165 lines
6.7 KiB
Text
165 lines
6.7 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.ProtoOperatorAlgebra.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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/-!
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## Time contract.
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-/
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/-- Given a Wick contraction `c` 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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(c : WickContraction φs.length) :
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Subalgebra.center ℂ 𝓞.A :=
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∏ (a : c.1), ⟨𝓞.timeContract (φs.get (c.fstFieldOfContract a)) (φs.get (c.sndFieldOfContract a)),
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𝓞.timeContract_mem_center _ _⟩
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@[simp]
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lemma timeContract_insertList_none (𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) :
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(c.insertList φ φs i none).timeContract 𝓞 = c.timeContract 𝓞 := by
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rw [timeContract, insertList_none_prod_contractions]
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congr
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ext a
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simp
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lemma timeConract_insertList_some (𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
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(c.insertList φ φs i (some j)).timeContract 𝓞 =
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(if i < i.succAbove j then
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⟨𝓞.timeContract φ φs[j.1], 𝓞.timeContract_mem_center _ _⟩
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else ⟨𝓞.timeContract φs[j.1] φ, 𝓞.timeContract_mem_center _ _⟩) * c.timeContract 𝓞 := by
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rw [timeContract, insertList_some_prod_contractions]
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congr 1
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· simp only [Nat.succ_eq_add_one, insertList_fstFieldOfContract_some_incl, finCongr_apply,
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List.get_eq_getElem, insertList_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 timeConract_insertList_some_eq_mul_contractStateAtIndex_lt
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(𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (k : c.uncontracted)
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(ht : 𝓕.timeOrderRel φ φs[k.1]) (hik : i < i.succAbove k) :
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(c.insertList φ φs i (some k)).timeContract 𝓞 =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (c.uncontracted.filter (fun x => x < k))⟩)
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• (𝓞.contractStateAtIndex φ (List.map φs.get c.uncontractedList)
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((uncontractedStatesEquiv φs c) (some k)) * c.timeContract 𝓞) := by
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rw [timeConract_insertList_some]
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simp only [Nat.succ_eq_add_one, Fin.getElem_fin, ite_mul, instCommGroup.eq_1,
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ProtoOperatorAlgebra.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]
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· simp only [hik, ↓reduceIte, MulMemClass.coe_mul]
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rw [𝓞.timeContract_of_timeOrderRel]
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trans (1 : ℂ) • (𝓞.crAnF ((CrAnAlgebra.superCommute
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(CrAnAlgebra.anPart (StateAlgebra.ofState φ))) (CrAnAlgebra.ofState φs[k.1])) *
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↑(timeContract 𝓞 c))
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· simp
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simp only [smul_smul]
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congr
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have h1 : ofList 𝓕.statesStatistic (List.take (↑(c.uncontractedIndexEquiv.symm k))
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(List.map φs.get c.uncontractedList))
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= (𝓕 |>ₛ ⟨φs.get, (Finset.filter (fun x => x < k) c.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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· exact ht
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lemma timeConract_insertList_some_eq_mul_contractStateAtIndex_not_lt
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(𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (k : c.uncontracted)
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(ht : ¬ 𝓕.timeOrderRel φs[k.1] φ) (hik : ¬ i < i.succAbove k) :
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(c.insertList φ φs i (some k)).timeContract 𝓞 =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (c.uncontracted.filter (fun x => x ≤ k))⟩)
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• (𝓞.contractStateAtIndex φ (List.map φs.get c.uncontractedList)
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((uncontractedStatesEquiv φs c) (some k)) * c.timeContract 𝓞) := by
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rw [timeConract_insertList_some]
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simp only [Nat.succ_eq_add_one, Fin.getElem_fin, ite_mul, instCommGroup.eq_1,
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ProtoOperatorAlgebra.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]
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simp only [hik, ↓reduceIte, MulMemClass.coe_mul]
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rw [𝓞.timeContract_of_not_timeOrderRel, 𝓞.timeContract_of_timeOrderRel]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, smul_smul]
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congr
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have h1 : ofList 𝓕.statesStatistic (List.take (↑(c.uncontractedIndexEquiv.symm k))
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(List.map φs.get c.uncontractedList))
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= (𝓕 |>ₛ ⟨φs.get, (Finset.filter (fun x => x < k) c.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, filter_uncontractedList]
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rw [h1]
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trans 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, {k.1}⟩)
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· rw [exchangeSign_symm, ofFinset_singleton]
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simp
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rw [← map_mul]
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congr
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rw [ofFinset_union]
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congr
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ext a
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simp only [Finset.mem_singleton, Finset.mem_sdiff, Finset.mem_union, Finset.mem_filter,
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Finset.mem_inter, not_and, not_lt, and_imp]
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apply Iff.intro
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· intro h
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subst h
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simp
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· intro h
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have h1 := h.1
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rcases h1 with h1 | h1
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· have h2' := h.2 h1.1 h1.2 h1.1
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omega
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· have h2' := h.2 h1.1 (by omega) h1.1
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omega
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have ht := IsTotal.total (r := timeOrderRel) φs[k.1] φ
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simp_all only [Fin.getElem_fin, Nat.succ_eq_add_one, not_lt, false_or]
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exact ht
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lemma timeContract_of_not_gradingCompliant (𝓞 : 𝓕.ProtoOperatorAlgebra) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (h : ¬ GradingCompliant φs c) :
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c.timeContract 𝓞 = 0 := by
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rw [timeContract]
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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 [ProtoOperatorAlgebra.timeContract_zero_of_diff_grade]
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simp [ha2]
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end WickContraction
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