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doc: Edits to Wick theorem docs

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jstoobysmith 2025-02-10 10:21:57 +00:00
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42
HepLean/PerturbationTheory/FieldOpFreeAlgebra/Basic.lean
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@ -41,13 +41,14 @@ abbrev FieldOpFreeAlgebra (𝓕 : FieldSpecification) : Type := FreeAlgebra ℂ
namespace FieldOpFreeAlgebra
remark naming_convention := "
For mathematicial objects defined in relation to `FieldOpFreeAlgebra` we will often postfix
their names with an `F` to indicate that they are related to the free algebra.
For mathematicial objects defined in relation to `FieldOpFreeAlgebra` the postfix `F`
may be given to
their names to indicate that they are related to the free algebra.
This is to avoid confusion when working within the context of `FieldOpAlgebra` which is defined
as a quotient of `FieldOpFreeAlgebra`."
/-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
single `𝓕.CrAnFieldOp`. -/
/-- For a field specification `𝓕`, and a element `φ` of `𝓕.CrAnFieldOp`,
`ofCrAnOpF φ` is defined as the element of `𝓕.FieldOpFreeAlgebra` formed by `φ`. -/
def ofCrAnOpF (φ : 𝓕.CrAnFieldOp) : FieldOpFreeAlgebra 𝓕 :=
FreeAlgebra.ι ℂ φ
@ -70,9 +71,11 @@ lemma universality {A : Type} [Semiring A] [Algebra ℂ A] (f : 𝓕.CrAnFieldOp
ext x
simpa using congrFun hg x
/-- For a field specification `𝓕`, `ofCrAnListF φs` of `𝓕.FieldOpFreeAlgebra` formed by a
list `φs` of `𝓕.CrAnFieldOp`. For example for the list `[φ₁ᶜ, φ₂ᵃ, φ₃ᶜ]` we schematically
get `φ₁ᶜφ₂ᵃφ₃ᶜ`. The set of all `ofCrAnListF φs` forms a basis of `FieldOpFreeAlgebra 𝓕`. -/
/-- For a field specification `𝓕`, and a list `φs` of `𝓕.CrAnFieldOp`,
`ofCrAnListF φs` is defined as the element of `𝓕.FieldOpFreeAlgebra`
obtained by the product of `ofCrAnListF φ` for each `φ` in `φs`.
For example `ofCrAnListF [φ₁, φ₂, φ₃] = ofCrAnOpF φ₁ * ofCrAnOpF φ₂ * ofCrAnOpF φ₃`.
The set of all `ofCrAnListF φs` forms a basis of `FieldOpFreeAlgebra 𝓕`. -/
def ofCrAnListF (φs : List 𝓕.CrAnFieldOp) : FieldOpFreeAlgebra 𝓕 := (List.map ofCrAnOpF φs).prod
@[simp]
@ -88,22 +91,25 @@ lemma ofCrAnListF_append (φs φs' : List 𝓕.CrAnFieldOp) :
lemma ofCrAnListF_singleton (φ : 𝓕.CrAnFieldOp) :
ofCrAnListF [φ] = ofCrAnOpF φ := by simp [ofCrAnListF]
/-- 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
`ofCrAnOpF φ₁ᶜ`, and for `φ₁` a
position field operator we get `ofCrAnOpF φ₁ᶜ + ofCrAnOpF φ₁ᵃ`. -/
/-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`,
`ofFieldOpF φ` is the element of `𝓕.FieldOpFreeAlgebra` formed by summing over
`ofCrAnOpF` of the
creation and annihilation parts of `φ`.
For example for `φ` an incoming asymptotic field operator we get
`ofCrAnOpF ⟨φ, ()⟩`, and for `φ` a
position field operator we get `ofCrAnOpF ⟨φ, .create⟩ + ofCrAnOpF ⟨φ, .annihilate⟩`. -/
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
`(ofCrAnOpF φ1ᶜ + ofCrAnOpF φ1ᵃ) * (ofCrAnOpF φ2ᶜ + ofCrAnOpF φ2ᵃ)`. -/
/-- For a field specification `𝓕`, and a list `φs` of `𝓕.FieldOp`,
`𝓕.ofFieldOpListF φs` is defined as the element of `𝓕.FieldOpFreeAlgebra`
obtained by the product of `ofFieldOpF φ` for each `φ` in `φs`.
For example `ofFieldOpListF [φ₁, φ₂, φ₃] = ofFieldOpF φ₁ * ofFieldOpF φ₂ * ofFieldOpF φ₃`. -/
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`"
remark notation_drop := "In doc-strings explicit applications of `ofCrAnOpF`,
`ofCrAnListF`, `ofFieldOpF`, and `ofFieldOpListF` will often be dropped."
/-- Coercion from `List 𝓕.FieldOp` to `FieldOpFreeAlgebra 𝓕` through `ofFieldOpListF`. -/
instance : Coe (List 𝓕.FieldOp) (FieldOpFreeAlgebra 𝓕) := ⟨ofFieldOpListF⟩