114 lines
4.8 KiB
Text
114 lines
4.8 KiB
Text
/-
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Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.WickContraction.Sign
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import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.TimeContraction
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/-!
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# Time contractions
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-/
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open FieldSpecification
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variable {𝓕 : FieldSpecification}
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namespace WickContraction
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variable {n : ℕ} (c : WickContraction n)
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open HepLean.List
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open FieldOpAlgebra
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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 staticContract {φs : List 𝓕.States}
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(φsΛ : WickContraction φs.length) :
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Subalgebra.center ℂ 𝓕.FieldOpAlgebra :=
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∏ (a : φsΛ.1), ⟨[anPart (φs.get (φsΛ.fstFieldOfContract a)),
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ofFieldOp (φs.get (φsΛ.sndFieldOfContract a))]ₛ,
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superCommute_anPart_ofFieldOp_mem_center _ _⟩
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@[simp]
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lemma staticContract_insertAndContract_none (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).staticContract = φsΛ.staticContract := by
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rw [staticContract, insertAndContract_none_prod_contractions]
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congr
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ext a
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simp
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/-- For `φsΛ` a Wick contraction for `φs = φ₀…φₙ`, the time contraction
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`(φsΛ ↩Λ φ i (some j)).timeContract 𝓞` is equal to the multiple of
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- the time contraction of `φ` with `φⱼ` if `i < i.succAbove j` else
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`φⱼ` with `φ`.
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- `φsΛ.timeContract 𝓞`.
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This follows from the fact that `(φsΛ ↩Λ φ i (some j))` has one more contracted pair than `φsΛ`,
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corresponding to `φ` contracted with `φⱼ`. The order depends on whether we insert `φ` before
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or after `φⱼ`. -/
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lemma staticContract_insertAndContract_some
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(φ : 𝓕.States) (φs : List 𝓕.States)
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
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(φsΛ ↩Λ φ i (some j)).staticContract =
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(if i < i.succAbove j then
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⟨[anPart φ, ofFieldOp φs[j.1]]ₛ, superCommute_anPart_ofFieldOp_mem_center _ _⟩
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else ⟨[anPart φs[j.1], ofFieldOp φ]ₛ, superCommute_anPart_ofFieldOp_mem_center _ _⟩) *
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φsΛ.staticContract := by
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rw [staticContract, insertAndContract_some_prod_contractions]
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congr 1
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· simp only [Nat.succ_eq_add_one, insertAndContract_fstFieldOfContract_some_incl, finCongr_apply,
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List.get_eq_getElem, insertAndContract_sndFieldOfContract_some_incl, Fin.getElem_fin]
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split
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· simp
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· simp
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· congr
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ext a
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simp
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open FieldStatistic
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lemma staticConract_insertAndContract_some_eq_mul_contractStateAtIndex_lt
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(φ : 𝓕.States) (φs : List 𝓕.States)
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (k : φsΛ.uncontracted)
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(hik : i < i.succAbove k) :
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(φsΛ ↩Λ φ i (some k)).staticContract =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (φsΛ.uncontracted.filter (fun x => x < k))⟩)
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• (contractStateAtIndex φ [φsΛ]ᵘᶜ ((uncontractedStatesEquiv φs φsΛ) (some k)) *
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φsΛ.staticContract) := by
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rw [staticContract_insertAndContract_some]
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simp only [Nat.succ_eq_add_one, Fin.getElem_fin, ite_mul, instCommGroup.eq_1,
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contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
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Equiv.coe_trans, Option.map_some', Function.comp_apply, finCongr_apply, Fin.coe_cast,
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List.getElem_map, uncontractedList_getElem_uncontractedIndexEquiv_symm, List.get_eq_getElem,
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Algebra.smul_mul_assoc, uncontractedListGet]
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· simp only [hik, ↓reduceIte, MulMemClass.coe_mul]
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trans (1 : ℂ) • ((superCommute (anPart φ)) (ofFieldOp φs[k.1]) * ↑φsΛ.staticContract)
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· simp
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simp only [smul_smul]
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congr 1
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have h1 : ofList 𝓕.statesStatistic (List.take (↑(φsΛ.uncontractedIndexEquiv.symm k))
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(List.map φs.get φsΛ.uncontractedList))
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= (𝓕 |>ₛ ⟨φs.get, (Finset.filter (fun x => x < k) φsΛ.uncontracted)⟩) := by
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simp only [ofFinset]
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congr
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rw [← List.map_take]
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congr
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rw [take_uncontractedIndexEquiv_symm]
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rw [filter_uncontractedList]
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rw [h1]
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simp only [exchangeSign_mul_self]
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lemma staticContract_of_not_gradingCompliant (φs : List 𝓕.States)
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(φsΛ : WickContraction φs.length) (h : ¬ GradingCompliant φs φsΛ) :
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φsΛ.staticContract = 0 := by
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rw [staticContract]
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simp only [GradingCompliant, Fin.getElem_fin, Subtype.forall, not_forall] at h
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obtain ⟨a, ha⟩ := h
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obtain ⟨ha, ha2⟩ := ha
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apply Finset.prod_eq_zero (i := ⟨a, ha⟩)
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simp only [Finset.univ_eq_attach, Finset.mem_attach]
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apply Subtype.eq
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simp only [List.get_eq_getElem, ZeroMemClass.coe_zero]
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rw [superCommute_anPart_ofState_diff_grade_zero]
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simp [ha2]
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end WickContraction
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