175 lines
6.7 KiB
Text
175 lines
6.7 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.CrAnAlgebra.TimeOrder
|
||
import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.Basic
|
||
/-!
|
||
|
||
# Time Ordering on Field operator algebra
|
||
|
||
-/
|
||
|
||
namespace FieldSpecification
|
||
open CrAnAlgebra
|
||
open HepLean.List
|
||
open FieldStatistic
|
||
|
||
namespace FieldOpAlgebra
|
||
variable {𝓕 : FieldSpecification}
|
||
|
||
lemma ι_timeOrder_superCommute_time {φ ψ : 𝓕.CrAnStates}
|
||
(hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.CrAnAlgebra) :
|
||
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
|
||
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b)) := by
|
||
let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
|
||
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b))
|
||
change pb b (Basis.mem_span _ b)
|
||
apply Submodule.span_induction
|
||
· intro x hx
|
||
obtain ⟨φs, rfl⟩ := hx
|
||
simp [pb]
|
||
let pa (a : 𝓕.CrAnAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * ofCrAnList φs) =
|
||
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a* ofCrAnList φs))
|
||
change pa a (Basis.mem_span _ a)
|
||
apply Submodule.span_induction
|
||
· intro x hx
|
||
obtain ⟨φs', rfl⟩ := hx
|
||
simp [pa]
|
||
conv_lhs =>
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||
simp [mul_sub, sub_mul, ← ofCrAnList_append]
|
||
rw [timeOrder_ofCrAnList, timeOrder_ofCrAnList]
|
||
have h1 : crAnTimeOrderSign (φs' ++ φ :: ψ :: φs) = crAnTimeOrderSign (φs' ++ ψ :: φ :: φs) := by
|
||
trans crAnTimeOrderSign (φs' ++ [φ, ψ] ++ φs)
|
||
simp
|
||
rw [crAnTimeOrderSign]
|
||
have hp : List.Perm [φ,ψ] [ψ,φ] := by exact List.Perm.swap ψ φ []
|
||
rw [Wick.koszulSign_perm_eq _ _ φ _ _ _ _ _ hp]
|
||
simp
|
||
rfl
|
||
simp_all
|
||
rw [h1]
|
||
simp
|
||
have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [φ, ψ] φs
|
||
(by simp_all)
|
||
rw [crAnTimeOrderList, show φs' ++ φ :: ψ :: φs = φs' ++ [φ, ψ] ++ φs by simp, h1]
|
||
have h2 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [ψ, φ] φs
|
||
(by simp_all)
|
||
rw [crAnTimeOrderList, show φs' ++ ψ :: φ :: φs = φs' ++ [ψ, φ] ++ φs by simp, h2]
|
||
repeat rw [ofCrAnList_append]
|
||
rw [smul_smul, mul_comm, ← smul_smul, ← smul_sub]
|
||
rw [map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul]
|
||
rw [← mul_smul_comm]
|
||
rw [mul_assoc, mul_assoc, mul_assoc ,mul_assoc ,mul_assoc ,mul_assoc]
|
||
rw [← mul_sub, ← mul_sub, mul_smul_comm, mul_smul_comm, ← smul_mul_assoc,
|
||
← smul_mul_assoc]
|
||
rw [← sub_mul]
|
||
have h1 : (ι (ofCrAnList [φ, ψ]) - (exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnList [ψ, φ])) =
|
||
ι [ofCrAnState φ, ofCrAnState ψ]ₛca := by
|
||
rw [superCommute_ofCrAnState_ofCrAnState]
|
||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_append]
|
||
simp only [instCommGroup.eq_1, List.singleton_append, Algebra.smul_mul_assoc, map_sub,
|
||
map_smul]
|
||
rw [← ofCrAnList_append]
|
||
simp
|
||
rw [h1]
|
||
have hc : ι ((superCommute (ofCrAnState φ)) (ofCrAnState ψ)) ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
|
||
apply ι_superCommute_ofCrAnState_ofCrAnState_mem_center
|
||
rw [Subalgebra.mem_center_iff] at hc
|
||
repeat rw [← mul_assoc]
|
||
rw [hc]
|
||
repeat rw [mul_assoc]
|
||
rw [smul_mul_assoc]
|
||
rw [← map_mul, ← map_mul, ← map_mul, ← map_mul]
|
||
rw [← ofCrAnList_append, ← ofCrAnList_append, ← ofCrAnList_append, ← ofCrAnList_append]
|
||
have h1 := insertionSort_of_takeWhile_filter 𝓕.crAnTimeOrderRel φ φs' φs
|
||
simp at h1 ⊢
|
||
rw [← h1]
|
||
rw [← crAnTimeOrderList]
|
||
by_cases hq : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)
|
||
· rw [ι_superCommute_of_diff_statistic hq]
|
||
simp
|
||
· rw [crAnTimeOrderSign, Wick.koszulSign_eq_rel_eq_stat _ _, ← crAnTimeOrderSign]
|
||
rw [timeOrder_ofCrAnList]
|
||
simp
|
||
exact hψφ
|
||
exact hφψ
|
||
simpa using hq
|
||
· simp [pa]
|
||
· intro x y hx hy hpx hpy
|
||
simp_all [pa,mul_add, add_mul]
|
||
· intro x hx hpx
|
||
simp_all [pa, hpx]
|
||
· simp [pb]
|
||
· intro x y hx hy hpx hpy
|
||
simp_all [pb,mul_add, add_mul]
|
||
· intro x hx hpx
|
||
simp_all [pb, hpx]
|
||
|
||
/-!
|
||
|
||
## Defining normal order for `FiedOpAlgebra`.
|
||
|
||
-/
|
||
|
||
lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
|
||
(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓣ᶠ(a) = 0 := by
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at h
|
||
let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓣ᶠ(a) = 0
|
||
change p a h
|
||
apply AddSubgroup.closure_induction
|
||
· intro x hx
|
||
obtain ⟨a, ha, b, hb, rfl⟩ := Set.mem_mul.mp hx
|
||
obtain ⟨a, ha, c, hc, rfl⟩ := ha
|
||
simp only [p]
|
||
simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq] at hc
|
||
match hc with
|
||
| Or.inl hc =>
|
||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||
sorry
|
||
| Or.inr (Or.inl hc) =>
|
||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||
sorry
|
||
| Or.inr (Or.inr (Or.inl hc)) =>
|
||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||
sorry
|
||
| Or.inr (Or.inr (Or.inr hc)) =>
|
||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||
sorry
|
||
· simp [p]
|
||
· intro x y hx hy
|
||
simp only [map_add, p]
|
||
intro h1 h2
|
||
simp [h1, h2]
|
||
· intro x hx
|
||
simp [p]
|
||
|
||
lemma ι_timeOrder_eq_of_equiv (a b : 𝓕.CrAnAlgebra) (h : a ≈ b) :
|
||
ι 𝓣ᶠ(a) = ι 𝓣ᶠ(b) := by
|
||
rw [equiv_iff_sub_mem_ideal] at h
|
||
rw [LinearMap.sub_mem_ker_iff.mp]
|
||
simp only [LinearMap.mem_ker, ← map_sub]
|
||
exact ι_timeOrder_zero_of_mem_ideal (a - b) h
|
||
|
||
/-- Normal ordering on `FieldOpAlgebra`. -/
|
||
noncomputable def timeOrder : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
|
||
toFun := Quotient.lift (ι.toLinearMap ∘ₗ CrAnAlgebra.timeOrder) ι_timeOrder_eq_of_equiv
|
||
map_add' x y := by
|
||
obtain ⟨x, hx⟩ := ι_surjective x
|
||
obtain ⟨y, hy⟩ := ι_surjective y
|
||
subst hx hy
|
||
rw [← map_add, ι_apply, ι_apply, ι_apply]
|
||
rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
|
||
simp
|
||
map_smul' c y := by
|
||
obtain ⟨y, hy⟩ := ι_surjective y
|
||
subst hy
|
||
rw [← map_smul, ι_apply, ι_apply]
|
||
simp
|
||
|
||
end FieldOpAlgebra
|
||
end FieldSpecification
|