doc: Edits to Wick theorem docs

HepLean/PerturbationTheory/FieldOpAlgebra/Basic.lean

@@ -33,15 +33,18 @@ def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
       x = [ofCrAnOpF φ, ofCrAnOpF φ']ₛca)}
 
 /-- For a field specification `𝓕`, the algebra `𝓕.FieldOpAlgebra` is defined as the quotient
-  of the free algebra `𝓕.FieldOpFreeAlgebra` by the ideal generated by elements of the form
-- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca`, which (with the other conditions)
-  gives us that super-commutors are in the center.
-- `[ofCrAnOpF φc, ofCrAnOpF φc']ₛca` for `φc` and `φc'` creation operators. I.e two creation
-  operators always super-commute.
-- `[ofCrAnOpF φa, ofCrAnOpF φa']ₛca` for `φa` and `φa'` annihilation operators. I.e two
-  annihilation operators always super-commute.
-- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛca` for `φ` and `φ'` field operators with different statistics.
-  I.e. Fermions super-commute with bosons. -/
+  of the free algebra `𝓕.FieldOpFreeAlgebra` by the ideal generated by
+- `[ofCrAnOpF φc, ofCrAnOpF φc']ₛca` for `φc` and `φc'` field creation operators.
+  This corresponds to the condition that two creation operators always super-commute.
+- `[ofCrAnOpF φa, ofCrAnOpF φa']ₛca` for `φa` and `φa'` field annihilation operators.
+  This corresponds to the condition that two annihilation operators always super-commute.
+- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛca` for `φ` and `φ'` operators with different statistics.
+  This corresponds to the condition that two operators with different statistics always
+  super-commute. In otherwords, fermions and bosons always super-commute.
+- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca`. This corresponds to the condition,
+  when combined with the conditions above, that the super-commutor is in the center of the
+  of the algebra.
+-/
 abbrev FieldOpAlgebra : Type := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.Quotient
 
 namespace FieldOpAlgebra
@@ -457,13 +460,15 @@ lemma ι_eq_zero_iff_ι_bosonicProjF_fermonicProj_zero (x : FieldOpFreeAlgebra
 
 -/
 
-/-- For a field specification `𝓕` and an element `φ` of `𝓕.FieldOp`, the element of
+/-- For a field specification `𝓕` and an element `φ` of `𝓕.FieldOp`,
+  `ofFieldOp φ` is defined as the element of
   `𝓕.FieldOpAlgebra` given by `ι (ofFieldOpF φ)`. -/
 def ofFieldOp (φ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (ofFieldOpF φ)
 
 lemma ofFieldOp_eq_ι_ofFieldOpF (φ : 𝓕.FieldOp) : ofFieldOp φ = ι (ofFieldOpF φ) := rfl
 
-/-- For a field specification `𝓕` and a list `φs` of `𝓕.FieldOp`, the element of
+/-- For a field specification `𝓕` and a list `φs` of `𝓕.FieldOp`,
+  `ofFieldOpList φs` is defined as the element of
   `𝓕.FieldOpAlgebra` given by `ι (ofFieldOpListF φ)`. -/
 def ofFieldOpList (φs : List 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (ofFieldOpListF φs)
 
@@ -487,7 +492,8 @@ lemma ofFieldOpList_singleton (φ : 𝓕.FieldOp) :
     ofFieldOpList [φ] = ofFieldOp φ := by
   simp only [ofFieldOpList, ofFieldOp, ofFieldOpListF_singleton]
 
-/-- For a field specification `𝓕` and an element `φ` of `𝓕.CrAnFieldOp`, the element of
+/-- For a field specification `𝓕` and an element `φ` of `𝓕.CrAnFieldOp`,
+  `ofCrAnOp φ` is defined as the element of
   `𝓕.FieldOpAlgebra` given by `ι (ofCrAnOpF φ)`. -/
 def ofCrAnOp (φ : 𝓕.CrAnFieldOp) : 𝓕.FieldOpAlgebra := ι (ofCrAnOpF φ)
 
@@ -500,7 +506,8 @@ lemma ofFieldOp_eq_sum (φ : 𝓕.FieldOp) :
   simp only [map_sum]
   rfl
 
-/-- For a field specification `𝓕` and a list `φs` of `𝓕.CrAnFieldOp`, the element of
+/-- For a field specification `𝓕` and a list `φs` of `𝓕.CrAnFieldOp`,
+  `ofCrAnList φs` is defined as the element of
   `𝓕.FieldOpAlgebra` given by `ι (ofCrAnListF φ)`. -/
 def ofCrAnList (φs : List 𝓕.CrAnFieldOp) : 𝓕.FieldOpAlgebra := ι (ofCrAnListF φs)
 
@@ -528,8 +535,10 @@ remark notation_drop := "In doc-strings we will often drop explicit applications
 
 /-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`, the
   annihilation part of `𝓕.FieldOp` as an element of `𝓕.FieldOpAlgebra`.
-  If `φ` is an incoming asymptotic state this is zero by definition, otherwise
-  it is of the form `ofCrAnOp _`. -/
+  Thus for `φ`
+- an incoming asymptotic state this is `0`.
+- a position based state this is `ofCrAnOp ⟨φ, .create⟩`.
+- an outgoing asymptotic state this is `ofCrAnOp ⟨φ, ()⟩`. -/
 def anPart (φ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (anPartF φ)
 
 lemma anPart_eq_ι_anPartF (φ : 𝓕.FieldOp) : anPart φ = ι (anPartF φ) := rfl
@@ -552,8 +561,10 @@ lemma anPart_posAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × (Fin 3 → ℝ)) :
 
 /-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`, the
   creation part of `𝓕.FieldOp` as an element of `𝓕.FieldOpAlgebra`.
-  If `φ` is an outgoing asymptotic state this is zero by definition, otherwise
-  it is of the form `ofCrAnOp _`. -/
+  Thus for `φ`
+- an incoming asymptotic state this is `ofCrAnOp ⟨φ, ()⟩`.
+- a position based state this is `ofCrAnOp ⟨φ, .create⟩`.
+- an outgoing asymptotic state this is `0`. -/
 def crPart (φ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra := ι (crPartF φ)
 
 lemma crPart_eq_ι_crPartF (φ : 𝓕.FieldOp) : crPart φ = ι (crPartF φ) := rfl