92 lines
3.7 KiB
Text
92 lines
3.7 KiB
Text
/-
|
||
Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Authors: Joseph Tooby-Smith
|
||
-/
|
||
import HepLean.PerturbationTheory.Algebras.OperatorAlgebra.NormalOrder
|
||
import HepLean.PerturbationTheory.Algebras.StateAlgebra.TimeOrder
|
||
/-!
|
||
|
||
# Time contractions
|
||
|
||
We define the state algebra of a field structure to be the free algebra
|
||
generated by the states.
|
||
|
||
-/
|
||
|
||
namespace FieldStruct
|
||
variable {𝓕 : FieldStruct}
|
||
open CrAnAlgebra
|
||
noncomputable section
|
||
|
||
namespace OperatorAlgebra
|
||
|
||
variable (𝓞 : 𝓕.OperatorAlgebra)
|
||
open FieldStatistic
|
||
|
||
/-- The time contraction of two States as an element of `𝓞.A` defined
|
||
as their time ordering in the state algebra minus their normal ordering in the
|
||
creation and annihlation algebra, both mapped to `𝓞.A`.. -/
|
||
def timeContract (φ ψ : 𝓕.States) : 𝓞.A :=
|
||
𝓞.crAnF (ofStateAlgebra (StateAlgebra.timeOrder (StateAlgebra.ofState φ * StateAlgebra.ofState ψ))
|
||
- normalOrder (ofState φ * ofState ψ))
|
||
|
||
lemma timeContract_eq_smul (φ ψ : 𝓕.States) : 𝓞.timeContract φ ψ =
|
||
𝓞.crAnF (ofStateAlgebra (StateAlgebra.timeOrder
|
||
(StateAlgebra.ofState φ * StateAlgebra.ofState ψ))
|
||
+ (-1 : ℂ) • normalOrder (ofState φ * ofState ψ)) := by rfl
|
||
|
||
lemma timeContract_of_timeOrderRel (φ ψ : 𝓕.States) (h : timeOrderRel φ ψ) :
|
||
𝓞.timeContract φ ψ = 𝓞.crAnF (⟨anPart (StateAlgebra.ofState φ), ofState ψ⟩ₛca) := by
|
||
conv_rhs =>
|
||
rw [ofState_eq_crPart_add_anPart]
|
||
rw [map_add, map_add, crAnF_superCommute_anPart_anPart, superCommute_anPart_crPart]
|
||
simp only [timeContract, instCommGroup.eq_1, Algebra.smul_mul_assoc, add_zero]
|
||
rw [StateAlgebra.timeOrder_ofState_ofState_ordered h]
|
||
rw [normalOrder_ofState_mul_ofState]
|
||
rw [map_mul]
|
||
simp only [ofStateAlgebra_ofState, instCommGroup.eq_1]
|
||
rw [ofState_eq_crPart_add_anPart, ofState_eq_crPart_add_anPart]
|
||
simp only [mul_add, add_mul]
|
||
abel_nf
|
||
|
||
lemma timeContract_of_not_timeOrderRel (φ ψ : 𝓕.States) (h : ¬ timeOrderRel φ ψ) :
|
||
𝓞.timeContract φ ψ = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓞.timeContract ψ φ := by
|
||
rw [timeContract_eq_smul]
|
||
simp only [Int.reduceNeg, one_smul, map_add]
|
||
rw [map_smul]
|
||
rw [crAnF_normalOrder_ofState_ofState_swap]
|
||
rw [StateAlgebra.timeOrder_ofState_ofState_not_ordered_eq_timeOrder h]
|
||
rw [timeContract_eq_smul]
|
||
simp only [FieldStatistic.instCommGroup.eq_1, map_smul, one_smul, map_add, smul_add]
|
||
rw [smul_smul, smul_smul, mul_comm]
|
||
|
||
lemma timeContract_mem_center (φ ψ : 𝓕.States) : 𝓞.timeContract φ ψ ∈ Subalgebra.center ℂ 𝓞.A := by
|
||
by_cases h : timeOrderRel φ ψ
|
||
· rw [timeContract_of_timeOrderRel _ _ _ h]
|
||
exact 𝓞.crAnF_superCommute_anPart_ofState_mem_center _ _
|
||
· rw [timeContract_of_not_timeOrderRel _ _ _ h]
|
||
refine Subalgebra.smul_mem (Subalgebra.center ℂ 𝓞.A) ?_ 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ)
|
||
rw [timeContract_of_timeOrderRel]
|
||
exact 𝓞.crAnF_superCommute_anPart_ofState_mem_center _ _
|
||
have h1 := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ
|
||
simp_all
|
||
|
||
lemma timeContract_zero_of_diff_grade (φ ψ : 𝓕.States) (h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) :
|
||
𝓞.timeContract φ ψ = 0 := by
|
||
by_cases h1 : timeOrderRel φ ψ
|
||
· rw [timeContract_of_timeOrderRel _ _ _ h1]
|
||
rw [crAnF_superCommute_anPart_ofState_diff_grade_zero]
|
||
exact h
|
||
· rw [timeContract_of_not_timeOrderRel _ _ _ h1]
|
||
rw [timeContract_of_timeOrderRel _ _ _]
|
||
rw [crAnF_superCommute_anPart_ofState_diff_grade_zero]
|
||
simp only [instCommGroup.eq_1, smul_zero]
|
||
exact h.symm
|
||
have ht := IsTotal.total (r := 𝓕.timeOrderRel) φ ψ
|
||
simp_all
|
||
|
||
end OperatorAlgebra
|
||
|
||
end
|
||
end FieldStruct
|