426 lines
17 KiB
Text
426 lines
17 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.CrAnAlgebra.SuperCommute
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import Mathlib.Algebra.RingQuot
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import Mathlib.RingTheory.TwoSidedIdeal.Operations
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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 HepLean.List
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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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(∃ (φ1 φ2 φ3 : 𝓕.CrAnStates),
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x = [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca)
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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_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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lemma ι_superCommute_zero_of_fermionic (φ ψ : 𝓕.CrAnStates)
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(h : [ofCrAnState φ, ofCrAnState ψ]ₛca ∈ statisticSubmodule fermionic) :
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ι [ofCrAnState φ, ofCrAnState ψ]ₛca = 0 := by
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton] at h ⊢
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rcases statistic_neq_of_superCommute_fermionic h with h | h
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· simp only [ofCrAnList_singleton]
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apply ι_superCommute_of_diff_statistic
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simpa using h
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· simp [h]
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lemma ι_superCommute_ofCrAnState_ofCrAnState_bosonic_or_zero (φ ψ : 𝓕.CrAnStates) :
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[ofCrAnState φ, ofCrAnState ψ]ₛca ∈ statisticSubmodule bosonic ∨
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ι [ofCrAnState φ, ofCrAnState ψ]ₛca = 0 := by
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rcases superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ] [ψ] with h | h
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· simp_all [ofCrAnList_singleton]
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· simp_all only [ofCrAnList_singleton]
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right
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exact ι_superCommute_zero_of_fermionic _ _ h
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/-!
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## Super-commutes are in the center
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-/
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@[simp]
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lemma ι_superCommute_ofCrAnState_superCommute_ofCrAnState_ofCrAnState (φ1 φ2 φ3 : 𝓕.CrAnStates) :
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ι [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]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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left
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use φ1, φ2, φ3
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lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState (φ1 φ2 φ3 : 𝓕.CrAnStates) :
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ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, ofCrAnState φ3]ₛca = 0 := by
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rcases superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h | h
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· rw [bonsonic_superCommute_symm h]
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simp [ofCrAnList_singleton]
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· rcases ofCrAnList_bosonic_or_fermionic [φ3] with h' | h'
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· rw [superCommute_bonsonic_symm h']
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simp [ofCrAnList_singleton]
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· rw [superCommute_fermionic_fermionic_symm h h']
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simp [ofCrAnList_singleton]
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lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnList (φ1 φ2 : 𝓕.CrAnStates)
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(φs : List 𝓕.CrAnStates) :
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ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, ofCrAnList φs]ₛca = 0 := by
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rcases superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h | h
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· rw [superCommute_bosonic_ofCrAnList_eq_sum _ _ h]
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simp [ofCrAnList_singleton, ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState]
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· rw [superCommute_fermionic_ofCrAnList_eq_sum _ _ h]
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simp [ofCrAnList_singleton, ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState]
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@[simp]
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lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_crAnAlgebra (φ1 φ2 : 𝓕.CrAnStates)
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(a : 𝓕.CrAnAlgebra) : ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, a]ₛca = 0 := by
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change (ι.toLinearMap ∘ₗ superCommute [ofCrAnState φ1, ofCrAnState φ2]ₛca) a = _
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have h1 : (ι.toLinearMap ∘ₗ superCommute [ofCrAnState φ1, ofCrAnState φ2]ₛca) = 0 := by
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apply (ofCrAnListBasis.ext fun l ↦ ?_)
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simp [ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnList]
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rw [h1]
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simp
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lemma ι_commute_crAnAlgebra_superCommute_ofCrAnState_ofCrAnState (φ1 φ2 : 𝓕.CrAnStates)
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(a : 𝓕.CrAnAlgebra) : ι a * ι [ofCrAnState φ1, ofCrAnState φ2]ₛca -
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ι [ofCrAnState φ1, ofCrAnState φ2]ₛca * ι a = 0 := by
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rcases ι_superCommute_ofCrAnState_ofCrAnState_bosonic_or_zero φ1 φ2 with h | h
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swap
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· simp [h]
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trans - ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, a]ₛca
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· rw [bosonic_superCommute h]
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simp
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· simp
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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 a
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obtain ⟨a, rfl⟩ := ι_surjective a
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have h0 := ι_commute_crAnAlgebra_superCommute_ofCrAnState_ofCrAnState φ ψ a
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trans ι ((superCommute (ofCrAnState φ)) (ofCrAnState ψ)) * ι a + 0
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swap
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simp only [add_zero]
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rw [← h0]
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abel
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/-!
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## The kernal of ι
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-/
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lemma ι_eq_zero_iff_mem_ideal (x : CrAnAlgebra 𝓕) :
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ι x = 0 ↔ x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := 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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rw [TwoSidedIdeal.mem_iff]
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simp only
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rfl
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lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
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x.bosonicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.bosonicProj = 0 := by
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have hx' := hx
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simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
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rcases hx with ⟨φ1, φ2, φ3, rfl⟩ | ⟨φc, φc', hφc, hφc', rfl⟩ | ⟨φa, φa', hφa, hφa', rfl⟩ |
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⟨φ, φ', hdiff, rfl⟩
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· rcases superCommute_superCommute_ofCrAnState_bosonic_or_fermionic φ1 φ2 φ3 with h | h
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· left
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rw [bosonicProj_of_mem_bosonic _ h]
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simpa using hx'
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· right
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rw [bosonicProj_of_mem_fermionic _ h]
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φc φc' with h | h
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· left
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rw [bosonicProj_of_mem_bosonic _ h]
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simpa using hx'
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· right
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rw [bosonicProj_of_mem_fermionic _ h]
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φa φa' with h | h
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· left
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rw [bosonicProj_of_mem_bosonic _ h]
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simpa using hx'
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· right
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rw [bosonicProj_of_mem_fermionic _ h]
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ φ' with h | h
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· left
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rw [bosonicProj_of_mem_bosonic _ h]
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simpa using hx'
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· right
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rw [bosonicProj_of_mem_fermionic _ h]
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lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
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x.fermionicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.fermionicProj = 0 := by
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have hx' := hx
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simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
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rcases hx with ⟨φ1, φ2, φ3, rfl⟩ | ⟨φc, φc', hφc, hφc', rfl⟩ | ⟨φa, φa', hφa, hφa', rfl⟩ |
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⟨φ, φ', hdiff, rfl⟩
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· rcases superCommute_superCommute_ofCrAnState_bosonic_or_fermionic φ1 φ2 φ3 with h | h
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· right
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rw [fermionicProj_of_mem_bosonic _ h]
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· left
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rw [fermionicProj_of_mem_fermionic _ h]
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simpa using hx'
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φc φc' with h | h
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· right
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rw [fermionicProj_of_mem_bosonic _ h]
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· left
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rw [fermionicProj_of_mem_fermionic _ h]
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simpa using hx'
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φa φa' with h | h
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· right
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rw [fermionicProj_of_mem_bosonic _ h]
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· left
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rw [fermionicProj_of_mem_fermionic _ h]
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simpa using hx'
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· rcases superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic φ φ' with h | h
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· right
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rw [fermionicProj_of_mem_bosonic _ h]
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· left
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rw [fermionicProj_of_mem_fermionic _ h]
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simpa using hx'
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lemma bosonicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
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x.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at hx
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let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) : Prop :=
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a.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet
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change p x hx
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apply AddSubgroup.closure_induction
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· intro x hx
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simp only [p]
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obtain ⟨a, ha, b, hb, rfl⟩ := Set.mem_mul.mp hx
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obtain ⟨d, hd, y, hy, rfl⟩ := Set.mem_mul.mp ha
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rw [bosonicProj_mul, bosonicProj_mul, fermionicProj_mul]
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simp only [add_mul]
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rcases fermionicProj_mem_fieldOpIdealSet_or_zero y hy with hfy | hfy
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<;> rcases bosonicProj_mem_fieldOpIdealSet_or_zero y hy with hby | hby
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· apply TwoSidedIdeal.add_mem
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apply TwoSidedIdeal.add_mem
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· /- boson, boson, boson mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(bosonicProj d) * ↑(bosonicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use bosonicProj d
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simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
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use (bosonicProj y).1
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simp [hby]
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· use ↑(bosonicProj b)
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simp
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· /- fermion, fermion, boson mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(fermionicProj d) * ↑(fermionicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use fermionicProj d
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simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
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use (fermionicProj y).1
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simp [hby, hfy]
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· use ↑(bosonicProj b)
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simp
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apply TwoSidedIdeal.add_mem
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· /- boson, fermion, fermion mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(bosonicProj d) * ↑(fermionicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use bosonicProj d
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simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
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use (fermionicProj y).1
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simp [hby, hfy]
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· use ↑(fermionicProj b)
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simp
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· /- fermion, boson, fermion mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(fermionicProj d) * ↑(bosonicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use fermionicProj d
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simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
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use (bosonicProj y).1
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simp [hby, hfy]
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· use ↑(fermionicProj b)
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simp
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· simp only [hby, ZeroMemClass.coe_zero, mul_zero, zero_mul, zero_add, add_zero]
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apply TwoSidedIdeal.add_mem
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· /- fermion, fermion, boson mem-/
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rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
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refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
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apply Set.mem_mul.mpr
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use ↑(fermionicProj d) * ↑(fermionicProj y)
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apply And.intro
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· apply Set.mem_mul.mpr
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use fermionicProj d
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simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
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use (fermionicProj y).1
|
||
simp [hby, hfy]
|
||
· use ↑(bosonicProj b)
|
||
simp
|
||
· /- boson, fermion, fermion mem-/
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||
apply Set.mem_mul.mpr
|
||
use ↑(bosonicProj d) * ↑(fermionicProj y)
|
||
apply And.intro
|
||
· apply Set.mem_mul.mpr
|
||
use bosonicProj d
|
||
simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
|
||
use (fermionicProj y).1
|
||
simp [hby, hfy]
|
||
· use ↑(fermionicProj b)
|
||
simp
|
||
· simp only [hfy, ZeroMemClass.coe_zero, mul_zero, zero_mul, add_zero, zero_add]
|
||
apply TwoSidedIdeal.add_mem
|
||
· /- boson, boson, boson mem-/
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||
apply Set.mem_mul.mpr
|
||
use ↑(bosonicProj d) * ↑(bosonicProj y)
|
||
apply And.intro
|
||
· apply Set.mem_mul.mpr
|
||
use bosonicProj d
|
||
simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
|
||
use (bosonicProj y).1
|
||
simp [hby]
|
||
· use ↑(bosonicProj b)
|
||
simp
|
||
· /- fermion, boson, fermion mem-/
|
||
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure]
|
||
refine Set.mem_of_mem_of_subset ?_ AddSubgroup.subset_closure
|
||
apply Set.mem_mul.mpr
|
||
use ↑(fermionicProj d) * ↑(bosonicProj y)
|
||
apply And.intro
|
||
· apply Set.mem_mul.mpr
|
||
use fermionicProj d
|
||
simp only [Set.mem_univ, mul_eq_mul_left_iff, ZeroMemClass.coe_eq_zero, true_and]
|
||
use (bosonicProj y).1
|
||
simp [hby, hfy]
|
||
· use ↑(fermionicProj b)
|
||
simp
|
||
· simp [hfy, hby]
|
||
· simp [p]
|
||
· intro x y hx hy hpx hpy
|
||
simp_all only [map_add, Submodule.coe_add, p]
|
||
apply TwoSidedIdeal.add_mem
|
||
exact hpx
|
||
exact hpy
|
||
· intro x hx
|
||
simp [p]
|
||
|
||
lemma fermionicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
|
||
x.fermionicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
|
||
have hb := bosonicProj_mem_ideal x hx
|
||
rw [← ι_eq_zero_iff_mem_ideal] at hx hb ⊢
|
||
rw [← bosonicProj_add_fermionicProj x] at hx
|
||
simp only [map_add] at hx
|
||
simp_all
|
||
|
||
lemma ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero (x : CrAnAlgebra 𝓕) :
|
||
ι x = 0 ↔ ι x.bosonicProj.1 = 0 ∧ ι x.fermionicProj.1 = 0 := by
|
||
apply Iff.intro
|
||
· intro h
|
||
rw [@ι_eq_zero_iff_mem_ideal] at h ⊢
|
||
rw [ι_eq_zero_iff_mem_ideal]
|
||
apply And.intro
|
||
· exact bosonicProj_mem_ideal x h
|
||
· exact fermionicProj_mem_ideal x h
|
||
· intro h
|
||
rw [← bosonicProj_add_fermionicProj x]
|
||
simp_all
|
||
|
||
end FieldOpAlgebra
|
||
end FieldSpecification
|