refactor: Simplify some notation
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jstoobysmith
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8 changed files with 106 additions and 111 deletions
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@ -27,20 +27,20 @@ open FieldStatistic
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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.insertList φ φs i none).uncontractedList = s • N(φ * c.uncontractedList)`
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We have (roughly) `N(c.insertList φ i none).uncontractedList = s • N(φ * c.uncontractedList)`
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Where `s` is the exchange sign for `φ` and the uncontracted fields in `φ₀φ₁…φᵢ`.
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-/
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lemma insertList_none_normalOrder (φ : 𝓕.States) (φs : List 𝓕.States)
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(i : Fin φs.length.succ) (c : WickContraction φs.length) :
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𝓞.crAnF (normalOrder (ofStateList (List.map (φs.insertIdx i φ).get
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(c.insertList φ φs i none).uncontractedList)))
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𝓞.crAnF (𝓝(ofStateList (List.map (φs.insertIdx i φ).get
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(c.insertList φ i none).uncontractedList)))
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= 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, c.uncontracted.filter (fun x => i.succAbove x < i)⟩) •
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𝓞.crAnF (normalOrder (ofStateList (optionEraseZ (c.uncontractedList.map φs.get) φ none))) := by
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𝓞.crAnF (𝓝(ofStateList (optionEraseZ (c.uncontractedList.map φs.get) φ none))) := by
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simp only [Nat.succ_eq_add_one, instCommGroup.eq_1, optionEraseZ]
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rw [crAnF_ofState_normalOrder_insert φ (c.uncontractedList.map φs.get)
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⟨(c.uncontractedListOrderPos i), by simp⟩, smul_smul]
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trans (1 : ℂ) • 𝓞.crAnF (normalOrder (ofStateList
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(List.map (List.insertIdx (↑i) φ φs).get (insertList φ φs c i none).uncontractedList)))
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trans (1 : ℂ) • 𝓞.crAnF (𝓝(ofStateList
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(List.map (List.insertIdx (↑i) φ φs).get (insertList φ c i none).uncontractedList)))
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· simp
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congr 1
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simp only [instCommGroup.eq_1]
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@ -98,16 +98,16 @@ lemma insertList_none_normalOrder (φ : 𝓕.States) (φs : List 𝓕.States)
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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.insertList φ φs i k).uncontractedList`
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We have (roughly) `N(c.insertList 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 insertList_some_normalOrder (φ : 𝓕.States) (φs : List 𝓕.States)
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(i : Fin φs.length.succ) (c : WickContraction φs.length) (k : c.uncontracted) :
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𝓞.crAnF (normalOrder (ofStateList (List.map (φs.insertIdx i φ).get
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(c.insertList φ φs i (some k)).uncontractedList)))
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= 𝓞.crAnF (normalOrder (ofStateList (optionEraseZ (c.uncontractedList.map φs.get) φ
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((uncontractedStatesEquiv φs c) k)))) := by
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𝓞.crAnF 𝓝(ofStateList (List.map (φs.insertIdx i φ).get
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(c.insertList φ i (some k)).uncontractedList))
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= 𝓞.crAnF 𝓝(ofStateList (optionEraseZ (c.uncontractedList.map φs.get) φ
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((uncontractedStatesEquiv φs c) k))) := by
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simp only [Nat.succ_eq_add_one, insertList, optionEraseZ, uncontractedStatesEquiv,
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Equiv.optionCongr_apply, Equiv.coe_trans, Option.map_some', Function.comp_apply, finCongr_apply,
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Fin.coe_cast]
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@ -121,17 +121,17 @@ lemma insertList_some_normalOrder (φ : 𝓕.States) (φs : List 𝓕.States)
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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 `(c.sign • c.timeContract 𝓞) * N(c.uncontracted)`
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for `c` equal to `c.insertList φ φs i none` is equal to that for just `c`
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for `c` equal to `c.insertList φ i none` 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_insertList_none (φ : 𝓕.States) (φs : List 𝓕.States)
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(i : Fin φs.length.succ) (c : WickContraction φs.length) :
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(c.insertList φ φs i none).sign • (c.insertList φ φs i none).timeContract 𝓞
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* 𝓞.crAnF (normalOrder (ofStateList (List.map (φs.insertIdx i φ).get
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(c.insertList φ φs i none).uncontractedList))) =
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(c.insertList φ i none).sign • (c.insertList φ i none).timeContract 𝓞
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* 𝓞.crAnF 𝓝(ofStateList (List.map (φs.insertIdx i φ).get
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(c.insertList φ i none).uncontractedList)) =
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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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• (c.sign • c.timeContract 𝓞 * 𝓞.crAnF (normalOrder
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(ofStateList (optionEraseZ (c.uncontractedList.map φs.get) φ none)))) := by
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• (c.sign • c.timeContract 𝓞 *
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𝓞.crAnF 𝓝(ofStateList (optionEraseZ (c.uncontractedList.map φs.get) φ none))) := by
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by_cases hg : GradingCompliant φs c
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· rw [insertList_none_normalOrder, sign_insert_none]
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simp only [Nat.succ_eq_add_one, timeContract_insertList_none, instCommGroup.eq_1,
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@ -171,21 +171,21 @@ lemma sign_timeContract_normalOrder_insertList_none (φ : 𝓕.States) (φs : Li
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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.insertList φ φs i (some k)` is equal to that for just `c`
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for `c` equal to `c.insertList 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_insertList_some (φ : 𝓕.States) (φs : List 𝓕.States)
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(i : Fin φs.length.succ) (c : WickContraction φs.length) (k : c.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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(c.insertList φ φs i (some k)).sign • (c.insertList φ φs i (some k)).timeContract 𝓞
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* 𝓞.crAnF (normalOrder (ofStateList (List.map (φs.insertIdx i φ).get
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(c.insertList φ φs i (some k)).uncontractedList))) =
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(c.insertList φ i (some k)).sign • (c.insertList φ i (some k)).timeContract 𝓞
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* 𝓞.crAnF 𝓝(ofStateList (List.map (φs.insertIdx i φ).get
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(c.insertList φ i (some k)).uncontractedList)) =
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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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• (c.sign • (𝓞.contractStateAtIndex φ (List.map φs.get c.uncontractedList)
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((uncontractedStatesEquiv φs c) (some k)) * c.timeContract 𝓞)
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* 𝓞.crAnF (normalOrder (ofStateList (optionEraseZ (c.uncontractedList.map φs.get) φ
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((uncontractedStatesEquiv φs c) k))))) := by
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* 𝓞.crAnF 𝓝(ofStateList (optionEraseZ (c.uncontractedList.map φs.get) φ
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((uncontractedStatesEquiv φs c) k)))) := by
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by_cases hg : GradingCompliant φs c ∧ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[k.1])
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· by_cases hk : i.succAbove k < i
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· rw [WickContraction.timeConract_insertList_some_eq_mul_contractStateAtIndex_not_lt]
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@ -244,19 +244,19 @@ 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.insertList φ φs i k)` for `k : Option (c.uncontracted)`, multiplied by
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for all `c' = (c.insertList 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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-/
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lemma mul_sum_contractions (φ : 𝓕.States) (φs : List 𝓕.States) (i : Fin φs.length.succ)
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(c : 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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(c.sign • c.timeContract 𝓞) * 𝓞.crAnF ((CrAnAlgebra.ofState φ) *
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normalOrder (ofStateList (c.uncontractedList.map φs.get))) =
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𝓝(ofStateList (c.uncontractedList.map φs.get))) =
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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 (c.uncontracted)),
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((c.insertList φ φs i k).sign • (c.insertList φ φs i k).timeContract 𝓞
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* 𝓞.crAnF (normalOrder
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(ofStateList ((c.insertList φ φs i k).uncontractedList.map (φs.insertIdx i φ).get)))) := by
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((c.insertList φ i k).sign • (c.insertList φ i k).timeContract 𝓞
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* 𝓞.crAnF (𝓝(ofStateList
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((c.insertList φ i k).uncontractedList.map (φs.insertIdx i φ).get)))) := 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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@ -287,9 +287,9 @@ lemma mul_sum_contractions (φ : 𝓕.States) (φs : List 𝓕.States) (i : Fin
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lemma wicks_theorem_congr {φs φs' : List 𝓕.States} (h : φs = φs') :
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∑ (c : WickContraction φs.length), (c.sign • c.timeContract 𝓞) *
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𝓞.crAnF (normalOrder (ofStateList (c.uncontractedList.map φs.get)))
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𝓞.crAnF 𝓝(ofStateList (c.uncontractedList.map φs.get))
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= ∑ (c : WickContraction φs'.length), (c.sign • c.timeContract 𝓞) *
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𝓞.crAnF (normalOrder (ofStateList (c.uncontractedList.map φs'.get))) := by
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𝓞.crAnF 𝓝(ofStateList (c.uncontractedList.map φs'.get)) := by
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subst h
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simp
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@ -303,7 +303,7 @@ lemma wicks_theorem_congr {φs φs' : List 𝓕.States} (h : φs = φs') :
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lemma wicks_theorem_nil :
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𝓞.crAnF (ofStateAlgebra (timeOrder (ofList []))) = ∑ (c : WickContraction [].length),
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(c.sign [] • c.timeContract 𝓞) *
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𝓞.crAnF (normalOrder (ofStateList (c.uncontractedList.map [].get))) := by
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𝓞.crAnF 𝓝(ofStateList (c.uncontractedList.map [].get)) := by
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rw [timeOrder_ofList_nil]
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simp only [map_one, List.length_nil, Algebra.smul_mul_assoc]
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rw [sum_WickContraction_nil, nil_zero_uncontractedList]
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@ -330,7 +330,7 @@ remark wicks_theorem_context := "
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-/
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theorem wicks_theorem : (φs : List 𝓕.States) → 𝓞.crAnF (ofStateAlgebra (timeOrder (ofList φs))) =
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∑ (c : WickContraction φs.length), (c.sign φs • c.timeContract 𝓞) *
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𝓞.crAnF (normalOrder (ofStateList (c.uncontractedList.map φs.get)))
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𝓞.crAnF 𝓝(ofStateList (c.uncontractedList.map φs.get))
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| [] => wicks_theorem_nil
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| φ :: φs => by
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have ih := wicks_theorem (eraseMaxTimeField φ φs)
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@ -351,13 +351,12 @@ 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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(insertList (maxTimeField φ φs) (eraseMaxTimeField φ φs) c (maxTimeFieldPosFin φ φs) k) •
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↑(WickContraction.timeContract 𝓞 (insertList (maxTimeField φ φs) (eraseMaxTimeField φ φs) c
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(maxTimeFieldPosFin φ φs) k)) *
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𝓞.crAnF (normalOrder (ofStateList (List.map (List.insertIdx (↑(maxTimeFieldPosFin φ φs))
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(insertList (maxTimeField φ φs) c (maxTimeFieldPosFin φ φs) k) •
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↑((c.insertList (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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(insertList (maxTimeField φ φs) (eraseMaxTimeField φ φs) c
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(maxTimeFieldPosFin φ φs) k).uncontractedList)))
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(insertList (maxTimeField φ φs) c
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(maxTimeFieldPosFin φ φs) k).uncontractedList))
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swap
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· simp
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rw [smul_smul]
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