refactor: Docs for FieldOpFreeAlgebra

HepLean/PerturbationTheory/FieldOpFreeAlgebra/Basic.lean

@@ -35,11 +35,13 @@ namespace FieldSpecification
 variable {𝓕 : FieldSpecification}
 
 /-- For a field specification `𝓕`, the algebra `𝓕.FieldOpFreeAlgebra` is
-  the free algebra generated by creation and annihilation parts of field operators defined in
-  `𝓕.CrAnFieldOp`.
-  It represents the algebra containing all possible products and linear combinations
-  of creation and annihilation parts of field operators, without imposing any conditions.
--/
+  the free algebra generated by `𝓕.CrAnFieldOp`.
+
+  The algebra `𝓕.FieldOpFreeAlgebra` satisfies the universal property that for any other algebra
+  `A` (e.g. the operator algebra of the theory) with a map `f : 𝓕.CrAnFieldOp → A` (e.g.
+  the inclusion of the creation and annihilation parts of field operators into the
+  operator algebra) there is a unqiue algebra map `g : 𝓕.FieldOpFreeAlgebra → A`
+  through which `f` factors. -/
 abbrev FieldOpFreeAlgebra (𝓕 : FieldSpecification) : Type := FreeAlgebra ℂ 𝓕.CrAnFieldOp
 
 namespace FieldOpFreeAlgebra
@@ -75,16 +77,21 @@ lemma ofCrAnListF_singleton (φ : 𝓕.CrAnFieldOp) :
 
 /-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
   `𝓕.FieldOp` by summing over the creation and annihilation components of `𝓕.FieldOp`.
-  For example for `φ₁` an incoming asymptotic field operator we get `φ₁ᶜ`, and for `φ₁` a
-  position field operator we get `φ₁ᶜ + φ₁ᵃ`. -/
+  For example for `φ₁` an incoming asymptotic field operator we get
+  `ofCrAnOpF φ₁ᶜ`, and for `φ₁` a
+  position field operator we get `ofCrAnOpF φ₁ᶜ + ofCrAnOpF φ₁ᵃ`. -/
 def ofFieldOpF (φ : 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 :=
   ∑ (i : 𝓕.fieldOpToCrAnType φ), ofCrAnOpF ⟨φ, i⟩
 
 /-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
   list of `𝓕.FieldOp` by summing over the creation and annihilation components.
-  For example, `φ₁` and `φ₂` position states `[φ1, φ2]` gets sent to `(φ1ᶜ+ φ1ᵃ) * (φ2ᶜ+ φ2ᵃ)`. -/
+  For example, `φ₁` and `φ₂` position states `[φ1, φ2]` gets sent to
+  `(ofCrAnOpF φ1ᶜ + ofCrAnOpF φ1ᵃ) * (ofCrAnOpF φ2ᶜ + ofCrAnOpF φ2ᵃ)`. -/
 def ofFieldOpListF (φs : List 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 := (List.map ofFieldOpF φs).prod
 
+remark notation_drop := "In doc-strings we will often drop explicit applications of `ofCrAnOpF`,
+`ofCrAnListF`, `ofFieldOpF`, and `ofFieldOpListF`"
+
 /-- Coercion from `List 𝓕.FieldOp` to `FieldOpFreeAlgebra 𝓕` through `ofFieldOpListF`. -/
 instance : Coe (List 𝓕.FieldOp) (FieldOpFreeAlgebra 𝓕) := ⟨ofFieldOpListF⟩
 

HepLean/PerturbationTheory/FieldOpFreeAlgebra/Grading.lean

@@ -237,8 +237,8 @@ lemma directSum_eq_bosonic_plus_fermionic
     abel
 
 /-- For a field statistic `𝓕`, the algebra `𝓕.FieldOpFreeAlgebra` is graded by `FieldStatistic`.
-  Those `ofCrAnListF φs` for which `φs` has `bosonic` statistics form one part of the grading,
-  whilst those where `φs` has `fermionic` statistics form the other part of the grading. -/
+  Those `ofCrAnListF φs` for which `φs` has `bosonic` statistics span one part  of the grading,
+  whilst those where `φs` has `fermionic` statistics span the other part of the grading. -/
 instance fieldOpFreeAlgebraGrade :
     GradedAlgebra (A := 𝓕.FieldOpFreeAlgebra) statisticSubmodule where
   one_mem := by

HepLean/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean

@@ -24,10 +24,11 @@ namespace FieldOpFreeAlgebra
 open FieldStatistic
 
 /-- For a field specification `𝓕`, the super commutator `superCommuteF` is defined as the linear
-  map `𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra` such that
-  `superCommuteF (φ₀ᶜ…φₙᵃ) (φ₀'ᶜ…φₙ'ᶜ)` is equal to
-  `φ₀ᶜ…φₙᵃ * φ₀'ᶜ…φₙ'ᶜ - 𝓢(φ₀ᶜ…φₙᵃ, φ₀'ᶜ…φₙ'ᶜ) φ₀'ᶜ…φₙ'ᶜ * φ₀ᶜ…φₙᵃ`.
-  The notation `[a, b]ₛca` is used for this super commutator. -/
+  map `𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra`
+  which on the lists `φs` and `φs'` of `𝓕.CrAnFieldOp` gives
+  `superCommuteF φs φs' = φs * φs' - 𝓢(φs, φs') • φs' * φs`.
+
+  The notation `[a, b]ₛca` can be used for  `superCommuteF a b`. -/
 noncomputable def superCommuteF : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ]
     𝓕.FieldOpFreeAlgebra :=
   Basis.constr ofCrAnListFBasis ℂ fun φs =>
@@ -404,9 +405,12 @@ lemma superCommuteF_ofCrAnListF_ofFieldOpListF_cons (φ : 𝓕.FieldOp) (φs : L
   simp [mul_comm]
 
 /--
-Within the creation and annihilation algebra, we have that
-`[φᶜᵃs, φᶜᵃ₀ … φᶜᵃₙ]ₛca = ∑ i, sᵢ • φᶜᵃs₀ … φᶜᵃᵢ₋₁ * [φᶜᵃs, φᶜᵃᵢ]ₛca * φᶜᵃᵢ₊₁ … φᶜᵃₙ`
-where `sᵢ` is the exchange sign for `φᶜᵃs` and `φᶜᵃs₀ … φᶜᵃᵢ₋₁`.
+For a field specification `𝓕`, and to lists `φs = φ₀…φₙ` and `φs'` of `𝓕.CrAnFieldOp`
+the following super commutation relation holds:
+
+`[φs', φ₀…φₙ]ₛca = ∑ i, 𝓢(φs', φ₀…φᵢ₋₁) • φ₀…φᵢ₋₁ * [φs', φᵢ]ₛca * φᵢ₊₁ … φₙ`
+
+The proof of this relation is via induction on the length of `φs`.
 -/
 lemma superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum (φs : List 𝓕.CrAnFieldOp) :
     (φs' : List 𝓕.CrAnFieldOp) → [ofCrAnListF φs, ofCrAnListF φs']ₛca =