93 lines
3.7 KiB
Text
93 lines
3.7 KiB
Text
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/-
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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.Algebras.OperatorAlgebra.NormalOrder
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import HepLean.PerturbationTheory.Algebras.StateAlgebra.TimeOrder
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/-!
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# Time contractions
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We define the state algebra of a field structure to be the free algebra
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generated by the states.
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-/
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namespace FieldStruct
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variable {𝓕 : FieldStruct}
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open CrAnAlgebra
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noncomputable section
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namespace OperatorAlgebra
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variable (𝓞 : 𝓕.OperatorAlgebra)
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open FieldStatistic
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/-- The time contraction of two States as an element of `𝓞.A` defined
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as their time ordering in the state algebra minus their normal ordering in the
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creation and annihlation algebra, both mapped to `𝓞.A`.. -/
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def timeContract (φ ψ : 𝓕.States) : 𝓞.A :=
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𝓞.crAnF (ofStateAlgebra (StateAlgebra.timeOrder (StateAlgebra.ofState φ * StateAlgebra.ofState ψ))
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- normalOrder (ofState φ * ofState ψ))
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lemma timeContract_eq_smul (φ ψ : 𝓕.States) : 𝓞.timeContract φ ψ =
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𝓞.crAnF (ofStateAlgebra (StateAlgebra.timeOrder
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(StateAlgebra.ofState φ * StateAlgebra.ofState ψ))
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+ (-1 : ℂ) • normalOrder (ofState φ * ofState ψ)) := by rfl
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lemma timeContract_of_timeOrderRel (φ ψ : 𝓕.States) (h : timeOrderRel φ ψ) :
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𝓞.timeContract φ ψ = 𝓞.crAnF (⟨anPart (StateAlgebra.ofState φ), ofState ψ⟩ₛca) := by
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conv_rhs =>
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rw [ofState_eq_crPart_add_anPart]
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rw [map_add, map_add, crAnF_superCommute_anPart_anPart, superCommute_anPart_crPart]
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simp only [timeContract, instCommGroup.eq_1, Algebra.smul_mul_assoc, add_zero]
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rw [StateAlgebra.timeOrder_ofState_ofState_ordered h]
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rw [normalOrder_ofState_mul_ofState]
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rw [map_mul]
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simp only [ofStateAlgebra_ofState, instCommGroup.eq_1]
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rw [ofState_eq_crPart_add_anPart, ofState_eq_crPart_add_anPart]
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simp only [mul_add, add_mul]
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abel_nf
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lemma timeContract_of_not_timeOrderRel (φ ψ : 𝓕.States) (h : ¬ timeOrderRel φ ψ) :
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𝓞.timeContract φ ψ = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓞.timeContract ψ φ := by
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rw [timeContract_eq_smul]
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simp only [Int.reduceNeg, one_smul, map_add]
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rw [map_smul]
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rw [crAnF_normalOrder_ofState_ofState_swap]
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rw [StateAlgebra.timeOrder_ofState_ofState_not_ordered_eq_timeOrder h]
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rw [timeContract_eq_smul]
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simp only [FieldStatistic.instCommGroup.eq_1, map_smul, one_smul, map_add, smul_add]
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rw [smul_smul, smul_smul, mul_comm]
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lemma timeContract_mem_center (φ ψ : 𝓕.States) : 𝓞.timeContract φ ψ ∈ Subalgebra.center ℂ 𝓞.A := by
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by_cases h : timeOrderRel φ ψ
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· rw [timeContract_of_timeOrderRel _ _ _ h]
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exact 𝓞.crAnF_superCommute_anPart_ofState_mem_center _ _
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· rw [timeContract_of_not_timeOrderRel _ _ _ h]
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refine Subalgebra.smul_mem (Subalgebra.center ℂ 𝓞.A) ?_ 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ)
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rw [timeContract_of_timeOrderRel]
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exact 𝓞.crAnF_superCommute_anPart_ofState_mem_center _ _
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have h1 := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ
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simp_all
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lemma timeContract_zero_of_diff_grade (φ ψ : 𝓕.States) (h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) :
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𝓞.timeContract φ ψ = 0 := by
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by_cases h1 : timeOrderRel φ ψ
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· rw [timeContract_of_timeOrderRel _ _ _ h1]
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rw [crAnF_superCommute_anPart_ofState_diff_grade_zero]
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exact h
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· rw [timeContract_of_not_timeOrderRel _ _ _ h1]
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rw [timeContract_of_timeOrderRel _ _ _]
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rw [crAnF_superCommute_anPart_ofState_diff_grade_zero]
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simp only [instCommGroup.eq_1, smul_zero]
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exact h.symm
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have ht := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ
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simp_all
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end OperatorAlgebra
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end
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end FieldStruct
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