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feat: More notes

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jstoobysmith 2025-02-05 05:44:40 +00:00
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20
HepLean/PerturbationTheory/FieldOpFreeAlgebra/Basic.lean
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@ -50,12 +50,14 @@ remark naming_convention := "
This is to avoid confusion when working within the context of `FieldOpAlgebra` which is defined
as a quotient of `FieldOpFreeAlgebra`."
/-- Maps a creation and annihlation state to the creation and annihlation free-algebra. -/
/-- For a field specification `𝓕`, the element of `𝓕.FieldOpFreeAlgebra` formed by a
single `𝓕.CrAnFieldOp`. -/
def ofCrAnOpF (φ : 𝓕.CrAnFieldOp) : FieldOpFreeAlgebra 𝓕 :=
FreeAlgebra.ι ℂ φ
/-- Maps a list creation and annihlation state to the creation and annihlation free-algebra
by taking their product. -/
/-- 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`. -/
def ofCrAnListF (φs : List 𝓕.CrAnFieldOp) : FieldOpFreeAlgebra 𝓕 := (List.map ofCrAnOpF φs).prod
@[simp]
@ -71,14 +73,16 @@ lemma ofCrAnListF_append (φs φs' : List 𝓕.CrAnFieldOp) :
lemma ofCrAnListF_singleton (φ : 𝓕.CrAnFieldOp) :
ofCrAnListF [φ] = ofCrAnOpF φ := by simp [ofCrAnListF]
/-- Maps a state to the sum of creation and annihilation operators in
creation and annihilation free-algebra. -/
/-- 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`. -/
def ofFieldOpF (φ : 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 :=
∑ (i : 𝓕.fieldOpToCrAnType φ), ofCrAnOpF ⟨φ, i⟩
/-- Maps a list of states to the creation and annihilation free-algebra by taking
the product of their sums of creation and annihlation operators.
Roughly `[φ1, φ2]` gets sent to `(φ1ᶜ+ φ1ᵃ) * (φ2ᶜ+ φ2ᵃ)` etc. -/
/-- 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)`. -/
def ofFieldOpListF (φs : List 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 := (List.map ofFieldOpF φs).prod
/-- Coercion from `List 𝓕.FieldOp` to `FieldOpFreeAlgebra 𝓕` through `ofFieldOpListF`. -/