refactor: Remove ProtoOperatorAlgebra
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jstoobysmith
parent
c18b4850e5
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fc20099282
15 changed files with 615 additions and 792 deletions
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@ -243,6 +243,33 @@ scoped[FieldSpecification.FieldOpAlgebra] notation "𝓝(" a ")" => normalOrder
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lemma normalOrder_eq_ι_normalOrderF (a : 𝓕.CrAnAlgebra) :
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𝓝(ι a) = ι 𝓝ᶠ(a) := rfl
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lemma normalOrder_ofCrAnFieldOpList (φs : List 𝓕.CrAnStates) :
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𝓝(ofCrAnFieldOpList φs) = normalOrderSign φs • ofCrAnFieldOpList (normalOrderList φs) := by
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rw [ofCrAnFieldOpList, normalOrder_eq_ι_normalOrderF, normalOrderF_ofCrAnList]
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rfl
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lemma ofCrAnFieldOpList_eq_normalOrder (φs : List 𝓕.CrAnStates) :
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ofCrAnFieldOpList (normalOrderList φs) = normalOrderSign φs • 𝓝(ofCrAnFieldOpList φs) := by
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rw [normalOrder_ofCrAnFieldOpList, smul_smul, normalOrderSign, Wick.koszulSign_mul_self,
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one_smul]
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/-!
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## mul anpart and crpart
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-/
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lemma normalOrder_mul_anPart (φ : 𝓕.States) (a : 𝓕.FieldOpAlgebra) :
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𝓝(a * anPart φ) = 𝓝(a) * anPart φ := by
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obtain ⟨a, rfl⟩ := ι_surjective a
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rw [anPart, ← map_mul, normalOrder_eq_ι_normalOrderF, normalOrderF_mul_anPartF]
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rfl
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lemma crPart_mul_normalOrder (φ : 𝓕.States) (a : 𝓕.FieldOpAlgebra) :
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𝓝(crPart φ * a) = crPart φ * 𝓝(a) := by
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obtain ⟨a, rfl⟩ := ι_surjective a
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rw [crPart, ← map_mul, normalOrder_eq_ι_normalOrderF, normalOrderF_crPartF_mul]
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rfl
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/-!
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### Normal order and super commutes
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@ -286,5 +313,252 @@ lemma normalOrder_superCommute_mid_eq_zero (a b c d : 𝓕.FieldOpAlgebra) :
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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 normalOrder_ofFieldOp_ofFieldOp_swap (φ φ' : 𝓕.States) :
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𝓝(ofFieldOp φ * ofFieldOp φ') = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓝(ofFieldOp φ' * ofFieldOp φ) := by
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rw [ofFieldOp_mul_ofFieldOp_eq_superCommute]
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simp
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lemma normalOrder_ofCrAnFieldOp_ofCrAnFieldOpList (φ : 𝓕.CrAnStates)
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(φs : List 𝓕.CrAnStates) : 𝓝(ofCrAnFieldOp φ * ofCrAnFieldOpList φs) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • 𝓝(ofCrAnFieldOpList φs * ofCrAnFieldOp φ) := by
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rw [← ofCrAnFieldOpList_singleton, ofCrAnFieldOpList_mul_ofCrAnFieldOpList_eq_superCommute]
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simp
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lemma normalOrder_ofCrAnFieldOp_ofFieldOpList_swap (φ : 𝓕.CrAnStates) (φ' : List 𝓕.States) :
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𝓝(ofCrAnFieldOp φ * ofFieldOpList φ') = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
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𝓝(ofFieldOpList φ' * ofCrAnFieldOp φ) := by
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rw [← ofCrAnFieldOpList_singleton, ofCrAnFieldOpList_mul_ofFieldOpList_eq_superCommute]
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simp
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lemma normalOrder_anPart_ofFieldOpList_swap (φ : 𝓕.States) (φ' : List 𝓕.States) :
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𝓝(anPart φ * ofFieldOpList φ') = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓝(ofFieldOpList 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 [normalOrder_ofCrAnFieldOp_ofFieldOpList_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 [normalOrder_ofCrAnFieldOp_ofFieldOpList_swap]
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rfl
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lemma normalOrder_ofFieldOpList_anPart_swap (φ : 𝓕.States) (φ' : List 𝓕.States) :
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𝓝(ofFieldOpList φ' * anPart φ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓝(anPart φ * ofFieldOpList φ') := by
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rw [normalOrder_anPart_ofFieldOpList_swap]
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simp [smul_smul, FieldStatistic.exchangeSign_mul_self]
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lemma normalOrder_ofFieldOpList_mul_anPart_swap (φ : 𝓕.States) (φs : List 𝓕.States) :
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𝓝(ofFieldOpList φs) * anPart φ = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • 𝓝(anPart φ * ofFieldOpList φs) := by
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rw [← normalOrder_mul_anPart]
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rw [normalOrder_ofFieldOpList_anPart_swap]
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lemma anPart_mul_normalOrder_ofFieldOpList_eq_superCommute (φ : 𝓕.States)
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(φs' : List 𝓕.States) : anPart φ * 𝓝(ofFieldOpList φs') =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝(ofFieldOpList φs' * anPart φ) + [anPart φ, 𝓝(ofFieldOpList φs')]ₛ := by
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rw [anPart, ofFieldOpList, normalOrder_eq_ι_normalOrderF, ← map_mul]
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rw [anPartF_mul_normalOrderF_ofStateList_eq_superCommuteF]
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simp only [instCommGroup.eq_1, map_add, map_smul]
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rfl
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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 ofCrAnFieldOp_superCommute_normalOrder_ofCrAnFieldOpList_sum (φ : 𝓕.CrAnStates)
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(φs : List 𝓕.CrAnStates) : [ofCrAnFieldOp φ, 𝓝(ofCrAnFieldOpList φs)]ₛ = ∑ n : Fin φs.length,
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) • [ofCrAnFieldOp φ, ofCrAnFieldOp φs[n]]ₛ
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* 𝓝(ofCrAnFieldOpList (φs.eraseIdx n)) := by
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rw [normalOrder_ofCrAnFieldOpList, map_smul]
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rw [superCommute_ofCrAnFieldOp_ofCrAnFieldOpList_eq_sum, Finset.smul_sum,
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sum_normalOrderList_length]
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congr
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funext n
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simp
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rw [ofCrAnFieldOpList_eq_normalOrder, 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 [superCommute_diff_statistic hs]
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simp
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lemma ofCrAnFieldOp_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.CrAnStates) (φs : List 𝓕.States) :
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[ofCrAnFieldOp φ, 𝓝(ofFieldOpList φs)]ₛ = ∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
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[ofCrAnFieldOp φ, ofFieldOp φs[n]]ₛ * 𝓝(ofFieldOpList (φs.eraseIdx n)) := by
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conv_lhs =>
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rw [ofFieldOpList_eq_sum, map_sum, map_sum]
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enter [2, s]
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rw [ofCrAnFieldOp_superCommute_normalOrder_ofCrAnFieldOpList_sum, 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, ← ofFieldOp_eq_sum, ← ofFieldOpList_eq_sum]
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/--
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Within a proto-operator algebra we have that
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`[anPartF φ, 𝓝(φs)] = ∑ i, sᵢ • [anPartF φ, φᵢ]ₛ * 𝓝(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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-/
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lemma anPart_superCommute_normalOrder_ofFieldOpList_sum (φ : 𝓕.States) (φs : List 𝓕.States) :
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[anPart φ, 𝓝(ofFieldOpList φs)]ₛ = ∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
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[anPart φ, ofState φs[n]]ₛ * 𝓝(ofFieldOpList (φs.eraseIdx n)) := 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, Fin.getElem_fin, Algebra.smul_mul_assoc]
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rw [ofCrAnFieldOp_superCommute_normalOrder_ofFieldOpList_sum]
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simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnStatesToStates_prod,
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Fin.getElem_fin, Algebra.smul_mul_assoc]
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rfl
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| .outAsymp φ =>
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simp only [anPart_posAsymp, instCommGroup.eq_1, Fin.getElem_fin, Algebra.smul_mul_assoc]
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rw [ofCrAnFieldOp_superCommute_normalOrder_ofFieldOpList_sum]
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simp only [instCommGroup.eq_1, crAnStatistics, Function.comp_apply, crAnStatesToStates_prod,
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Fin.getElem_fin, Algebra.smul_mul_assoc]
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rfl
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/-!
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## Multiplying with normal ordered terms
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-/
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/--
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Within a proto-operator algebra we have that
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`anPartF φ * 𝓝(φ₀φ₁…φₙ) = 𝓝((anPart φ)φ₀φ₁…φₙ) + [anpart φ, 𝓝(φ₀φ₁…φₙ)]ₛ`.
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-/
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lemma anPart_mul_normalOrder_ofFieldOpList_eq_superCommute_reorder (φ : 𝓕.States)
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(φs : List 𝓕.States) : anPart φ * 𝓝(ofFieldOpList φs) =
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𝓝(anPart φ * ofFieldOpList φs) + [anPart φ, 𝓝(ofFieldOpList φs)]ₛ := by
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rw [anPart_mul_normalOrder_ofFieldOpList_eq_superCommute]
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simp [instCommGroup.eq_1, map_add, map_smul]
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rw [normalOrder_anPart_ofFieldOpList_swap]
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/--
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Within a proto-operator algebra we have that
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`φ * 𝓝ᶠ(φ₀φ₁…φₙ) = 𝓝ᶠ(φφ₀φ₁…φₙ) + [anpart φ, 𝓝ᶠ(φ₀φ₁…φₙ)]ₛca`.
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-/
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lemma ofFieldOp_mul_normalOrder_ofFieldOpList_eq_superCommute (φ : 𝓕.States)
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(φs : List 𝓕.States) : ofFieldOp φ * 𝓝(ofFieldOpList φs) =
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𝓝(ofFieldOp φ * ofFieldOpList φs) + [anPart φ, 𝓝(ofFieldOpList φs)]ₛ := by
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conv_lhs => rw [ofFieldOp_eq_crPart_add_anPart]
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rw [add_mul, anPart_mul_normalOrder_ofFieldOpList_eq_superCommute_reorder, ← add_assoc,
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← crPart_mul_normalOrder, ← map_add]
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conv_lhs =>
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lhs
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rw [← add_mul, ← ofFieldOp_eq_crPart_add_anPart]
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/-- In the expansion of `ofState φ * normalOrderF (ofStateList φs)` the element
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of `𝓞.A` associated with contracting `φ` with the (optional) `n`th element of `φs`. -/
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noncomputable def contractStateAtIndex (φ : 𝓕.States) (φs : List 𝓕.States)
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(n : Option (Fin φs.length)) : 𝓕.FieldOpAlgebra :=
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match n with
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| none => 1
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| some n => 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) • [anPart φ, ofFieldOp φs[n]]ₛ
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/--
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Within a proto-operator algebra,
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`φ * N(φ₀φ₁…φₙ) = N(φφ₀φ₁…φₙ) + ∑ i, (sᵢ • [anPartF φ, φᵢ]ₛ) * N(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)`,
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where `sₙ` is the exchange sign for `φ` and `φ₀φ₁…φᵢ₋₁`.
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-/
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lemma ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum (φ : 𝓕.States)
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(φs : List 𝓕.States) : ofFieldOp φ * 𝓝(ofFieldOpList φs) =
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∑ n : Option (Fin φs.length), contractStateAtIndex φ φs n *
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𝓝(ofFieldOpList (HepLean.List.optionEraseZ φs φ n)) := by
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rw [ofFieldOp_mul_normalOrder_ofFieldOpList_eq_superCommute]
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rw [anPart_superCommute_normalOrder_ofFieldOpList_sum]
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simp only [instCommGroup.eq_1, Fin.getElem_fin, Algebra.smul_mul_assoc, contractStateAtIndex,
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Fintype.sum_option, one_mul]
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rfl
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/-!
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## Cons vs insertIdx for a normal ordered term.
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-/
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/--
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Within a proto-operator algebra, `N(φφ₀φ₁…φₙ) = s • N(φ₀…φₖ₋₁φφₖ…φₙ)`, where
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`s` is the exchange sign for `φ` and `φ₀…φₖ₋₁`.
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-/
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lemma ofFieldOpList_normalOrder_insert (φ : 𝓕.States) (φs : List 𝓕.States)
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(k : Fin φs.length.succ) : 𝓝(ofFieldOpList (φ :: φs)) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs.take k) • 𝓝(ofFieldOpList (φs.insertIdx k φ)) := by
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have hl : φs.insertIdx k φ = φs.take k ++ [φ] ++ φs.drop k := by
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rw [HepLean.List.insertIdx_eq_take_drop]
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simp
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rw [hl]
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rw [ofFieldOpList_append, ofFieldOpList_append]
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rw [ofFieldOpList_mul_ofFieldOpList_eq_superCommute, add_mul]
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simp [instCommGroup.eq_1, Nat.succ_eq_add_one, ofList_singleton, Algebra.smul_mul_assoc,
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map_add, map_smul, add_zero, smul_smul,
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exchangeSign_mul_self_swap, one_smul]
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rw [← ofFieldOpList_append, ← ofFieldOpList_append]
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simp
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/-!
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## The normal ordering of a product of two states
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-/
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@[simp]
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lemma normalOrder_crPart_mul_crPart (φ φ' : 𝓕.States) :
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𝓝(crPart φ * crPart φ') = crPart φ * crPart φ' := by
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rw [crPart, crPart, ← map_mul, normalOrder_eq_ι_normalOrderF, normalOrderF_crPartF_mul_crPartF]
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@[simp]
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lemma normalOrder_anPart_mul_anPart (φ φ' : 𝓕.States) :
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𝓝(anPart φ * anPart φ') = anPart φ * anPart φ' := by
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rw [anPart, anPart, ← map_mul, normalOrder_eq_ι_normalOrderF, normalOrderF_anPartF_mul_anPartF]
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@[simp]
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lemma normalOrder_crPart_mul_anPart (φ φ' : 𝓕.States) :
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𝓝(crPart φ * anPart φ') = crPart φ * anPart φ' := by
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rw [crPart, anPart, ← map_mul, normalOrder_eq_ι_normalOrderF, normalOrderF_crPartF_mul_anPartF]
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@[simp]
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lemma normalOrder_anPart_mul_crPart (φ φ' : 𝓕.States) :
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𝓝(anPart φ * crPart φ') = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ := by
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rw [anPart, crPart, ← map_mul, normalOrder_eq_ι_normalOrderF, normalOrderF_anPartF_mul_crPartF]
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simp
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lemma normalOrder_ofFieldOp_mul_ofFieldOp (φ φ' : 𝓕.States) : 𝓝(ofFieldOp φ * ofFieldOp φ') =
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crPart φ * crPart φ' + 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • (crPart φ' * anPart φ) +
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crPart φ * anPart φ' + anPart φ * anPart φ' := by
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rw [ofFieldOp, ofFieldOp, ← map_mul, normalOrder_eq_ι_normalOrderF,
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normalOrderF_ofState_mul_ofState]
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rfl
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end FieldOpAlgebra
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end FieldSpecification
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