257 lines
10 KiB
Text
257 lines
10 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.FieldSpecification.CrAnFieldOp
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import HepLean.PerturbationTheory.FieldSpecification.CrAnSection
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/-!
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# Creation and annihilation free-algebra
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This module defines the creation and annihilation algebra for a field structure.
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The creation and annihilation algebra extends from the state algebra by adding information about
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whether a state is a creation or annihilation operator.
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The algebra is spanned by lists of creation/annihilation states.
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The main structures defined in this module are:
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* `FieldOpFreeAlgebra` - The creation and annihilation algebra
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* `ofCrAnOpF` - Maps a creation/annihilation state to the algebra
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* `ofCrAnListF` - Maps a list of creation/annihilation states to the algebra
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* `ofFieldOpF` - Maps a state to a sum of creation and annihilation operators
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* `crPartF` - The creation part of a state in the algebra
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* `anPartF` - The annihilation part of a state in the algebra
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* `superCommuteF` - The super commutator on the algebra
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The key lemmas show how these operators interact, particularly focusing on the
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super commutation relations between creation and annihilation operators.
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-/
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namespace FieldSpecification
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variable {𝓕 : FieldSpecification}
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/-- For a field specification `𝓕`, the algebra `𝓕.FieldOpFreeAlgebra` is
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the free algebra generated by `𝓕.CrAnFieldOp`. -/
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abbrev FieldOpFreeAlgebra (𝓕 : FieldSpecification) : Type := FreeAlgebra ℂ 𝓕.CrAnFieldOp
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namespace FieldOpFreeAlgebra
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remark naming_convention := "
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For mathematicial objects defined in relation to `FieldOpFreeAlgebra` we will often postfix
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their names with an `F` to indicate that they are related to the free algebra.
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This is to avoid confusion when working within the context of `FieldOpAlgebra` which is defined
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as a quotient of `FieldOpFreeAlgebra`."
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/-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
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single `𝓕.CrAnFieldOp`. -/
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def ofCrAnOpF (φ : 𝓕.CrAnFieldOp) : FieldOpFreeAlgebra 𝓕 :=
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FreeAlgebra.ι ℂ φ
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/--
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The algebra `𝓕.FieldOpFreeAlgebra` satisfies the universal property that for any other algebra
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`A` (e.g. the operator algebra of the theory) with a map `f : 𝓕.CrAnFieldOp → A` (e.g.
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the inclusion of the creation and annihilation parts of field operators into the
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operator algebra) there is a unique algebra map `g : 𝓕.FieldOpFreeAlgebra → A`
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such that `g ∘ ofCrAnOpF = f`.
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The unique `g` is given by `FreeAlgebra.lift ℂ f`.
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-/
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lemma universality {A : Type} [Semiring A] [Algebra ℂ A] (f : 𝓕.CrAnFieldOp → A) :
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∃! g : FieldOpFreeAlgebra 𝓕 →ₐ[ℂ] A, g ∘ ofCrAnOpF = f := by
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use FreeAlgebra.lift ℂ f
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apply And.intro
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· funext x
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simp [ofCrAnOpF]
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· intro g hg
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ext x
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simpa using congrFun hg x
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/-- For a field specification `𝓕`, `ofCrAnListF φs` of `𝓕.FieldOpFreeAlgebra` formed by a
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list `φs` of `𝓕.CrAnFieldOp`. For example for the list `[φ₁ᶜ, φ₂ᵃ, φ₃ᶜ]` we schematically
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get `φ₁ᶜφ₂ᵃφ₃ᶜ`. The set of all `ofCrAnListF φs` forms a basis of `FieldOpFreeAlgebra 𝓕`. -/
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def ofCrAnListF (φs : List 𝓕.CrAnFieldOp) : FieldOpFreeAlgebra 𝓕 := (List.map ofCrAnOpF φs).prod
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@[simp]
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lemma ofCrAnListF_nil : ofCrAnListF ([] : List 𝓕.CrAnFieldOp) = 1 := rfl
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lemma ofCrAnListF_cons (φ : 𝓕.CrAnFieldOp) (φs : List 𝓕.CrAnFieldOp) :
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ofCrAnListF (φ :: φs) = ofCrAnOpF φ * ofCrAnListF φs := rfl
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lemma ofCrAnListF_append (φs φs' : List 𝓕.CrAnFieldOp) :
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ofCrAnListF (φs ++ φs') = ofCrAnListF φs * ofCrAnListF φs' := by
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simp [ofCrAnListF, List.map_append]
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lemma ofCrAnListF_singleton (φ : 𝓕.CrAnFieldOp) :
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ofCrAnListF [φ] = ofCrAnOpF φ := by simp [ofCrAnListF]
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/-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
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`𝓕.FieldOp` by summing over the creation and annihilation components of `𝓕.FieldOp`.
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For example for `φ₁` an incoming asymptotic field operator we get
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`ofCrAnOpF φ₁ᶜ`, and for `φ₁` a
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position field operator we get `ofCrAnOpF φ₁ᶜ + ofCrAnOpF φ₁ᵃ`. -/
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def ofFieldOpF (φ : 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 :=
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∑ (i : 𝓕.fieldOpToCrAnType φ), ofCrAnOpF ⟨φ, i⟩
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/-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
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list of `𝓕.FieldOp` by summing over the creation and annihilation components.
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For example, `φ₁` and `φ₂` position states `[φ1, φ2]` gets sent to
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`(ofCrAnOpF φ1ᶜ + ofCrAnOpF φ1ᵃ) * (ofCrAnOpF φ2ᶜ + ofCrAnOpF φ2ᵃ)`. -/
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def ofFieldOpListF (φs : List 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 := (List.map ofFieldOpF φs).prod
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remark notation_drop := "In doc-strings we will often drop explicit applications of `ofCrAnOpF`,
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`ofCrAnListF`, `ofFieldOpF`, and `ofFieldOpListF`"
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/-- Coercion from `List 𝓕.FieldOp` to `FieldOpFreeAlgebra 𝓕` through `ofFieldOpListF`. -/
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instance : Coe (List 𝓕.FieldOp) (FieldOpFreeAlgebra 𝓕) := ⟨ofFieldOpListF⟩
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@[simp]
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lemma ofFieldOpListF_nil : ofFieldOpListF ([] : List 𝓕.FieldOp) = 1 := rfl
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lemma ofFieldOpListF_cons (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
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ofFieldOpListF (φ :: φs) = ofFieldOpF φ * ofFieldOpListF φs := rfl
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lemma ofFieldOpListF_singleton (φ : 𝓕.FieldOp) :
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ofFieldOpListF [φ] = ofFieldOpF φ := by simp [ofFieldOpListF]
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lemma ofFieldOpListF_append (φs φs' : List 𝓕.FieldOp) :
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ofFieldOpListF (φs ++ φs') = ofFieldOpListF φs * ofFieldOpListF φs' := by
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dsimp only [ofFieldOpListF]
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rw [List.map_append, List.prod_append]
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lemma ofFieldOpListF_sum (φs : List 𝓕.FieldOp) :
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ofFieldOpListF φs = ∑ (s : CrAnSection φs), ofCrAnListF s.1 := by
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induction φs with
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| nil => simp
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| cons φ φs ih =>
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rw [CrAnSection.sum_cons]
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dsimp only [CrAnSection.cons, ofCrAnListF_cons]
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conv_rhs =>
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enter [2, x]
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rw [← Finset.mul_sum]
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rw [← Finset.sum_mul, ofFieldOpListF_cons, ← ih]
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rfl
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/-!
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## Creation and annihilation parts of a state
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-/
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/-- The algebra map taking an element of the free-state algebra to
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the part of it in the creation and annihilation free algebra
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spanned by creation operators. -/
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def crPartF : 𝓕.FieldOp → 𝓕.FieldOpFreeAlgebra := fun φ =>
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match φ with
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| FieldOp.inAsymp φ => ofCrAnOpF ⟨FieldOp.inAsymp φ, ()⟩
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| FieldOp.position φ => ofCrAnOpF ⟨FieldOp.position φ, CreateAnnihilate.create⟩
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| FieldOp.outAsymp _ => 0
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@[simp]
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lemma crPartF_negAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ)) :
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crPartF (FieldOp.inAsymp φ) = ofCrAnOpF ⟨FieldOp.inAsymp φ, ()⟩ := by
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simp [crPartF]
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@[simp]
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lemma crPartF_position (φ : (Σ f, 𝓕.PositionLabel f) × SpaceTime) :
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crPartF (FieldOp.position φ) =
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ofCrAnOpF ⟨FieldOp.position φ, CreateAnnihilate.create⟩ := by
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simp [crPartF]
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@[simp]
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lemma crPartF_posAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ)) :
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crPartF (FieldOp.outAsymp φ) = 0 := by
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simp [crPartF]
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/-- The algebra map taking an element of the free-state algebra to
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the part of it in the creation and annihilation free algebra
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spanned by annihilation operators. -/
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def anPartF : 𝓕.FieldOp → 𝓕.FieldOpFreeAlgebra := fun φ =>
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match φ with
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| FieldOp.inAsymp _ => 0
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| FieldOp.position φ => ofCrAnOpF ⟨FieldOp.position φ, CreateAnnihilate.annihilate⟩
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| FieldOp.outAsymp φ => ofCrAnOpF ⟨FieldOp.outAsymp φ, ()⟩
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@[simp]
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lemma anPartF_negAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ)) :
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anPartF (FieldOp.inAsymp φ) = 0 := by
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simp [anPartF]
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@[simp]
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lemma anPartF_position (φ : (Σ f, 𝓕.PositionLabel f) × SpaceTime) :
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anPartF (FieldOp.position φ) =
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ofCrAnOpF ⟨FieldOp.position φ, CreateAnnihilate.annihilate⟩ := by
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simp [anPartF]
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@[simp]
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lemma anPartF_posAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ)) :
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anPartF (FieldOp.outAsymp φ) = ofCrAnOpF ⟨FieldOp.outAsymp φ, ()⟩ := by
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simp [anPartF]
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lemma ofFieldOpF_eq_crPartF_add_anPartF (φ : 𝓕.FieldOp) :
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ofFieldOpF φ = crPartF φ + anPartF φ := by
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rw [ofFieldOpF]
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cases φ with
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| inAsymp φ => simp [fieldOpToCrAnType]
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| position φ => simp [fieldOpToCrAnType, CreateAnnihilate.sum_eq]
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| outAsymp φ => simp [fieldOpToCrAnType]
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/-!
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## The basis of the creation and annihilation free-algebra.
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-/
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/-- The basis of the free creation and annihilation algebra formed by lists of CrAnFieldOp. -/
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noncomputable def ofCrAnListFBasis : Basis (List 𝓕.CrAnFieldOp) ℂ (FieldOpFreeAlgebra 𝓕) where
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repr := FreeAlgebra.equivMonoidAlgebraFreeMonoid.toLinearEquiv
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@[simp]
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lemma ofListBasis_eq_ofList (φs : List 𝓕.CrAnFieldOp) :
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ofCrAnListFBasis φs = ofCrAnListF φs := by
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simp only [ofCrAnListFBasis, FreeAlgebra.equivMonoidAlgebraFreeMonoid, MonoidAlgebra.of_apply,
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Basis.coe_ofRepr, AlgEquiv.toLinearEquiv_symm, AlgEquiv.toLinearEquiv_apply,
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AlgEquiv.ofAlgHom_symm_apply, ofCrAnListF]
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erw [MonoidAlgebra.lift_apply]
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simp only [zero_smul, Finsupp.sum_single_index, one_smul]
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rw [@FreeMonoid.lift_apply]
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match φs with
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| [] => rfl
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| φ :: φs => erw [List.map_cons]
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/-!
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## Some useful multi-linear maps.
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-/
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/-- The bi-linear map associated with multiplication in `FieldOpFreeAlgebra`. -/
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noncomputable def mulLinearMap : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 →ₗ[ℂ]
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FieldOpFreeAlgebra 𝓕 where
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toFun a := {
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toFun := fun b => a * b,
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map_add' := fun c d => by simp [mul_add]
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map_smul' := by simp}
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map_add' := fun a b => by
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ext c
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simp [add_mul]
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map_smul' := by
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intros
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ext c
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simp [smul_mul']
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lemma mulLinearMap_apply (a b : FieldOpFreeAlgebra 𝓕) :
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mulLinearMap a b = a * b := rfl
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/-- The linear map associated with scalar-multiplication in `FieldOpFreeAlgebra`. -/
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noncomputable def smulLinearMap (c : ℂ) : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 where
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toFun a := c • a
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map_add' := by simp
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map_smul' m x := by simp [smul_smul, RingHom.id_apply, NonUnitalNormedCommRing.mul_comm]
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end FieldOpFreeAlgebra
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end FieldSpecification
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