403 lines
14 KiB
Text
403 lines
14 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.Basic
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import HepLean.PerturbationTheory.Koszul.KoszulSign
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import Mathlib.RingTheory.GradedAlgebra.Basic
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/-!
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# Grading on the CrAnAlgebra
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-/
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namespace FieldSpecification
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variable {𝓕 : FieldSpecification}
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open FieldStatistic
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namespace CrAnAlgebra
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noncomputable section
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/-- The submodule of `CrAnAlgebra` spanned by lists of field statistic `f`. -/
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def statisticSubmodule (f : FieldStatistic) : Submodule ℂ 𝓕.CrAnAlgebra :=
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Submodule.span ℂ {a | ∃ φs, a = ofCrAnList φs ∧ (𝓕 |>ₛ φs) = f}
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lemma ofCrAnList_mem_statisticSubmodule_of (φs : List 𝓕.CrAnStates) (f : FieldStatistic)
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(h : (𝓕 |>ₛ φs) = f) :
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ofCrAnList φs ∈ statisticSubmodule f := by
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refine Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩
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lemma ofCrAnList_bosonic_or_fermionic (φs : List 𝓕.CrAnStates) :
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ofCrAnList φs ∈ statisticSubmodule bosonic ∨ ofCrAnList φs ∈ statisticSubmodule fermionic := by
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by_cases h : (𝓕 |>ₛ φs) = bosonic
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· left
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exact ofCrAnList_mem_statisticSubmodule_of φs bosonic h
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· right
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exact ofCrAnList_mem_statisticSubmodule_of φs fermionic (by simpa using h)
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lemma ofCrAnState_bosonic_or_fermionic (φ : 𝓕.CrAnStates) :
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ofCrAnState φ ∈ statisticSubmodule bosonic ∨ ofCrAnState φ ∈ statisticSubmodule fermionic := by
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rw [← ofCrAnList_singleton]
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exact ofCrAnList_bosonic_or_fermionic [φ]
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/-- The projection of an element of `CrAnAlgebra` onto it's bosonic part. -/
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def bosonicProj : 𝓕.CrAnAlgebra →ₗ[ℂ] statisticSubmodule (𝓕 := 𝓕) bosonic :=
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Basis.constr ofCrAnListBasis ℂ fun φs =>
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if h : (𝓕 |>ₛ φs) = bosonic then
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⟨ofCrAnList φs, Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩⟩
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else
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0
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lemma bosonicProj_ofCrAnList (φs : List 𝓕.CrAnStates) :
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bosonicProj (ofCrAnList φs) = if h : (𝓕 |>ₛ φs) = bosonic then
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⟨ofCrAnList φs, Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩⟩ else 0 := by
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conv_lhs =>
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rw [← ofListBasis_eq_ofList, bosonicProj, Basis.constr_basis]
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lemma bosonicProj_of_mem_bosonic (a : 𝓕.CrAnAlgebra) (h : a ∈ statisticSubmodule bosonic) :
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bosonicProj a = ⟨a, h⟩ := by
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let p (a : 𝓕.CrAnAlgebra) (hx : a ∈ statisticSubmodule bosonic) : Prop :=
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bosonicProj a = ⟨a, hx⟩
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change p a h
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apply Submodule.span_induction
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· intro x hx
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simp only [Set.mem_setOf_eq] at hx
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obtain ⟨φs, rfl, h⟩ := hx
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simp [p, bosonicProj_ofCrAnList, h]
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· simp only [map_zero, p]
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rfl
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· intro x y hx hy hpx hpy
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simp_all [p]
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· intro a x hx hy
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simp_all [p]
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lemma bosonicProj_of_mem_fermionic (a : 𝓕.CrAnAlgebra) (h : a ∈ statisticSubmodule fermionic) :
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bosonicProj a = 0 := by
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let p (a : 𝓕.CrAnAlgebra) (hx : a ∈ statisticSubmodule fermionic) : Prop :=
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bosonicProj a = 0
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change p a h
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apply Submodule.span_induction
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· intro x hx
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simp only [Set.mem_setOf_eq] at hx
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obtain ⟨φs, rfl, h⟩ := hx
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simp [p, bosonicProj_ofCrAnList, h]
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· simp [p]
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· intro x y hx hy hpx hpy
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simp_all [p]
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· intro a x hx hy
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simp_all [p]
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@[simp]
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lemma bosonicProj_of_bonosic_part
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(a : DirectSum FieldStatistic (fun i => (statisticSubmodule (𝓕 := 𝓕) i))) :
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bosonicProj (a bosonic) = a bosonic := by
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apply bosonicProj_of_mem_bosonic
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@[simp]
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lemma bosonicProj_of_fermionic_part
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(a : DirectSum FieldStatistic (fun i => (statisticSubmodule (𝓕 := 𝓕) i))) :
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bosonicProj (a fermionic).1 = 0 := by
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apply bosonicProj_of_mem_fermionic
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exact Submodule.coe_mem (a.toFun fermionic)
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/-- The projection of an element of `CrAnAlgebra` onto it's fermionic part. -/
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def fermionicProj : 𝓕.CrAnAlgebra →ₗ[ℂ] statisticSubmodule (𝓕 := 𝓕) fermionic :=
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Basis.constr ofCrAnListBasis ℂ fun φs =>
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if h : (𝓕 |>ₛ φs) = fermionic then
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⟨ofCrAnList φs, Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩⟩
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else
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0
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lemma fermionicProj_ofCrAnList (φs : List 𝓕.CrAnStates) :
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fermionicProj (ofCrAnList φs) = if h : (𝓕 |>ₛ φs) = fermionic then
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⟨ofCrAnList φs, Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩⟩ else 0 := by
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conv_lhs =>
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rw [← ofListBasis_eq_ofList, fermionicProj, Basis.constr_basis]
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lemma fermionicProj_ofCrAnList_if_bosonic (φs : List 𝓕.CrAnStates) :
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fermionicProj (ofCrAnList φs) = if h : (𝓕 |>ₛ φs) = bosonic then
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0 else ⟨ofCrAnList φs, Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl,
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by simpa using h⟩⟩⟩ := by
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rw [fermionicProj_ofCrAnList]
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by_cases h1 : (𝓕 |>ₛ φs) = fermionic
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· simp [h1]
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· simp only [h1, ↓reduceDIte]
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simp only [neq_fermionic_iff_eq_bosonic] at h1
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simp [h1]
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lemma fermionicProj_of_mem_fermionic (a : 𝓕.CrAnAlgebra) (h : a ∈ statisticSubmodule fermionic) :
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fermionicProj a = ⟨a, h⟩ := by
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let p (a : 𝓕.CrAnAlgebra) (hx : a ∈ statisticSubmodule fermionic) : Prop :=
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fermionicProj a = ⟨a, hx⟩
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change p a h
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apply Submodule.span_induction
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· intro x hx
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simp only [Set.mem_setOf_eq] at hx
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obtain ⟨φs, rfl, h⟩ := hx
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simp [p, fermionicProj_ofCrAnList, h]
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· simp only [map_zero, p]
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rfl
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· intro x y hx hy hpx hpy
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simp_all [p]
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· intro a x hx hy
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simp_all [p]
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lemma fermionicProj_of_mem_bosonic (a : 𝓕.CrAnAlgebra) (h : a ∈ statisticSubmodule bosonic) :
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fermionicProj a = 0 := by
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let p (a : 𝓕.CrAnAlgebra) (hx : a ∈ statisticSubmodule bosonic) : Prop :=
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fermionicProj a = 0
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change p a h
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apply Submodule.span_induction
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· intro x hx
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simp only [Set.mem_setOf_eq] at hx
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obtain ⟨φs, rfl, h⟩ := hx
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simp [p, fermionicProj_ofCrAnList, h]
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· simp [p]
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· intro x y hx hy hpx hpy
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simp_all [p]
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· intro a x hx hy
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simp_all [p]
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@[simp]
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lemma fermionicProj_of_bosonic_part
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(a : DirectSum FieldStatistic (fun i => (statisticSubmodule (𝓕 := 𝓕) i))) :
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fermionicProj (a bosonic).1 = 0 := by
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apply fermionicProj_of_mem_bosonic
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exact Submodule.coe_mem (a.toFun bosonic)
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@[simp]
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lemma fermionicProj_of_fermionic_part
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(a : DirectSum FieldStatistic (fun i => (statisticSubmodule (𝓕 := 𝓕) i))) :
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fermionicProj (a fermionic) = a fermionic := by
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apply fermionicProj_of_mem_fermionic
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lemma bosonicProj_add_fermionicProj (a : 𝓕.CrAnAlgebra) :
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a.bosonicProj + (a.fermionicProj).1 = a := by
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let f1 :𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra :=
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(statisticSubmodule bosonic).subtype ∘ₗ bosonicProj
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let f2 :𝓕.CrAnAlgebra →ₗ[ℂ] 𝓕.CrAnAlgebra :=
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(statisticSubmodule fermionic).subtype ∘ₗ fermionicProj
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change (f1 + f2) a = LinearMap.id (R := ℂ) a
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refine LinearMap.congr_fun (ofCrAnListBasis.ext fun φs ↦ ?_) a
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simp only [ofListBasis_eq_ofList, LinearMap.add_apply, LinearMap.coe_comp, Submodule.coe_subtype,
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Function.comp_apply, LinearMap.id_coe, id_eq, f1, f2]
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rw [bosonicProj_ofCrAnList, fermionicProj_ofCrAnList_if_bosonic]
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by_cases h : (𝓕 |>ₛ φs) = bosonic
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· simp [h]
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· simp [h]
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lemma coeAddMonoidHom_apply_eq_bosonic_plus_fermionic
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(a : DirectSum FieldStatistic (fun i => (statisticSubmodule (𝓕 := 𝓕) i))) :
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DirectSum.coeAddMonoidHom statisticSubmodule a = a.1 bosonic + a.1 fermionic := by
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let C : DirectSum FieldStatistic (fun i => (statisticSubmodule (𝓕 := 𝓕) i)) → Prop :=
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fun a => DirectSum.coeAddMonoidHom statisticSubmodule a = a.1 bosonic + a.1 fermionic
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change C a
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apply DirectSum.induction_on
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· simp [C]
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· intro i x
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simp only [DFinsupp.toFun_eq_coe, DirectSum.coeAddMonoidHom_of, C]
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rw [DirectSum.of_apply, DirectSum.of_apply]
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match i with
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| bosonic => simp
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| fermionic => simp
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· intro x y hx hy
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simp_all only [C, DFinsupp.toFun_eq_coe, map_add, DirectSum.add_apply, Submodule.coe_add]
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abel
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lemma directSum_eq_bosonic_plus_fermionic
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(a : DirectSum FieldStatistic (fun i => (statisticSubmodule (𝓕 := 𝓕) i))) :
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a = (DirectSum.of (fun i => ↥(statisticSubmodule i)) bosonic) (a bosonic) +
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(DirectSum.of (fun i => ↥(statisticSubmodule i)) fermionic) (a fermionic) := by
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let C : DirectSum FieldStatistic (fun i => (statisticSubmodule (𝓕 := 𝓕) i)) → Prop :=
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fun a => a = (DirectSum.of (fun i => ↥(statisticSubmodule i)) bosonic) (a bosonic) +
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(DirectSum.of (fun i => ↥(statisticSubmodule i)) fermionic) (a fermionic)
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change C a
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apply DirectSum.induction_on
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· simp [C]
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· intro i x
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simp only [C]
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match i with
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| bosonic =>
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simp only [DirectSum.of_eq_same, self_eq_add_right]
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rw [DirectSum.of_eq_of_ne]
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simp only [map_zero]
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simp
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| fermionic =>
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simp only [DirectSum.of_eq_same, add_zero]
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rw [DirectSum.of_eq_of_ne]
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simp only [map_zero, zero_add]
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simp
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· intro x y hx hy
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simp only [DirectSum.add_apply, map_add, C] at hx hy ⊢
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conv_lhs => rw [hx, hy]
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abel
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instance crAnAlgebraGrade : GradedAlgebra (A := 𝓕.CrAnAlgebra) statisticSubmodule where
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one_mem := by
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simp only [statisticSubmodule]
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refine Submodule.mem_span.mpr fun p a => a ?_
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simp only [Set.mem_setOf_eq]
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use []
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simp only [ofCrAnList_nil, ofList_empty, true_and]
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rfl
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mul_mem f1 f2 a1 a2 h1 h2 := by
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let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
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a1 * a2 ∈ statisticSubmodule (f1 + f2)
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change p a2 h2
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apply Submodule.span_induction (p := p)
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· intro x hx
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simp only [Set.mem_setOf_eq] at hx
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obtain ⟨φs, rfl, h⟩ := hx
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simp only [p]
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let p (a1 : 𝓕.CrAnAlgebra) (hx : a1 ∈ statisticSubmodule f1) : Prop :=
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a1 * ofCrAnList φs ∈ statisticSubmodule (f1 + f2)
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change p a1 h1
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apply Submodule.span_induction (p := p)
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· intro y hy
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obtain ⟨φs', rfl, h'⟩ := hy
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simp only [p]
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rw [← ofCrAnList_append]
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refine Submodule.mem_span.mpr fun p a => a ?_
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simp only [Set.mem_setOf_eq]
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use φs' ++ φs
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simp only [ofList_append, h', h, true_and]
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cases f1 <;> cases f2 <;> rfl
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· simp [p]
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· intro x y hx hy hx1 hx2
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simp only [add_mul, p]
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exact Submodule.add_mem _ hx1 hx2
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· intro c a hx h1
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simp only [Algebra.smul_mul_assoc, p]
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exact Submodule.smul_mem _ _ h1
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· exact h1
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· simp [p]
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· intro x y hx hy hx1 hx2
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simp only [mul_add, p]
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exact Submodule.add_mem _ hx1 hx2
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· intro c a hx h1
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simp only [Algebra.mul_smul_comm, p]
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exact Submodule.smul_mem _ _ h1
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· exact h2
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decompose' a := DirectSum.of (fun i => (statisticSubmodule (𝓕 := 𝓕) i)) bosonic (bosonicProj a)
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+ DirectSum.of (fun i => (statisticSubmodule (𝓕 := 𝓕) i)) fermionic (fermionicProj a)
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left_inv a := by
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trans a.bosonicProj + fermionicProj a
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· simp
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· exact bosonicProj_add_fermionicProj a
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right_inv a := by
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rw [coeAddMonoidHom_apply_eq_bosonic_plus_fermionic]
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simp only [DFinsupp.toFun_eq_coe, map_add, bosonicProj_of_bonosic_part,
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bosonicProj_of_fermionic_part, add_zero, fermionicProj_of_bosonic_part,
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fermionicProj_of_fermionic_part, zero_add]
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conv_rhs => rw [directSum_eq_bosonic_plus_fermionic a]
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lemma eq_zero_of_bosonic_and_fermionic {a : 𝓕.CrAnAlgebra}
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(hb : a ∈ statisticSubmodule bosonic) (hf : a ∈ statisticSubmodule fermionic) : a = 0 := by
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have ha := bosonicProj_of_mem_bosonic a hb
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have hb := fermionicProj_of_mem_fermionic a hf
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have hc := (bosonicProj_add_fermionicProj a)
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rw [ha, hb] at hc
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simpa using hc
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lemma bosonicProj_mul (a b : 𝓕.CrAnAlgebra) :
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(a * b).bosonicProj.1 = a.bosonicProj.1 * b.bosonicProj.1
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+ a.fermionicProj.1 * b.fermionicProj.1 := by
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conv_lhs =>
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rw [← bosonicProj_add_fermionicProj a]
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rw [← bosonicProj_add_fermionicProj b]
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simp [mul_add, add_mul]
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rw [bosonicProj_of_mem_bosonic]
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conv_lhs =>
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left
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right
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rw [bosonicProj_of_mem_fermionic _
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(by
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have h1 : fermionic = fermionic + bosonic := by simp
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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conv_lhs =>
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right
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left
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rw [bosonicProj_of_mem_fermionic _
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(by
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have h1 : fermionic = bosonic + fermionic := by simp
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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conv_lhs =>
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right
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right
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rw [bosonicProj_of_mem_bosonic _
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(by
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have h1 : bosonic = fermionic + fermionic := by simp; rfl
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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simp
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· have h1 : bosonic = bosonic + bosonic := by simp; rfl
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp
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lemma fermionicProj_mul (a b : 𝓕.CrAnAlgebra) :
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(a * b).fermionicProj.1 = a.bosonicProj.1 * b.fermionicProj.1
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+ a.fermionicProj.1 * b.bosonicProj.1 := by
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conv_lhs =>
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rw [← bosonicProj_add_fermionicProj a]
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rw [← bosonicProj_add_fermionicProj b]
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simp [mul_add, add_mul]
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conv_lhs =>
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left
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left
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rw [fermionicProj_of_mem_bosonic _
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(by
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have h1 : bosonic = bosonic + bosonic := by simp; rfl
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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conv_lhs =>
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left
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right
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rw [fermionicProj_of_mem_fermionic _
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(by
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have h1 : fermionic = fermionic + bosonic := by simp
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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conv_lhs =>
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right
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left
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rw [fermionicProj_of_mem_fermionic _
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(by
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have h1 : fermionic = bosonic + fermionic := by simp
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conv_lhs => rw [h1]
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apply crAnAlgebraGrade.mul_mem
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simp
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simp)]
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conv_lhs =>
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right
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right
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rw [fermionicProj_of_mem_bosonic _
|
||
(by
|
||
have h1 : bosonic = fermionic + fermionic := by simp; rfl
|
||
conv_lhs => rw [h1]
|
||
apply crAnAlgebraGrade.mul_mem
|
||
simp
|
||
simp)]
|
||
simp
|
||
abel
|
||
|
||
end
|
||
|
||
end CrAnAlgebra
|
||
|
||
end FieldSpecification
|