feat: Time order for CrAnAlgebra
Also remove StateAlgebra
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jstoobysmith
parent
3abc31af98
commit
21f81a9331
19 changed files with 493 additions and 290 deletions
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@ -93,27 +93,27 @@ lemma normalOrder_mul_annihilate (φ : 𝓕.CrAnStates)
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normalOrder_ofCrAnList_append_annihilate φ hφ]
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lemma normalOrder_crPart_mul (φ : 𝓕.States) (a : CrAnAlgebra 𝓕) :
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𝓝(crPart (StateAlgebra.ofState φ) * a) =
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crPart (StateAlgebra.ofState φ) * 𝓝(a) := by
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𝓝(crPart φ * a) =
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crPart φ * 𝓝(a) := by
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match φ with
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| .inAsymp φ =>
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rw [crPart, StateAlgebra.ofState, FreeAlgebra.lift_ι_apply]
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rw [crPart]
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exact normalOrder_create_mul ⟨States.inAsymp φ, ()⟩ rfl a
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| .position φ =>
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rw [crPart, StateAlgebra.ofState, FreeAlgebra.lift_ι_apply]
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rw [crPart]
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exact normalOrder_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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𝓝(a * anPart (StateAlgebra.ofState φ)) =
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𝓝(a) * anPart (StateAlgebra.ofState φ) := by
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𝓝(a * anPart φ) =
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𝓝(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, StateAlgebra.ofState, FreeAlgebra.lift_ι_apply]
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rw [anPart]
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exact normalOrder_mul_annihilate _ rfl _
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| .outAsymp φ =>
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rw [anPart, StateAlgebra.ofState, FreeAlgebra.lift_ι_apply]
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rw [anPart]
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refine normalOrder_mul_annihilate _ rfl _
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/-!
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@ -182,9 +182,9 @@ lemma normalOrder_superCommute_annihilate_create (φc φa : 𝓕.CrAnStates)
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exact Or.inr (normalOrder_superCommute_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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𝓝(a * (crPart (StateAlgebra.ofState φ)) * (anPart (StateAlgebra.ofState φ')) * b) =
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𝓝(a * (crPart φ) * (anPart φ') * b) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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𝓝(a * (anPart (StateAlgebra.ofState φ')) * (crPart (StateAlgebra.ofState φ)) * b) := by
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𝓝(a * (anPart φ') * (crPart φ) * b) := 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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@ -218,14 +218,14 @@ 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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𝓝(a * (anPart (StateAlgebra.ofState φ)) * (crPart (StateAlgebra.ofState φ')) * b) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓝(a * (crPart (StateAlgebra.ofState φ')) *
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(anPart (StateAlgebra.ofState φ)) * b) := by
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𝓝(a * (anPart φ) * (crPart φ') * b) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓝(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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lemma normalOrder_superCommute_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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𝓝(a * superCommute
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(crPart (StateAlgebra.ofState φ)) (anPart (StateAlgebra.ofState φ')) * b) = 0 := by
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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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@ -244,7 +244,7 @@ lemma normalOrder_superCommute_crPart_anPart (φ φ' : 𝓕.States) (a b : CrAnA
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lemma normalOrder_superCommute_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnAlgebra 𝓕) :
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𝓝(a * superCommute
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(anPart (StateAlgebra.ofState φ)) (crPart (StateAlgebra.ofState φ')) * b) = 0 := by
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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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@ -269,49 +269,49 @@ lemma normalOrder_superCommute_anPart_crPart (φ φ' : 𝓕.States) (a b : CrAnA
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@[simp]
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lemma normalOrder_crPart_mul_crPart (φ φ' : 𝓕.States) :
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𝓝(crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ')) =
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crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ') := by
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𝓝(crPart φ * crPart φ') =
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crPart φ * crPart φ' := by
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rw [normalOrder_crPart_mul]
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conv_lhs => rw [← mul_one (crPart (StateAlgebra.ofState φ'))]
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conv_lhs => rw [← mul_one (crPart φ')]
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rw [normalOrder_crPart_mul, normalOrder_one]
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simp
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@[simp]
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lemma normalOrder_anPart_mul_anPart (φ φ' : 𝓕.States) :
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𝓝(anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ')) =
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anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') := by
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𝓝(anPart φ * anPart φ') =
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anPart φ * anPart φ' := by
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rw [normalOrder_mul_anPart]
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conv_lhs => rw [← one_mul (anPart (StateAlgebra.ofState φ))]
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conv_lhs => rw [← one_mul (anPart φ)]
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rw [normalOrder_mul_anPart, normalOrder_one]
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simp
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@[simp]
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lemma normalOrder_crPart_mul_anPart (φ φ' : 𝓕.States) :
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𝓝(crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ')) =
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crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') := by
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𝓝(crPart φ * anPart φ') =
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crPart φ * anPart φ' := by
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rw [normalOrder_crPart_mul]
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conv_lhs => rw [← one_mul (anPart (StateAlgebra.ofState φ'))]
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conv_lhs => rw [← one_mul (anPart φ')]
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rw [normalOrder_mul_anPart, normalOrder_one]
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simp
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@[simp]
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lemma normalOrder_anPart_mul_crPart (φ φ' : 𝓕.States) :
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𝓝(anPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ')) =
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𝓝(anPart φ * crPart φ') =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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(crPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ)) := by
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conv_lhs => rw [← one_mul (anPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ'))]
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conv_lhs => rw [← mul_one (1 * (anPart (StateAlgebra.ofState φ) *
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crPart (StateAlgebra.ofState φ')))]
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(crPart φ' * anPart φ) := by
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conv_lhs => rw [← one_mul (anPart φ * crPart φ')]
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conv_lhs => rw [← mul_one (1 * (anPart φ *
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crPart φ'))]
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rw [← mul_assoc, normalOrder_swap_anPart_crPart]
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simp
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lemma normalOrder_ofState_mul_ofState (φ φ' : 𝓕.States) :
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𝓝(ofState φ * ofState φ') =
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crPart (StateAlgebra.ofState φ) * crPart (StateAlgebra.ofState φ') +
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crPart φ * crPart φ' +
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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(crPart (StateAlgebra.ofState φ') * anPart (StateAlgebra.ofState φ)) +
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crPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') +
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anPart (StateAlgebra.ofState φ) * anPart (StateAlgebra.ofState φ') := by
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(crPart φ' * anPart φ) +
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crPart φ * anPart φ' +
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anPart φ * anPart φ' := by
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rw [ofState_eq_crPart_add_anPart, ofState_eq_crPart_add_anPart, mul_add, add_mul, add_mul]
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simp only [map_add, normalOrder_crPart_mul_crPart, normalOrder_anPart_mul_crPart,
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instCommGroup.eq_1, normalOrder_crPart_mul_anPart, normalOrder_anPart_mul_anPart]
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@ -497,9 +497,9 @@ lemma ofCrAnState_mul_normalOrder_ofStateList_eq_superCommute (φ : 𝓕.CrAnSta
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lemma anPart_mul_normalOrder_ofStateList_eq_superCommute (φ : 𝓕.States)
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(φs' : List 𝓕.States) :
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anPart (StateAlgebra.ofState φ) * 𝓝(ofStateList φs') =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝(ofStateList φs' * anPart (StateAlgebra.ofState φ))
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+ [anPart (StateAlgebra.ofState φ), 𝓝(ofStateList φs')]ₛca := by
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anPart φ * 𝓝(ofStateList φs') =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝(ofStateList φs' * anPart φ)
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+ [anPart φ, 𝓝(ofStateList φs')]ₛca := by
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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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