refactor: Rename superCommute for CrAnAlgebra
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jstoobysmith
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13 changed files with 428 additions and 425 deletions
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@ -120,7 +120,7 @@ lemma normalOrder_mul_anPart (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
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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_superCommute_annihilate_create`.
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The main result of this section is `normalOrder_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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@ -163,23 +163,23 @@ lemma normalOrder_swap_create_annihlate (φc φa : 𝓕.CrAnStates)
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normalOrder_swap_create_annihlate_ofCrAnList φc φa hφc hφa]
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rfl
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lemma normalOrder_superCommute_create_annihilate (φc φa : 𝓕.CrAnStates)
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lemma normalOrder_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 [superCommute_ofCrAnState_ofCrAnState, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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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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simp
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lemma normalOrder_superCommute_annihilate_create (φc φa : 𝓕.CrAnStates)
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lemma normalOrder_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 [superCommute_ofCrAnState_ofCrAnState_symm]
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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_superCommute_create_annihilate φc φa hφc hφa ..)
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exact Or.inr (normalOrder_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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𝓝ᶠ(a * (crPart φ) * (anPart φ') * b) =
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@ -223,43 +223,43 @@ lemma normalOrder_swap_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra
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(anPart φ) * b) := by
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simp [normalOrder_swap_crPart_anPart, smul_smul]
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lemma normalOrder_superCommute_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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𝓝ᶠ(a * superCommute
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lemma normalOrder_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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| _, .inAsymp φ' => simp
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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_superCommute_create_annihilate _ _ rfl rfl ..
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exact normalOrder_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_superCommute_create_annihilate _ _ rfl rfl ..
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exact normalOrder_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_superCommute_create_annihilate _ _ rfl rfl ..
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exact normalOrder_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_superCommute_create_annihilate _ _ rfl rfl ..
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exact normalOrder_superCommuteF_create_annihilate _ _ rfl rfl ..
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lemma normalOrder_superCommute_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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𝓝ᶠ(a * superCommute
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lemma normalOrder_superCommuteF_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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𝓝ᶠ(a * superCommuteF
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(anPart φ) (crPart φ') * b) = 0 := by
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match φ, φ' with
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| .inAsymp φ', _ => simp
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| _, .outAsymp φ' => simp
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| .position φ, .position φ' =>
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rw [anPart_position, crPart_position]
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exact normalOrder_superCommute_annihilate_create _ _ rfl rfl ..
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exact normalOrder_superCommuteF_annihilate_create _ _ rfl rfl ..
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| .outAsymp φ', .inAsymp φ =>
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simp only [anPart_posAsymp, crPart_negAsymp]
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exact normalOrder_superCommute_annihilate_create _ _ rfl rfl ..
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exact normalOrder_superCommuteF_annihilate_create _ _ rfl rfl ..
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| .position φ', .inAsymp φ =>
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simp only [anPart_position, crPart_negAsymp]
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exact normalOrder_superCommute_annihilate_create _ _ rfl rfl ..
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exact normalOrder_superCommuteF_annihilate_create _ _ rfl rfl ..
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| .outAsymp φ, .position φ' =>
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simp only [anPart_posAsymp, crPart_position]
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exact normalOrder_superCommute_annihilate_create _ _ rfl rfl ..
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exact normalOrder_superCommuteF_annihilate_create _ _ rfl rfl ..
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/-!
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@ -325,14 +325,14 @@ lemma normalOrder_ofState_mul_ofState (φ φ' : 𝓕.States) :
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TODO "Split the following two lemmas up into smaller parts."
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lemma normalOrder_superCommute_ofCrAnList_create_create_ofCrAnList
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lemma normalOrder_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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(𝓝ᶠ(ofCrAnList φs * [ofCrAnState φc, ofCrAnState φc']ₛca * ofCrAnList φs')) =
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normalOrderSign (φs ++ φc' :: φc :: φs') •
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(ofCrAnList (createFilter φs) * [ofCrAnState φc, ofCrAnState φc']ₛca *
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ofCrAnList (createFilter φs') * ofCrAnList (annihilateFilter (φs ++ φs'))) := by
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rw [superCommute_ofCrAnState_ofCrAnState, mul_sub, sub_mul, map_sub]
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rw [superCommuteF_ofCrAnState_ofCrAnState, mul_sub, sub_mul, map_sub]
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conv_lhs =>
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lhs; rhs
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_append, ← ofCrAnList_append,
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@ -384,7 +384,7 @@ lemma normalOrder_superCommute_ofCrAnList_create_create_ofCrAnList
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ofCrAnList_singleton]
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rw [ofCrAnList_append, ofCrAnList_singleton, ofCrAnList_singleton, smul_mul_assoc]
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lemma normalOrder_superCommute_ofCrAnList_annihilate_annihilate_ofCrAnList
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lemma normalOrder_superCommuteF_ofCrAnList_annihilate_annihilate_ofCrAnList
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(φa φa' : 𝓕.CrAnStates)
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(hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
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(hφa' : 𝓕 |>ᶜ φa' = CreateAnnihilate.annihilate)
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@ -394,7 +394,7 @@ lemma normalOrder_superCommute_ofCrAnList_annihilate_annihilate_ofCrAnList
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(ofCrAnList (createFilter (φs ++ φs'))
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* ofCrAnList (annihilateFilter φs) * [ofCrAnState φa, ofCrAnState φa']ₛca
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* ofCrAnList (annihilateFilter φs')) := by
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rw [superCommute_ofCrAnState_ofCrAnState, mul_sub, sub_mul, map_sub]
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rw [superCommuteF_ofCrAnState_ofCrAnState, mul_sub, sub_mul, map_sub]
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conv_lhs =>
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lhs; rhs
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_append, ← ofCrAnList_append,
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@ -458,14 +458,14 @@ lemma normalOrder_superCommute_ofCrAnList_annihilate_annihilate_ofCrAnList
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-/
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lemma ofCrAnList_superCommute_normalOrder_ofCrAnList (φs φs' : List 𝓕.CrAnStates) :
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lemma ofCrAnList_superCommuteF_normalOrder_ofCrAnList (φs φs' : List 𝓕.CrAnStates) :
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[ofCrAnList φs, 𝓝ᶠ(ofCrAnList φs')]ₛca =
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ofCrAnList φs * 𝓝ᶠ(ofCrAnList φs') -
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofCrAnList φs') * ofCrAnList φs := by
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simp [normalOrder_ofCrAnList, map_smul, superCommute_ofCrAnList_ofCrAnList, ofCrAnList_append,
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simp [normalOrder_ofCrAnList, map_smul, superCommuteF_ofCrAnList_ofCrAnList, ofCrAnList_append,
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smul_sub, smul_smul, mul_comm]
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lemma ofCrAnList_superCommute_normalOrder_ofStateList (φs : List 𝓕.CrAnStates)
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lemma ofCrAnList_superCommuteF_normalOrder_ofStateList (φs : List 𝓕.CrAnStates)
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(φs' : List 𝓕.States) : [ofCrAnList φs, 𝓝ᶠ(ofStateList φs')]ₛca =
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ofCrAnList φs * 𝓝ᶠ(ofStateList φs') -
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofStateList φs') * ofCrAnList φs := by
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@ -473,7 +473,7 @@ lemma ofCrAnList_superCommute_normalOrder_ofStateList (φs : List 𝓕.CrAnState
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← Finset.sum_sub_distrib, map_sum]
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congr
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funext n
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rw [ofCrAnList_superCommute_normalOrder_ofCrAnList,
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rw [ofCrAnList_superCommuteF_normalOrder_ofCrAnList,
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CrAnSection.statistics_eq_state_statistics]
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/-!
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@ -482,20 +482,20 @@ lemma ofCrAnList_superCommute_normalOrder_ofStateList (φs : List 𝓕.CrAnState
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-/
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lemma ofCrAnList_mul_normalOrder_ofStateList_eq_superCommute (φs : List 𝓕.CrAnStates)
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lemma ofCrAnList_mul_normalOrder_ofStateList_eq_superCommuteF (φs : List 𝓕.CrAnStates)
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(φs' : List 𝓕.States) :
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ofCrAnList φs * 𝓝ᶠ(ofStateList φs') =
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𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofStateList φs') * ofCrAnList φs
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+ [ofCrAnList φs, 𝓝ᶠ(ofStateList φs')]ₛca := by
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simp [ofCrAnList_superCommute_normalOrder_ofStateList]
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simp [ofCrAnList_superCommuteF_normalOrder_ofStateList]
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lemma ofCrAnState_mul_normalOrder_ofStateList_eq_superCommute (φ : 𝓕.CrAnStates)
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lemma ofCrAnState_mul_normalOrder_ofStateList_eq_superCommuteF (φ : 𝓕.CrAnStates)
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(φs' : List 𝓕.States) : ofCrAnState φ * 𝓝ᶠ(ofStateList φs') =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofStateList φs') * ofCrAnState φ
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+ [ofCrAnState φ, 𝓝ᶠ(ofStateList φs')]ₛca := by
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simp [← ofCrAnList_singleton, ofCrAnList_mul_normalOrder_ofStateList_eq_superCommute]
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simp [← ofCrAnList_singleton, ofCrAnList_mul_normalOrder_ofStateList_eq_superCommuteF]
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lemma anPart_mul_normalOrder_ofStateList_eq_superCommute (φ : 𝓕.States)
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lemma anPart_mul_normalOrder_ofStateList_eq_superCommuteF (φ : 𝓕.States)
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(φs' : List 𝓕.States) :
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anPart φ * 𝓝ᶠ(ofStateList φs') =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofStateList φs' * anPart φ)
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@ -503,8 +503,8 @@ lemma anPart_mul_normalOrder_ofStateList_eq_superCommute (φ : 𝓕.States)
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rw [normalOrder_mul_anPart]
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match φ with
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| .inAsymp φ => simp
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| .position φ => simp [ofCrAnState_mul_normalOrder_ofStateList_eq_superCommute, crAnStatistics]
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| .outAsymp φ => simp [ofCrAnState_mul_normalOrder_ofStateList_eq_superCommute, crAnStatistics]
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| .position φ => simp [ofCrAnState_mul_normalOrder_ofStateList_eq_superCommuteF, crAnStatistics]
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| .outAsymp φ => simp [ofCrAnState_mul_normalOrder_ofStateList_eq_superCommuteF, crAnStatistics]
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end
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