87 lines
3.3 KiB
Text
87 lines
3.3 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.Algebras.FieldOpAlgebra.NormalOrder
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import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.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 FieldSpecification
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variable {𝓕 : FieldSpecification}
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open CrAnAlgebra
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noncomputable section
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namespace FieldOpAlgebra
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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) : 𝓕.FieldOpAlgebra :=
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𝓣(ofFieldOp φ * ofFieldOp ψ) - 𝓝(ofFieldOp φ * ofFieldOp ψ)
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lemma timeContract_eq_smul (φ ψ : 𝓕.States) : timeContract φ ψ =
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𝓣(ofFieldOp φ * ofFieldOp ψ) + (-1 : ℂ) • 𝓝(ofFieldOp φ * ofFieldOp ψ) := by rfl
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lemma timeContract_of_timeOrderRel (φ ψ : 𝓕.States) (h : timeOrderRel φ ψ) :
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timeContract φ ψ = [anPart φ, ofFieldOp ψ]ₛ := by
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conv_rhs =>
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rw [ofFieldOp_eq_crPart_add_anPart]
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rw [map_add, 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 [timeOrder_ofFieldOp_ofFieldOp_ordered h]
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rw [normalOrder_ofFieldOp_mul_ofFieldOp]
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simp only [instCommGroup.eq_1]
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rw [ofFieldOp_eq_crPart_add_anPart, ofFieldOp_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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timeContract φ ψ = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 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 [normalOrder_ofFieldOp_ofFieldOp_swap]
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rw [timeOrder_ofFieldOp_ofFieldOp_not_ordered_eq_timeOrder h]
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rw [timeContract_eq_smul]
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simp only [instCommGroup.eq_1, map_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) :
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timeContract φ ψ ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := 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 superCommute_anPart_ofFieldOp_mem_center φ ψ
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· rw [timeContract_of_not_timeOrderRel _ _ h]
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refine Subalgebra.smul_mem (Subalgebra.center ℂ _) ?_ 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ)
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rw [timeContract_of_timeOrderRel]
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exact superCommute_anPart_ofFieldOp_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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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 [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 [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 FieldOpAlgebra
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end
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end FieldSpecification
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