refactor: improve docs
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6 changed files with 126 additions and 57 deletions
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@ -404,10 +404,14 @@ lemma superCommute_ofCrAnList_ofStateList_cons (φ : 𝓕.States) (φs : List
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rw [ofStateList_cons, mul_assoc, smul_smul, FieldStatistic.ofList_cons_eq_mul]
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simp [mul_comm]
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/--
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Within the creation and annihilation algebra, we have that
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`[φᶜᵃs, φᶜᵃ₀ … φᶜᵃₙ]ₛca = ∑ i, sᵢ • φᶜᵃs₀ … φᶜᵃᵢ₋₁ * [φᶜᵃs, φᶜᵃᵢ]ₛca * φᶜᵃᵢ₊₁ … φᶜᵃₙ`
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where `sᵢ` is the exchange sign for `φᶜᵃs` and `φᶜᵃs₀ … φᶜᵃᵢ₋₁`.
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-/
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lemma superCommute_ofCrAnList_ofCrAnList_eq_sum (φs : List 𝓕.CrAnStates) :
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(φs' : List 𝓕.CrAnStates) →
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[ofCrAnList φs, ofCrAnList φs']ₛca =
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∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ (List.take n φs')) •
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(φs' : List 𝓕.CrAnStates) → [ofCrAnList φs, ofCrAnList φs']ₛca =
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∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
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ofCrAnList (φs'.take n) * [ofCrAnList φs, ofCrAnState (φs'.get n)]ₛca *
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ofCrAnList (φs'.drop (n + 1))
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| [] => by
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@ -422,7 +426,7 @@ lemma superCommute_ofCrAnList_ofCrAnList_eq_sum (φs : List 𝓕.CrAnStates) :
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lemma superCommute_ofCrAnList_ofStateList_eq_sum (φs : List 𝓕.CrAnStates) : (φs' : List 𝓕.States) →
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[ofCrAnList φs, ofStateList φs']ₛca =
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∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ List.take n φs') •
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∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
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ofStateList (φs'.take n) * [ofCrAnList φs, ofState (φs'.get n)]ₛca *
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ofStateList (φs'.drop (n + 1))
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| [] => by
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@ -298,11 +298,18 @@ lemma crAnF_ofCrAnState_superCommute_normalOrder_ofStateList_eq_sum (φ : 𝓕.C
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rw [← Finset.mul_sum]
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rw [← Finset.sum_mul, ← map_sum, ← map_sum, ← ofState, ← map_sum, ← map_sum, ← ofStateList_sum]
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/--
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Within a proto-operator algebra we have that
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`[anPart φ, 𝓝(φs)] = ∑ i, sᵢ • [anPart φ, φᵢ]ₛca * 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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The origin of this result is
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- `superCommute_ofCrAnList_ofCrAnList_eq_sum`
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-/
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lemma crAnF_anPart_superCommute_normalOrder_ofStateList_eq_sum (φ : 𝓕.States) (φs : List 𝓕.States) :
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𝓞.crAnF ([anPart (StateAlgebra.ofState φ), normalOrder (ofStateList φs)]ₛca) =
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𝓞.crAnF ([anPart (StateAlgebra.ofState φ), 𝓝(φs)]ₛca) =
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∑ n : Fin φs.length, 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ (φs.take n)) •
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𝓞.crAnF ([anPart (StateAlgebra.ofState φ), ofState φs[n]]ₛca)
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* 𝓞.crAnF (normalOrder (ofStateList (φs.eraseIdx n))) := by
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𝓞.crAnF ([anPart (StateAlgebra.ofState φ), ofState φs[n]]ₛca) * 𝓞.crAnF 𝓝(φs.eraseIdx n) := by
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match φ with
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simp
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@ -319,6 +326,10 @@ lemma crAnF_anPart_superCommute_normalOrder_ofStateList_eq_sum (φ : 𝓕.States
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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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`anPart φ * 𝓝(φ₀φ₁…φₙ) = 𝓝((anPart φ)φ₀φ₁…φₙ) + [anpart φ, 𝓝(φ₀φ₁…φₙ)]ₛca`.
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-/
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lemma crAnF_anPart_mul_normalOrder_ofStatesList_eq_superCommute (φ : 𝓕.States)
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(φ' : List 𝓕.States) :
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@ -330,11 +341,14 @@ lemma crAnF_anPart_mul_normalOrder_ofStatesList_eq_superCommute (φ : 𝓕.State
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congr
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rw [crAnF_normalOrder_anPart_ofStatesList_swap]
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/--
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Within a proto-operator algebra we have that
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`φ * 𝓝(φ₀φ₁…φₙ) = 𝓝(φφ₀φ₁…φₙ) + [anpart φ, 𝓝(φ₀φ₁…φₙ)]ₛca`.
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-/
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lemma crAnF_ofState_mul_normalOrder_ofStatesList_eq_superCommute (φ : 𝓕.States)
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(φ' : List 𝓕.States) :
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𝓞.crAnF (ofState φ * normalOrder (ofStateList φ')) =
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𝓞.crAnF (normalOrder (ofState φ * ofStateList φ')) +
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𝓞.crAnF ([anPart (StateAlgebra.ofState φ), normalOrder (ofStateList φ')]ₛca) := by
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(φs : List 𝓕.States) : 𝓞.crAnF (ofState φ * 𝓝(φs)) =
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𝓞.crAnF (normalOrder (ofState φ * ofStateList φs)) +
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𝓞.crAnF ([anPart (StateAlgebra.ofState φ), 𝓝(φs)]ₛca) := by
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conv_lhs => rw [ofState_eq_crPart_add_anPart]
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rw [add_mul, map_add, crAnF_anPart_mul_normalOrder_ofStatesList_eq_superCommute, ← add_assoc,
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← normalOrder_crPart_mul, ← map_add]
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@ -268,6 +268,20 @@ lemma insert_fin_eq_self (φ : 𝓕.States) {φs : List 𝓕.States}
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use z
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rfl
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lemma insertLift_sum (φ : 𝓕.States) {φs : List 𝓕.States}
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(i : Fin φs.length.succ) [AddCommMonoid M] (f : WickContraction (φs.insertIdx i φ).length → M) :
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∑ c, f c =
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∑ (φsΛ : WickContraction φs.length), ∑ (k : Option φsΛ.uncontracted), f (φsΛ ↩Λ φ i k) := by
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rw [sum_extractEquiv_congr (finCongr (insertIdx_length_fin φ φs i).symm i) f
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(insertIdx_length_fin φ φs i)]
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rfl
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/-!
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## Uncontracted list
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-/
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lemma insertAndContract_uncontractedList_none_map (φ : 𝓕.States) {φs : List 𝓕.States}
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) :
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[φsΛ ↩Λ φ i none]ᵘᶜ = List.insertIdx (φsΛ.uncontractedListOrderPos i) φ [φsΛ]ᵘᶜ := by
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@ -283,12 +297,4 @@ lemma insertAndContract_uncontractedList_none_map (φ : 𝓕.States) {φs : List
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conv_rhs => rw [get_eq_insertIdx_succAbove φ φs i]
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rfl
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lemma insertLift_sum (φ : 𝓕.States) {φs : List 𝓕.States}
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(i : Fin φs.length.succ) [AddCommMonoid M] (f : WickContraction (φs.insertIdx i φ).length → M) :
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∑ c, f c =
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∑ (φsΛ : WickContraction φs.length), ∑ (k : Option φsΛ.uncontracted), f (φsΛ ↩Λ φ i k) := by
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rw [sum_extractEquiv_congr (finCongr (insertIdx_length_fin φ φs i).symm i) f
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(insertIdx_length_fin φ φs i)]
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rfl
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end WickContraction
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@ -18,7 +18,7 @@ namespace WickContraction
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variable {n : ℕ} (c : WickContraction n)
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open HepLean.List
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/-- Given a Wick contraction `c` associated with a list `φs`, the
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/-- Given a Wick contraction `φsΛ` associated with a list `φs`, the
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product of all time-contractions of pairs of contracted elements in `φs`,
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as a member of the center of `𝓞.A`. -/
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noncomputable def timeContract (𝓞 : 𝓕.ProtoOperatorAlgebra) {φs : List 𝓕.States}
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@ -28,6 +28,11 @@ noncomputable def timeContract (𝓞 : 𝓕.ProtoOperatorAlgebra) {φs : List
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(φs.get (φsΛ.fstFieldOfContract a)) (φs.get (φsΛ.sndFieldOfContract a)),
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𝓞.timeContract_mem_center _ _⟩
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/-- For `φsΛ` a Wick contraction for `φs`, the time contraction `(φsΛ ↩Λ φ i none).timeContract 𝓞`
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is equal to `φsΛ.timeContract 𝓞`.
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This result follows from the fact that `timeContract` only depends on contracted pairs,
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and `(φsΛ ↩Λ φ i none)` has the 'same' contracted pairs as `φsΛ`. -/
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@[simp]
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lemma timeContract_insertAndContract_none (𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) :
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@ -25,12 +25,18 @@ open HepLean.List
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open WickContraction
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open FieldStatistic
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/-!
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## Normal order of uncontracted terms within proto-algebra.
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-/
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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 ↩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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lemma normalOrder_uncontracted_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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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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@ -100,7 +106,7 @@ 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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lemma normalOrder_uncontracted_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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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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@ -114,20 +120,45 @@ lemma insertAndContract_some_normalOrder (φ : 𝓕.States) (φs : List 𝓕.Sta
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congr
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conv_rhs => rw [get_eq_insertIdx_succAbove φ φs i]
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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.insertAndContract 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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## Wick terms
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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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remark wick_term_terminology := "
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Let `φsΛ` be a Wick contraction. We informally call the term
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`(φsΛ.sign • φsΛ.timeContract 𝓞) * 𝓞.crAnF 𝓝([φsΛ]ᵘᶜ)` the Wick term
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associated with `φsΛ`. We do not make this a fully-fledge definition, as
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in most cases we want to consider slight modifications of this term."
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/--
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Let `φsΛ` be a Wick Contraction for `φs = φ₀φ₁…φₙ`. Then the wick-term of ` (φsΛ ↩Λ φ i none)`
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```(φsΛ ↩Λ φ i none).sign • (φsΛ ↩Λ φ i none).timeContract 𝓞 * 𝓞.crAnF 𝓝([φsΛ ↩Λ φ i none]ᵘᶜ)```
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is equal to
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`s • (φsΛ.sign • φsΛ.timeContract 𝓞 * 𝓞.crAnF 𝓝(φ :: [φsΛ]ᵘᶜ))`
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where `s` is the exchange sign of `φ` and the uncontracted fields in `φ₀φ₁…φᵢ₋₁`.
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The proof of this result relies primarily on:
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- `normalOrder_uncontracted_none` which replaces `𝓝([φsΛ ↩Λ φ i none]ᵘᶜ)` with
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`𝓝(φ :: [φsΛ]ᵘᶜ)` up to a sign.
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- `timeContract_insertAndContract_none` which replaces `(φsΛ ↩Λ φ i none).timeContract 𝓞` with
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`φsΛ.timeContract 𝓞`.
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- `sign_insert_none` and `signInsertNone_eq_filterset` which are used to take account of
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signs.
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-/
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lemma wick_term_none_eq_wick_term_cons (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 ↩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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· rw [insertAndContract_none_normalOrder, sign_insert_none]
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· rw [normalOrder_uncontracted_none, sign_insert_none]
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simp only [Nat.succ_eq_add_one, timeContract_insertAndContract_none, instCommGroup.eq_1,
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Algebra.mul_smul_comm, Algebra.smul_mul_assoc, smul_smul]
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congr 1
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@ -168,12 +199,12 @@ This lemma states that
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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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lemma wick_term_some_eq_wick_term_optionEraseZ (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 ↩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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* 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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@ -183,7 +214,7 @@ lemma sign_timeContract_normalOrder_insertAndContract_some (φ : 𝓕.States) (
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· rw [WickContraction.timeConract_insertAndContract_some_eq_mul_contractStateAtIndex_not_lt]
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swap
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· exact hn _ hk
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rw [insertAndContract_some_normalOrder, sign_insert_some]
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rw [normalOrder_uncontracted_some, sign_insert_some]
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simp only [instCommGroup.eq_1, smul_smul, Algebra.smul_mul_assoc]
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congr 1
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rw [mul_assoc, mul_comm (sign φs φsΛ), ← mul_assoc]
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@ -194,7 +225,7 @@ lemma sign_timeContract_normalOrder_insertAndContract_some (φ : 𝓕.States) (
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rw [WickContraction.timeConract_insertAndContract_some_eq_mul_contractStateAtIndex_lt]
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swap
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· exact hlt _
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rw [insertAndContract_some_normalOrder]
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rw [normalOrder_uncontracted_some]
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rw [sign_insert_some]
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simp only [instCommGroup.eq_1, smul_smul, Algebra.smul_mul_assoc]
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congr 1
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@ -236,11 +267,15 @@ Given a Wick contraction `φsΛ` of `φs = φ₀φ₁…φₙ` and an `i`, we h
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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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- the sum of `((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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The proof of this result primarily depends on
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- `crAnF_ofState_mul_normalOrder_ofStatesList_eq_sum` to rewrite `𝓞.crAnF (φ * 𝓝([φsΛ]ᵘᶜ))`
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- `wick_term_none_eq_wick_term_cons`
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- `wick_term_some_eq_wick_term_optionEraseZ`
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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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lemma wick_term_cons_eq_sum_wick_term (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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@ -254,7 +289,7 @@ lemma mul_sum_contractions (φ : 𝓕.States) (φs : List 𝓕.States) (i : Fin
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funext n
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match n with
|
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| none =>
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rw [sign_timeContract_normalOrder_insertAndContract_none]
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rw [wick_term_none_eq_wick_term_cons]
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simp only [contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
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||||
Equiv.coe_trans, Option.map_none', one_mul, Algebra.smul_mul_assoc, instCommGroup.eq_1,
|
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smul_smul]
|
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|
|
@ -262,7 +297,7 @@ lemma mul_sum_contractions (φ : 𝓕.States) (φs : List 𝓕.States) (i : Fin
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rw [← mul_assoc, exchangeSign_mul_self]
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simp
|
||||
| some n =>
|
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rw [sign_timeContract_normalOrder_insertAndContract_some _ _ _ _ _
|
||||
rw [wick_term_some_eq_wick_term_optionEraseZ _ _ _ _ _
|
||||
(fun k => hlt k) (fun k a => hn k a)]
|
||||
simp only [uncontractedStatesEquiv, Equiv.optionCongr_apply, Equiv.coe_trans, Option.map_some',
|
||||
Function.comp_apply, finCongr_apply, Algebra.smul_mul_assoc, instCommGroup.eq_1, smul_smul]
|
||||
|
|
@ -275,13 +310,6 @@ lemma mul_sum_contractions (φ : 𝓕.States) (φs : List 𝓕.States) (i : Fin
|
|||
rw [@Subalgebra.mem_center_iff] at ht
|
||||
rw [ht]
|
||||
|
||||
lemma wicks_theorem_congr {φs φs' : List 𝓕.States} (h : φs = φs') :
|
||||
∑ (φsΛ : WickContraction φs.length), (φsΛ.sign • φsΛ.timeContract 𝓞) *
|
||||
𝓞.crAnF 𝓝([φsΛ]ᵘᶜ)
|
||||
= ∑ (φs'Λ : WickContraction φs'.length), (φs'Λ.sign • φs'Λ.timeContract 𝓞) *
|
||||
𝓞.crAnF 𝓝([φs'Λ]ᵘᶜ) := by
|
||||
subst h
|
||||
simp
|
||||
|
||||
/-!
|
||||
|
||||
|
|
@ -301,6 +329,14 @@ lemma wicks_theorem_nil :
|
|||
rw [h1, normalOrder_ofCrAnList]
|
||||
simp [WickContraction.timeContract, empty, sign]
|
||||
|
||||
lemma wicks_theorem_congr {φs φs' : List 𝓕.States} (h : φs = φs') :
|
||||
∑ (φsΛ : WickContraction φs.length), (φsΛ.sign • φsΛ.timeContract 𝓞) *
|
||||
𝓞.crAnF 𝓝([φsΛ]ᵘᶜ)
|
||||
= ∑ (φs'Λ : WickContraction φs'.length), (φs'Λ.sign • φs'Λ.timeContract 𝓞) *
|
||||
𝓞.crAnF 𝓝([φs'Λ]ᵘᶜ) := by
|
||||
subst h
|
||||
simp
|
||||
|
||||
remark wicks_theorem_context := "
|
||||
Wick's theorem is one of the most important results in perturbative quantum field theory.
|
||||
It expresses a time-ordered product of fields as a sum of terms consisting of
|
||||
|
|
@ -335,7 +371,7 @@ theorem wicks_theorem : (φs : List 𝓕.States) → 𝓞.crAnF (ofStateAlgebra
|
|||
have ht := Subalgebra.mem_center_iff.mp (Subalgebra.smul_mem (Subalgebra.center ℂ 𝓞.A)
|
||||
(WickContraction.timeContract 𝓞 c).2 (sign (eraseMaxTimeField φ φs) c))
|
||||
rw [map_smul, map_smul, Algebra.smul_mul_assoc, ← mul_assoc, ht, mul_assoc, ← map_mul]
|
||||
rw [ofStateAlgebra_ofState, mul_sum_contractions (𝓞 := 𝓞)
|
||||
rw [ofStateAlgebra_ofState, wick_term_cons_eq_sum_wick_term (𝓞 := 𝓞)
|
||||
(maxTimeField φ φs) (eraseMaxTimeField φ φs) (maxTimeFieldPosFin φ φs) c]
|
||||
trans (1 : ℂ) • ∑ k : Option { x // x ∈ c.uncontracted }, sign
|
||||
(List.insertIdx (↑(maxTimeFieldPosFin φ φs)) (maxTimeField φ φs) (eraseMaxTimeField φ φs))
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue