_append,
|
||||
|
|
@ -458,14 +458,14 @@ lemma normalOrder_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList
|
|||
|
||||
-/
|
||||
|
||||
lemma ofCrAnList_superCommuteF_normalOrder_ofCrAnList (φs φs' : List 𝓕.CrAnStates) :
|
||||
lemma ofCrAnList_superCommuteF_normalOrderF_ofCrAnList (φs φs' : List 𝓕.CrAnStates) :
|
||||
[ofCrAnList φs, 𝓝ᶠ(ofCrAnList φs')]ₛca =
|
||||
ofCrAnList φs * 𝓝ᶠ(ofCrAnList φs') -
|
||||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofCrAnList φs') * ofCrAnList φs := by
|
||||
simp [normalOrder_ofCrAnList, map_smul, superCommuteF_ofCrAnList_ofCrAnList, ofCrAnList_append,
|
||||
simp [normalOrderF_ofCrAnList, map_smul, superCommuteF_ofCrAnList_ofCrAnList, ofCrAnList_append,
|
||||
smul_sub, smul_smul, mul_comm]
|
||||
|
||||
lemma ofCrAnList_superCommuteF_normalOrder_ofStateList (φs : List 𝓕.CrAnStates)
|
||||
lemma ofCrAnList_superCommuteF_normalOrderF_ofStateList (φs : List 𝓕.CrAnStates)
|
||||
(φs' : List 𝓕.States) : [ofCrAnList φs, 𝓝ᶠ(ofStateList φs')]ₛca =
|
||||
ofCrAnList φs * 𝓝ᶠ(ofStateList φs') -
|
||||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofStateList φs') * ofCrAnList φs := by
|
||||
|
|
@ -473,7 +473,7 @@ lemma ofCrAnList_superCommuteF_normalOrder_ofStateList (φs : List 𝓕.CrAnStat
|
|||
← Finset.sum_sub_distrib, map_sum]
|
||||
congr
|
||||
funext n
|
||||
rw [ofCrAnList_superCommuteF_normalOrder_ofCrAnList,
|
||||
rw [ofCrAnList_superCommuteF_normalOrderF_ofCrAnList,
|
||||
CrAnSection.statistics_eq_state_statistics]
|
||||
|
||||
/-!
|
||||
|
|
@ -482,29 +482,29 @@ lemma ofCrAnList_superCommuteF_normalOrder_ofStateList (φs : List 𝓕.CrAnStat
|
|||
|
||||
-/
|
||||
|
||||
lemma ofCrAnList_mul_normalOrder_ofStateList_eq_superCommuteF (φs : List 𝓕.CrAnStates)
|
||||
lemma ofCrAnList_mul_normalOrderF_ofStateList_eq_superCommuteF (φs : List 𝓕.CrAnStates)
|
||||
(φs' : List 𝓕.States) :
|
||||
ofCrAnList φs * 𝓝ᶠ(ofStateList φs') =
|
||||
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofStateList φs') * ofCrAnList φs
|
||||
+ [ofCrAnList φs, 𝓝ᶠ(ofStateList φs')]ₛca := by
|
||||
simp [ofCrAnList_superCommuteF_normalOrder_ofStateList]
|
||||
simp [ofCrAnList_superCommuteF_normalOrderF_ofStateList]
|
||||
|
||||
lemma ofCrAnState_mul_normalOrder_ofStateList_eq_superCommuteF (φ : 𝓕.CrAnStates)
|
||||
lemma ofCrAnState_mul_normalOrderF_ofStateList_eq_superCommuteF (φ : 𝓕.CrAnStates)
|
||||
(φs' : List 𝓕.States) : ofCrAnState φ * 𝓝ᶠ(ofStateList φs') =
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofStateList φs') * ofCrAnState φ
|
||||
+ [ofCrAnState φ, 𝓝ᶠ(ofStateList φs')]ₛca := by
|
||||
simp [← ofCrAnList_singleton, ofCrAnList_mul_normalOrder_ofStateList_eq_superCommuteF]
|
||||
simp [← ofCrAnList_singleton, ofCrAnList_mul_normalOrderF_ofStateList_eq_superCommuteF]
|
||||
|
||||
lemma anPart_mul_normalOrder_ofStateList_eq_superCommuteF (φ : 𝓕.States)
|
||||
lemma anPart_mul_normalOrderF_ofStateList_eq_superCommuteF (φ : 𝓕.States)
|
||||
(φs' : List 𝓕.States) :
|
||||
anPart φ * 𝓝ᶠ(ofStateList φs') =
|
||||
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofStateList φs' * anPart φ)
|
||||
+ [anPart φ, 𝓝ᶠ(ofStateList φs')]ₛca := by
|
||||
rw [normalOrder_mul_anPart]
|
||||
rw [normalOrderF_mul_anPart]
|
||||
match φ with
|
||||
| .inAsymp φ => simp
|
||||
| .position φ => simp [ofCrAnState_mul_normalOrder_ofStateList_eq_superCommuteF, crAnStatistics]
|
||||
| .outAsymp φ => simp [ofCrAnState_mul_normalOrder_ofStateList_eq_superCommuteF, crAnStatistics]
|
||||
| .position φ => simp [ofCrAnState_mul_normalOrderF_ofStateList_eq_superCommuteF, crAnStatistics]
|
||||
| .outAsymp φ => simp [ofCrAnState_mul_normalOrderF_ofStateList_eq_superCommuteF, crAnStatistics]
|
||||
|
||||
end
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue