b5c987180a
* refactor: Fix field struct defn. * rename: FieldStruct to FieldSpecification * feat: Add examples of field specifications * docs: Slight improvement of module docs
80 lines
3.2 KiB
Text
80 lines
3.2 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.FieldSpecification.TimeOrder
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import HepLean.PerturbationTheory.Koszul.KoszulSign
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/-!
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# Time ordering on the state algebra
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-/
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namespace FieldSpecification
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variable {𝓕 : FieldSpecification}
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noncomputable section
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namespace StateAlgebra
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open FieldStatistic
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/-- The linear map on the free state algebra defined as the map taking
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a list of states to the time-ordered list of states multiplied by
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the sign corresponding to the number of fermionic-fermionic
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exchanges done in ordering. -/
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def timeOrder : StateAlgebra 𝓕 →ₗ[ℂ] StateAlgebra 𝓕 :=
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Basis.constr ofListBasis ℂ fun φs =>
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timeOrderSign φs • ofList (timeOrderList φs)
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lemma timeOrder_ofList (φs : List 𝓕.States) :
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timeOrder (ofList φs) = timeOrderSign φs • ofList (timeOrderList φs) := by
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rw [← ofListBasis_eq_ofList]
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simp only [timeOrder, Basis.constr_basis]
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lemma timeOrder_ofList_nil : timeOrder (𝓕 := 𝓕) (ofList []) = 1 := by
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rw [timeOrder_ofList]
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simp [timeOrderSign, Wick.koszulSign, timeOrderList]
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@[simp]
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lemma timeOrder_ofList_singleton (φ : 𝓕.States) : timeOrder (ofList [φ]) = ofList [φ] := by
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simp [timeOrder_ofList, timeOrderSign, timeOrderList]
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lemma timeOrder_ofState_ofState_ordered {φ ψ : 𝓕.States} (h : timeOrderRel φ ψ) :
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timeOrder (ofState φ * ofState ψ) = ofState φ * ofState ψ := by
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rw [← ofList_singleton, ← ofList_singleton, ← ofList_append, timeOrder_ofList]
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simp only [List.singleton_append]
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rw [timeOrderSign_pair_ordered h, timeOrderList_pair_ordered h]
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simp
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lemma timeOrder_ofState_ofState_not_ordered {φ ψ : 𝓕.States} (h :¬ timeOrderRel φ ψ) :
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timeOrder (ofState φ * ofState ψ) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • ofState ψ * ofState φ := by
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rw [← ofList_singleton, ← ofList_singleton, ← ofList_append, timeOrder_ofList]
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simp only [List.singleton_append, instCommGroup.eq_1, Algebra.smul_mul_assoc]
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rw [timeOrderSign_pair_not_ordered h, timeOrderList_pair_not_ordered h]
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simp [← ofList_append]
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lemma timeOrder_ofState_ofState_not_ordered_eq_timeOrder {φ ψ : 𝓕.States} (h :¬ timeOrderRel φ ψ) :
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timeOrder (ofState φ * ofState ψ) =
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𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • timeOrder (ofState ψ * ofState φ) := by
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rw [timeOrder_ofState_ofState_not_ordered h]
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rw [timeOrder_ofState_ofState_ordered]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc]
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have hx := IsTotal.total (r := timeOrderRel) ψ φ
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simp_all
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lemma timeOrder_eq_maxTimeField_mul (φ : 𝓕.States) (φs : List 𝓕.States) :
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timeOrder (ofList (φ :: φs)) =
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𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ (φ :: φs).take (maxTimeFieldPos φ φs)) •
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ofState (maxTimeField φ φs) * timeOrder (ofList (eraseMaxTimeField φ φs)) := by
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rw [timeOrder_ofList, timeOrderList_eq_maxTimeField_timeOrderList]
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rw [ofList_cons, timeOrder_ofList]
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simp only [instCommGroup.eq_1, Algebra.mul_smul_comm, Algebra.smul_mul_assoc, smul_smul]
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congr
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rw [timerOrderSign_of_eraseMaxTimeField, mul_assoc]
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simp
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end StateAlgebra
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end
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end FieldSpecification
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