121 lines
4.3 KiB
Text
121 lines
4.3 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.WickContraction.TimeContract
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import HepLean.Meta.Remark.Basic
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import Mathlib.RingTheory.TwoSidedIdeal.Operations
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import Mathlib.Algebra.RingQuot
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/-!
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# Field operator algebra
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-/
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namespace FieldSpecification
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open CrAnAlgebra
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open ProtoOperatorAlgebra
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open HepLean.List
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open WickContraction
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open FieldStatistic
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variable (𝓕 : FieldSpecification)
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/-- The set contains the super-commutors equal to zero in the operator algebra.
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This contains e.g. the super-commutor of two creation operators. -/
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def fieldOpIdealSet : Set (CrAnAlgebra 𝓕) :=
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{ x |
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(∃ (φ ψ : 𝓕.CrAnStates) (a : CrAnAlgebra 𝓕),
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x = a * [ofCrAnState φ, ofCrAnState ψ]ₛca - [ofCrAnState φ, ofCrAnState ψ]ₛca * a)
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∨ (∃ (φc φc' : 𝓕.CrAnStates) (_ : 𝓕 |>ᶜ φc = .create) (_ : 𝓕 |>ᶜ φc' = .create),
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x = [ofCrAnState φc, ofCrAnState φc']ₛca)
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∨ (∃ (φa φa' : 𝓕.CrAnStates) (_ : 𝓕 |>ᶜ φa = .annihilate) (_ : 𝓕 |>ᶜ φa' = .annihilate),
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x = [ofCrAnState φa, ofCrAnState φa']ₛca)
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∨ (∃ (φ φ' : 𝓕.CrAnStates) (_ : ¬ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φ')),
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x = [ofCrAnState φ, ofCrAnState φ']ₛca)}
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/-- The algebra spanned by cr and an parts of fields, with appropriate super-commutors
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set to zero. -/
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abbrev FieldOpAlgebra : Type := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.Quotient
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namespace FieldOpAlgebra
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variable {𝓕 : FieldSpecification}
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/-- The instance of a setoid on `CrAnAlgebra` from the ideal `TwoSidedIdeal`. -/
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instance : Setoid (CrAnAlgebra 𝓕) := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.toSetoid
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lemma equiv_iff_sub_mem_ideal (x y : CrAnAlgebra 𝓕) :
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x ≈ y ↔ x - y ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
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rw [← TwoSidedIdeal.rel_iff]
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rfl
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/-- The projection of `CrAnAlgebra` down to `FieldOpAlgebra` as an algebra map. -/
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def ι : CrAnAlgebra 𝓕 →ₐ[ℂ] FieldOpAlgebra 𝓕 where
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toFun := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.mk'
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map_one' := by rfl
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map_mul' x y := by rfl
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map_zero' := by rfl
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map_add' x y := by rfl
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commutes' x := by rfl
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lemma ι_surjective : Function.Surjective (@ι 𝓕) := by
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intro x
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obtain ⟨x⟩ := x
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use x
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rfl
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lemma ι_apply (x : CrAnAlgebra 𝓕) : ι x = Quotient.mk _ x := rfl
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lemma ι_of_mem_fieldOpIdealSet (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
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ι x = 0 := by
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rw [ι_apply]
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change ⟦x⟧ = ⟦0⟧
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simp only [ringConGen, Quotient.eq]
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refine RingConGen.Rel.of x 0 ?_
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simpa using hx
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lemma ι_superCommute_ofCrAnState_ofCrAnState_mem_center (φ ψ : 𝓕.CrAnStates) :
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ι [ofCrAnState φ, ofCrAnState ψ]ₛca ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
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rw [Subalgebra.mem_center_iff]
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intro b
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obtain ⟨b, rfl⟩ := ι_surjective b
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rw [← map_mul, ← map_mul]
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rw [LinearMap.sub_mem_ker_iff.mp]
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simp only [LinearMap.mem_ker]
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apply ι_of_mem_fieldOpIdealSet
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simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq]
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left
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use φ, ψ, b
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lemma ι_superCommute_of_create_create (φc φc' : 𝓕.CrAnStates) (hφc : 𝓕 |>ᶜ φc = .create)
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(hφc' : 𝓕 |>ᶜ φc' = .create) : ι [ofCrAnState φc, ofCrAnState φc']ₛca = 0 := by
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apply ι_of_mem_fieldOpIdealSet
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simp only [fieldOpIdealSet, exists_and_left, Set.mem_setOf_eq]
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simp only [exists_prop]
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right
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left
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use φc, φc', hφc, hφc'
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lemma ι_superCommute_of_annihilate_annihilate (φa φa' : 𝓕.CrAnStates)
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(hφa : 𝓕 |>ᶜ φa = .annihilate) (hφa' : 𝓕 |>ᶜ φa' = .annihilate) :
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ι [ofCrAnState φa, ofCrAnState φa']ₛca = 0 := by
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apply ι_of_mem_fieldOpIdealSet
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simp only [fieldOpIdealSet, exists_and_left, Set.mem_setOf_eq]
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simp only [exists_prop]
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right
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right
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left
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use φa, φa', hφa, hφa'
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lemma ι_superCommute_of_diff_statistic (φ ψ : 𝓕.CrAnStates)
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(h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : ι [ofCrAnState φ, ofCrAnState ψ]ₛca = 0 := by
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apply ι_of_mem_fieldOpIdealSet
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simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq]
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right
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right
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right
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use φ, ψ
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end FieldOpAlgebra
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end FieldSpecification
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