refactor: Notation for insertAndContract
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jstoobysmith
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6 changed files with 69 additions and 65 deletions
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@ -27,18 +27,18 @@ open FieldStatistic
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/--
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Let `c` be a Wick Contraction for `φs := φ₀φ₁…φₙ`.
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We have (roughly) `𝓝([φsΛ.insertAndContract φ i none]ᵘᶜ) = s • 𝓝(φ :: [φsΛ]ᵘᶜ)`
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We have (roughly) `𝓝([φsΛ ↩Λ φ i none]ᵘᶜ) = s • 𝓝(φ :: [φsΛ]ᵘᶜ)`
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Where `s` is the exchange sign for `φ` and the uncontracted fields in `φ₀φ₁…φᵢ₋₁`.
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-/
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lemma insertAndContract_none_normalOrder (φ : 𝓕.States) (φs : List 𝓕.States)
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(i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) :
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𝓞.crAnF (𝓝([φsΛ.insertAndContract φ i none]ᵘᶜ))
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𝓞.crAnF (𝓝([φsΛ ↩Λ φ i none]ᵘᶜ))
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= 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, φsΛ.uncontracted.filter (fun x => i.succAbove x < i)⟩) •
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𝓞.crAnF 𝓝(φ :: [φsΛ]ᵘᶜ) := by
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simp only [Nat.succ_eq_add_one, instCommGroup.eq_1]
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rw [crAnF_ofState_normalOrder_insert φ [φsΛ]ᵘᶜ
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⟨(φsΛ.uncontractedListOrderPos i), by simp [uncontractedListGet]⟩, smul_smul]
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trans (1 : ℂ) • 𝓞.crAnF (𝓝(ofStateList [φsΛ.insertAndContract φ i none]ᵘᶜ))
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trans (1 : ℂ) • 𝓞.crAnF (𝓝(ofStateList [φsΛ ↩Λ φ i none]ᵘᶜ))
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· simp
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congr 1
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simp only [instCommGroup.eq_1, uncontractedListGet]
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@ -96,13 +96,13 @@ lemma insertAndContract_none_normalOrder (φ : 𝓕.States) (φs : List 𝓕.Sta
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/--
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Let `c` be a Wick Contraction for `φ₀φ₁…φₙ`.
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We have (roughly) `N(c.insertAndContract φ i k).uncontractedList`
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We have (roughly) `N(c ↩Λ φ i k).uncontractedList`
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is equal to `N((c.uncontractedList).eraseIdx k')`
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where `k'` is the position in `c.uncontractedList` corresponding to `k`.
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-/
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lemma insertAndContract_some_normalOrder (φ : 𝓕.States) (φs : List 𝓕.States)
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(i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) (k : φsΛ.uncontracted) :
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𝓞.crAnF 𝓝([φsΛ.insertAndContract φ i (some k)]ᵘᶜ)
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𝓞.crAnF 𝓝([φsΛ ↩Λ φ i (some k)]ᵘᶜ)
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= 𝓞.crAnF 𝓝((optionEraseZ [φsΛ]ᵘᶜ φ ((uncontractedStatesEquiv φs φsΛ) k))) := by
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simp only [Nat.succ_eq_add_one, insertAndContract, optionEraseZ, uncontractedStatesEquiv,
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Equiv.optionCongr_apply, Equiv.coe_trans, Option.map_some', Function.comp_apply, finCongr_apply,
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@ -122,8 +122,8 @@ mulitiplied by the exchange sign of `φ` and `φ₀φ₁…φᵢ₋₁`.
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-/
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lemma sign_timeContract_normalOrder_insertAndContract_none (φ : 𝓕.States) (φs : List 𝓕.States)
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(i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) :
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(φsΛ.insertAndContract φ i none).sign • (φsΛ.insertAndContract φ i none).timeContract 𝓞
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* 𝓞.crAnF 𝓝([φsΛ.insertAndContract φ i none]ᵘᶜ) =
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(φsΛ ↩Λ φ i none).sign • (φsΛ ↩Λ φ i none).timeContract 𝓞
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* 𝓞.crAnF 𝓝([φsΛ ↩Λ φ i none]ᵘᶜ) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (Finset.univ.filter (fun k => i.succAbove k < i))⟩)
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• (φsΛ.sign • φsΛ.timeContract 𝓞 * 𝓞.crAnF 𝓝(φ :: [φsΛ]ᵘᶜ)) := by
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by_cases hg : GradingCompliant φs φsΛ
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@ -165,15 +165,15 @@ lemma sign_timeContract_normalOrder_insertAndContract_none (φ : 𝓕.States) (
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Let `c` be a Wick Contraction for `φ₀φ₁…φₙ`.
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This lemma states that
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`(c.sign • c.timeContract 𝓞) * N(c.uncontracted)`
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for `c` equal to `c.insertAndContract φ i (some k)` is equal to that for just `c`
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for `c` equal to `c ↩Λ φ i (some k)` is equal to that for just `c`
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mulitiplied by the exchange sign of `φ` and `φ₀φ₁…φᵢ₋₁`.
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-/
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lemma sign_timeContract_normalOrder_insertAndContract_some (φ : 𝓕.States) (φs : List 𝓕.States)
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(i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) (k : φsΛ.uncontracted)
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(hlt : ∀ (k : Fin φs.length), timeOrderRel φ φs[k])
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(hn : ∀ (k : Fin φs.length), i.succAbove k < i → ¬ timeOrderRel φs[k] φ) :
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(φsΛ.insertAndContract φ i (some k)).sign • (φsΛ.insertAndContract φ i (some k)).timeContract 𝓞
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* 𝓞.crAnF 𝓝([φsΛ.insertAndContract φ i (some k)]ᵘᶜ) =
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(φsΛ ↩Λ φ i (some k)).sign • (φsΛ ↩Λ φ i (some k)).timeContract 𝓞
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* 𝓞.crAnF 𝓝([φsΛ ↩Λ φ i (some k)]ᵘᶜ) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (Finset.univ.filter (fun x => i.succAbove x < i))⟩)
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• (φsΛ.sign • (𝓞.contractStateAtIndex φ [φsΛ]ᵘᶜ
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((uncontractedStatesEquiv φs φsΛ) (some k)) * φsΛ.timeContract 𝓞)
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@ -232,21 +232,21 @@ lemma sign_timeContract_normalOrder_insertAndContract_some (φ : 𝓕.States) (
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exact hg'
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/--
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Let `c` be a Wick Contraction for `φ₀φ₁…φₙ`.
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This lemma states that
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`(c.sign • c.timeContract 𝓞) * N(c.uncontracted)`
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is equal to the sum of
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`(c'.sign • c'.timeContract 𝓞) * N(c'.uncontracted)`
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for all `c' = (c.insertAndContract φ i k)` for `k : Option c.uncontracted`, multiplied by
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the exchange sign of `φ` and `φ₀φ₁…φᵢ₋₁`.
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Given a Wick contraction `φsΛ` of `φs = φ₀φ₁…φₙ` and an `i`, we have that
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`(φsΛ.sign • φsΛ.timeContract 𝓞) * 𝓞.crAnF (φ * 𝓝([φsΛ]ᵘᶜ))`
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is equal to the product of
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- the exchange sign of `φ` and `φ₀φ₁…φᵢ₋₁`,
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- the sum of
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`((φsΛ ↩Λ φ i k).sign • (φsΛ ↩Λ φ i k).timeContract 𝓞) * 𝓞.crAnF 𝓝([φsΛ ↩Λ φ i k]ᵘᶜ)`
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over all `k` in `Option φsΛ.uncontracted`.
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-/
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lemma mul_sum_contractions (φ : 𝓕.States) (φs : List 𝓕.States) (i : Fin φs.length.succ)
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(φsΛ : WickContraction φs.length) (hlt : ∀ (k : Fin φs.length), timeOrderRel φ φs[k])
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(hn : ∀ (k : Fin φs.length), i.succAbove k < i → ¬timeOrderRel φs[k] φ) :
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(φsΛ.sign • φsΛ.timeContract 𝓞) * 𝓞.crAnF ((CrAnAlgebra.ofState φ) * 𝓝([φsΛ]ᵘᶜ)) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (Finset.univ.filter (fun x => i.succAbove x < i))⟩) •
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∑ (k : Option φsΛ.uncontracted), ((φsΛ.insertAndContract φ i k).sign •
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(φsΛ.insertAndContract φ i k).timeContract 𝓞 * 𝓞.crAnF (𝓝(ofStateList [φsΛ.insertAndContract φ i k]ᵘᶜ))) := by
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∑ (k : Option φsΛ.uncontracted), ((φsΛ ↩Λ φ i k).sign •
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(φsΛ ↩Λ φ i k).timeContract 𝓞 * 𝓞.crAnF (𝓝([φsΛ ↩Λ φ i k]ᵘᶜ))) := by
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rw [crAnF_ofState_mul_normalOrder_ofStatesList_eq_sum, Finset.mul_sum,
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uncontractedStatesEquiv_list_sum, Finset.smul_sum]
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simp only [instCommGroup.eq_1, Nat.succ_eq_add_one]
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@ -339,12 +339,11 @@ theorem wicks_theorem : (φs : List 𝓕.States) → 𝓞.crAnF (ofStateAlgebra
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(maxTimeField φ φs) (eraseMaxTimeField φ φs) (maxTimeFieldPosFin φ φs) c]
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trans (1 : ℂ) • ∑ k : Option { x // x ∈ c.uncontracted }, sign
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(List.insertIdx (↑(maxTimeFieldPosFin φ φs)) (maxTimeField φ φs) (eraseMaxTimeField φ φs))
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(insertAndContract (maxTimeField φ φs) c (maxTimeFieldPosFin φ φs) k) •
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↑((c.insertAndContract (maxTimeField φ φs) (maxTimeFieldPosFin φ φs) k).timeContract 𝓞) *
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(c ↩Λ (maxTimeField φ φs) (maxTimeFieldPosFin φ φs) k) •
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↑((c ↩Λ (maxTimeField φ φs) (maxTimeFieldPosFin φ φs) k).timeContract 𝓞) *
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𝓞.crAnF 𝓝(ofStateList (List.map (List.insertIdx (↑(maxTimeFieldPosFin φ φs))
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(maxTimeField φ φs) (eraseMaxTimeField φ φs)).get
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(insertAndContract (maxTimeField φ φs) c
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(maxTimeFieldPosFin φ φs) k).uncontractedList))
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(c ↩Λ (maxTimeField φ φs) (maxTimeFieldPosFin φ φs) k).uncontractedList))
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swap
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· simp [uncontractedListGet]
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rw [smul_smul]
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