112 lines
4.6 KiB
Text
112 lines
4.6 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.StaticContract
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import HepLean.PerturbationTheory.WicksTheorem
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import HepLean.Meta.Remark.Basic
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/-!
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# Static Wick's theorem
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-/
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namespace FieldSpecification
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variable {𝓕 : FieldSpecification}
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open FieldOpFreeAlgebra
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open HepLean.List
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open WickContraction
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open FieldStatistic
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namespace FieldOpAlgebra
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lemma static_wick_theorem_nil : ofFieldOpList [] = ∑ (φsΛ : WickContraction [].length),
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φsΛ.sign (𝓕 := 𝓕) • φsΛ.staticContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ) := by
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simp only [ofFieldOpList, ofStateList_nil, map_one, List.length_nil]
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rw [sum_WickContraction_nil, uncontractedListGet, nil_zero_uncontractedList]
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simp [sign, empty, staticContract]
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/--
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The static Wicks theorem states that
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`φ₀…φₙ` is equal to the sum of
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`φsΛ.1.sign • φsΛ.1.staticContract * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)`
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over all Wick contraction `φsΛ`.
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This is compared to the ordinary Wicks theorem in which `staticContract` is replaced with
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`timeContract`.
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-/
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theorem static_wick_theorem : (φs : List 𝓕.States) →
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ofFieldOpList φs = ∑ (φsΛ : WickContraction φs.length),
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φsΛ.sign • φsΛ.staticContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
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| [] => static_wick_theorem_nil
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| φ :: φs => by
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rw [ofFieldOpList_cons, static_wick_theorem φs]
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rw [show (φ :: φs) = φs.insertIdx (⟨0, Nat.zero_lt_succ φs.length⟩ : Fin φs.length.succ) φ
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from rfl]
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conv_rhs => rw [insertLift_sum]
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rw [Finset.mul_sum]
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apply Finset.sum_congr rfl
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intro c _
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trans (sign φs c • ↑c.staticContract * (ofFieldOp φ * normalOrder (ofFieldOpList [c]ᵘᶜ)))
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· have ht := Subalgebra.mem_center_iff.mp (Subalgebra.smul_mem (Subalgebra.center ℂ _)
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(c.staticContract).2 c.sign)
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conv_rhs => rw [← mul_assoc, ← ht]
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simp [mul_assoc]
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rw [ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum]
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rw [Finset.mul_sum]
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rw [uncontractedStatesEquiv_list_sum]
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refine Finset.sum_congr rfl (fun n _ => ?_)
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match n with
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| none =>
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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, Nat.succ_eq_add_one,
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Fin.zero_eta, Fin.val_zero, List.insertIdx_zero, staticContract_insertAndContract_none,
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insertAndContract_uncontractedList_none_zero]
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erw [sign_insert_none_zero]
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rfl
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| some n =>
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simp only [Algebra.smul_mul_assoc, Nat.succ_eq_add_one, Fin.zero_eta, Fin.val_zero,
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List.insertIdx_zero]
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rw [normalOrder_uncontracted_some]
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simp only [← mul_assoc]
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rw [← smul_mul_assoc]
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conv_rhs => rw [← smul_mul_assoc]
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congr 1
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rw [staticConract_insertAndContract_some_eq_mul_contractStateAtIndex_lt]
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swap
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· simp
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rw [smul_smul]
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by_cases hn : GradingCompliant φs c ∧ (𝓕|>ₛφ) = (𝓕|>ₛ φs[n.1])
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· congr 1
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swap
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· have h1 := c.staticContract.2
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rw [@Subalgebra.mem_center_iff] at h1
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rw [h1]
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erw [sign_insert_some]
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rw [mul_assoc, mul_comm c.sign, ← mul_assoc]
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rw [signInsertSome_mul_filter_contracted_of_not_lt]
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simp only [instCommGroup.eq_1, Fin.zero_succAbove, Fin.not_lt_zero, Finset.filter_False,
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ofFinset_empty, map_one, one_mul]
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simp only [Fin.zero_succAbove, Fin.not_lt_zero, not_false_eq_true]
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exact hn
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· simp only [Fin.getElem_fin, not_and] at hn
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by_cases h0 : ¬ GradingCompliant φs c
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· rw [staticContract_of_not_gradingCompliant]
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simp only [ZeroMemClass.coe_zero, zero_mul, smul_zero, instCommGroup.eq_1, mul_zero]
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exact h0
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· simp_all only [Finset.mem_univ, not_not, instCommGroup.eq_1, forall_const]
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have h1 : contractStateAtIndex φ [c]ᵘᶜ
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((uncontractedStatesEquiv φs c) (some n)) = 0 := by
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simp only [contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
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Equiv.coe_trans, Option.map_some', Function.comp_apply, finCongr_apply,
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instCommGroup.eq_1, Fin.coe_cast, Fin.getElem_fin, smul_eq_zero]
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right
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simp only [uncontractedListGet, List.getElem_map,
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uncontractedList_getElem_uncontractedIndexEquiv_symm, List.get_eq_getElem]
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rw [superCommute_anPart_ofState_diff_grade_zero]
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exact hn
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rw [h1]
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simp
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end FieldOpAlgebra
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end FieldSpecification
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