feat: Wick contraction docs
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jstoobysmith
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6 changed files with 154 additions and 65 deletions
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@ -356,6 +356,11 @@ noncomputable def contractStateAtIndex (φ : 𝓕.States) (φs : List 𝓕.State
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| some n => 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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/--
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Within a proto-operator algebra,
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`φ * N(φ₀φ₁…φₙ) = N(φφ₀φ₁…φₙ) + ∑ i, (sᵢ • [anPart 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 crAnF_ofState_mul_normalOrder_ofStatesList_eq_sum (φ : 𝓕.States)
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(φs : List 𝓕.States) :
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𝓞.crAnF (ofState φ * normalOrder (ofStateList φs)) =
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@ -374,6 +379,10 @@ lemma crAnF_ofState_mul_normalOrder_ofStatesList_eq_sum (φ : 𝓕.States)
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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 crAnF_ofState_normalOrder_insert (φ : 𝓕.States) (φs : List 𝓕.States)
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(k : Fin φs.length.succ) :
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𝓞.crAnF (normalOrder (ofStateList (φ :: φs))) =
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@ -18,6 +18,7 @@ noncomputable section
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namespace StateAlgebra
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open FieldStatistic
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open HepLean.List
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/-- The linear map on the free state algebra defined as the map taking
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a list of states to the time-ordered list of states multiplied by
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@ -64,6 +65,8 @@ lemma timeOrder_ofState_ofState_not_ordered_eq_timeOrder {φ ψ : 𝓕.States} (
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have hx := IsTotal.total (r := timeOrderRel) ψ φ
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simp_all
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/-- In the state algebra time, ordering obeys `T(φ₀φ₁…φₙ) = s * φᵢ * T(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)` where `φᵢ` is the state
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which has maximum time and `s` is the exchange sign of `φᵢ` and `φ₀φ₁…φᵢ₋₁`. -/
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lemma timeOrder_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States) :
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timeOrder (ofList (φ :: φs)) =
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𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ (φ :: φs).take (maxTimeFieldPos φ φs)) •
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@ -75,6 +78,56 @@ lemma timeOrder_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States)
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rw [timerOrderSign_of_eraseMaxTimeField, mul_assoc]
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simp
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/-- In the state algebra time, ordering obeys `T(φ₀φ₁…φₙ) = s * φᵢ * T(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)` where `φᵢ` is the state
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which has maximum time and `s` is the exchange sign of `φᵢ` and `φ₀φ₁…φᵢ₋₁`.
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Here `s` is written using finite sets. -/
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lemma timeOrder_eq_maxTimeField_mul_finset (φ : 𝓕.States) (φs : List 𝓕.States) :
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timeOrder (ofList (φ :: φs)) = 𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ ⟨(eraseMaxTimeField φ φs).get,
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(Finset.filter (fun x =>
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(maxTimeFieldPosFin φ φs).succAbove x < maxTimeFieldPosFin φ φs) Finset.univ)⟩) •
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StateAlgebra.ofState (maxTimeField φ φs) * timeOrder (ofList (eraseMaxTimeField φ φs)) := by
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rw [timeOrder_eq_maxTimeField_mul]
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congr 3
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apply FieldStatistic.ofList_perm
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nth_rewrite 1 [← List.finRange_map_get (φ :: φs)]
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simp only [List.length_cons, eraseMaxTimeField, insertionSortDropMinPos]
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rw [eraseIdx_get, ← List.map_take, ← List.map_map]
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refine List.Perm.map (φ :: φs).get ?_
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apply (List.perm_ext_iff_of_nodup _ _).mpr
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· intro i
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simp only [List.length_cons, maxTimeFieldPos, mem_take_finrange, Fin.val_fin_lt, List.mem_map,
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Finset.mem_sort, Finset.mem_filter, Finset.mem_univ, true_and, Function.comp_apply]
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refine Iff.intro (fun hi => ?_) (fun h => ?_)
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· have h2 := (maxTimeFieldPosFin φ φs).2
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simp only [eraseMaxTimeField, insertionSortDropMinPos, List.length_cons, Nat.succ_eq_add_one,
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maxTimeFieldPosFin, insertionSortMinPosFin] at h2
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use ⟨i, by omega⟩
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apply And.intro
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· simp only [Fin.succAbove, List.length_cons, Fin.castSucc_mk, maxTimeFieldPosFin,
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insertionSortMinPosFin, Nat.succ_eq_add_one, Fin.mk_lt_mk, Fin.val_fin_lt, Fin.succ_mk]
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rw [Fin.lt_def]
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split
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· simp only [Fin.val_fin_lt]
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omega
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· omega
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· simp only [Fin.succAbove, List.length_cons, Fin.castSucc_mk, Fin.succ_mk, Fin.ext_iff,
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Fin.coe_cast]
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split
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· simp
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· simp_all [Fin.lt_def]
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· obtain ⟨j, h1, h2⟩ := h
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subst h2
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simp only [Fin.lt_def, Fin.coe_cast]
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exact h1
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· exact List.Sublist.nodup (List.take_sublist _ _) <|
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List.nodup_finRange (φs.length + 1)
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· refine List.Nodup.map ?_ ?_
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· refine Function.Injective.comp ?hf.hg Fin.succAbove_right_injective
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exact Fin.cast_injective (eraseIdx_length (φ :: φs) (insertionSortMinPos timeOrderRel φ φs))
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· exact Finset.sort_nodup (fun x1 x2 => x1 ≤ x2)
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(Finset.filter (fun x => (maxTimeFieldPosFin φ φs).succAbove x < maxTimeFieldPosFin φ φs)
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Finset.univ)
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end StateAlgebra
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end
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end FieldSpecification
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@ -13,18 +13,23 @@ import HepLean.PerturbationTheory.FieldSpecification.Basic
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namespace FieldSpecification
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variable (𝓕 : FieldSpecification)
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/-- The Field specification corresponding to a single bosonic field. -/
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/-- The Field specification corresponding to a single bosonic field.
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The type of fields is the unit type, and the statistic of the unique element of the unit
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type is bosonic. -/
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def singleBoson : FieldSpecification where
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Fields := Unit
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statistics := fun _ => FieldStatistic.bosonic
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/-- The Field specification corresponding to a single fermionic field. -/
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/-- The Field specification corresponding to a single fermionic field.
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The type of fields is the unit type, and the statistic of the unique element of the unit
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type is fermionic. -/
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def singleFermion : FieldSpecification where
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Fields := Unit
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statistics := fun _ => FieldStatistic.fermionic
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/-- The Field specification corresponding to a two bosonic field and
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two fermionic fields. -/
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/-- The Field specification corresponding to two bosonic fields and two fermionic fields.
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The type of fields is `Fin 2 ⊕ Fin 2`, and the statistic of the two `.inl` (left) elements
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is bosonic and the statistic of the two `.inr` (right) elements is fermionic. -/
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def doubleBosonDoubleFermion : FieldSpecification where
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Fields := Fin 2 ⊕ Fin 2
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statistics := fun b =>
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@ -25,6 +25,11 @@ open HepLean.List
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open WickContraction
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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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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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@ -91,6 +96,12 @@ lemma insertList_none_normalOrder (φ : 𝓕.States) (φs : List 𝓕.States)
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simp only [Nat.succ_eq_add_one]
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rw [insertList_uncontractedList_none_map]
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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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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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@ -107,6 +118,12 @@ lemma insertList_some_normalOrder (φ : 𝓕.States) (φs : List 𝓕.States)
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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.insertList φ φs 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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@ -150,6 +167,13 @@ lemma sign_timeContract_normalOrder_insertList_none (φ : 𝓕.States) (φs : Li
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simp only [ZeroMemClass.coe_zero, zero_mul, smul_zero]
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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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for `c` equal to `c.insertList φ φs 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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@ -214,6 +238,15 @@ lemma sign_timeContract_normalOrder_insertList_some (φ : 𝓕.States) (φs : Li
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zero_mul, instCommGroup.eq_1]
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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.insertList φ φs 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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@ -226,8 +259,7 @@ lemma mul_sum_contractions (φ : 𝓕.States) (φs : List 𝓕.States) (i : Fin
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(ofStateList ((c.insertList φ φs 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 [Finset.univ_eq_attach, instCommGroup.eq_1,
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Nat.succ_eq_add_one, timeContract_insertList_none]
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simp only [instCommGroup.eq_1, Nat.succ_eq_add_one]
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congr 1
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funext n
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match n with
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@ -261,54 +293,6 @@ lemma wicks_theorem_congr {φs φs' : List 𝓕.States} (h : φs = φs') :
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subst h
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simp
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lemma timeOrder_eq_maxTimeField_mul_finset (φ : 𝓕.States) (φs : List 𝓕.States) :
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timeOrder (ofList (φ :: φs)) = 𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ ⟨(eraseMaxTimeField φ φs).get,
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(Finset.filter (fun x =>
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(maxTimeFieldPosFin φ φs).succAbove x < maxTimeFieldPosFin φ φs) Finset.univ)⟩) •
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StateAlgebra.ofState (maxTimeField φ φs) * timeOrder (ofList (eraseMaxTimeField φ φs)) := by
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rw [timeOrder_eq_maxTimeField_mul]
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congr 3
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apply FieldStatistic.ofList_perm
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nth_rewrite 1 [← List.finRange_map_get (φ :: φs)]
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simp only [List.length_cons, eraseMaxTimeField, insertionSortDropMinPos]
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rw [eraseIdx_get, ← List.map_take, ← List.map_map]
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refine List.Perm.map (φ :: φs).get ?_
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apply (List.perm_ext_iff_of_nodup _ _).mpr
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· intro i
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simp only [List.length_cons, maxTimeFieldPos, mem_take_finrange, Fin.val_fin_lt, List.mem_map,
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Finset.mem_sort, Finset.mem_filter, Finset.mem_univ, true_and, Function.comp_apply]
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refine Iff.intro (fun hi => ?_) (fun h => ?_)
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· have h2 := (maxTimeFieldPosFin φ φs).2
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simp only [eraseMaxTimeField, insertionSortDropMinPos, List.length_cons, Nat.succ_eq_add_one,
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maxTimeFieldPosFin, insertionSortMinPosFin] at h2
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use ⟨i, by omega⟩
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apply And.intro
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· simp only [Fin.succAbove, List.length_cons, Fin.castSucc_mk, maxTimeFieldPosFin,
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insertionSortMinPosFin, Nat.succ_eq_add_one, Fin.mk_lt_mk, Fin.val_fin_lt, Fin.succ_mk]
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rw [Fin.lt_def]
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split
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· simp only [Fin.val_fin_lt]
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omega
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· omega
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· simp only [Fin.succAbove, List.length_cons, Fin.castSucc_mk, Fin.succ_mk, Fin.ext_iff,
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Fin.coe_cast]
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split
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· simp
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· simp_all [Fin.lt_def]
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· obtain ⟨j, h1, h2⟩ := h
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subst h2
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simp only [Fin.lt_def, Fin.coe_cast]
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exact h1
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· exact List.Sublist.nodup (List.take_sublist _ _) <|
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List.nodup_finRange (φs.length + 1)
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· refine List.Nodup.map ?_ ?_
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· refine Function.Injective.comp ?hf.hg Fin.succAbove_right_injective
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exact Fin.cast_injective (eraseIdx_length (φ :: φs) (insertionSortMinPos timeOrderRel φ φs))
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· exact Finset.sort_nodup (fun x1 x2 => x1 ≤ x2)
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(Finset.filter (fun x => (maxTimeFieldPosFin φ φs).succAbove x < maxTimeFieldPosFin φ φs)
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Finset.univ)
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/-!
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## Wick's theorem
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@ -335,7 +319,15 @@ remark wicks_theorem_context := "
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the remaining fields. Wick's theorem is also the precursor to the diagrammatic
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approach to quantum field theory called Feynman diagrams."
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/-- Wick's theorem for time-ordered products of bosonic and fermionic fields. -/
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/-- Wick's theorem for time-ordered products of bosonic and fermionic fields.
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The time ordered product `T(φ₀φ₁…φₙ)` is equal to the sum of terms,
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for all possible Wick contractions `c` of the list of fields `φs := φ₀φ₁…φₙ`, given by
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the multiple of:
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- The sign corresponding to the number of fermionic-fermionic exchanges one must do
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to put elements in contracted pairs of `c` next to each other.
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- The product of time-contractions of the contracted pairs of `c`.
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- The normal-ordering of the uncontracted fields in `c`.
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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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