refactor: improve remarks

HepLean/PerturbationTheory/FieldSpecification/Basic.lean

@@ -28,13 +28,14 @@ These states carry the same field statistic as the field they are derived from.
 remark fieldSpecification_intro := "The raw ingredients of a field theory are:
   - The specification of the fields.
   - Whether each field is a boson or a fermion.
-  - Vertices present.
-  - Coefficents of each vertex.
+  - Vertices present in the Lagrangian.
+  - The coefficent of each vertex.
 
   We call the first two of these ingredients the `FieldSpecification` of the theory. "
 
-/-- A field specification is a type of fields plus a specification of the
-  statistics (fermionic or bosonic) of each field. -/
+/-- A field specification is a type, `Fields`, elements of which are fields
+  present in a theory, and a map `statistics` from `Fields` to `FieldStatistic` which
+  identifies each field as a boson or a fermion. -/
 structure FieldSpecification where
   /-- The type of fields. This also includes anti-states. -/
   Fields : Type
@@ -44,16 +45,16 @@ structure FieldSpecification where
 namespace FieldSpecification
 variable (𝓕 : FieldSpecification)
 
-/-- Incoming asymptotic states are specified by a field and a momentum. -/
+/-- An incoming asymptotic state is a field and a momentum. -/
 def IncomingAsymptotic : Type := 𝓕.Fields × Lorentz.Contr 4
 
-/-- Outgoing asymptotic states are specified by a field and a momentum. -/
+/-- An outgoing asymptotic states is a field and a momentum. -/
 def OutgoingAsymptotic : Type := 𝓕.Fields × Lorentz.Contr 4
 
-/-- States specified by a field and a space-time position. -/
+/-- A position state is a field and a space-time position. -/
 def PositionStates : Type := 𝓕.Fields × SpaceTime
 
-/-- The combination of asymptotic states and position states. -/
+/-- The type States is the inductive type combining the asymptotic states and position states. -/
 inductive States (𝓕 : FieldSpecification) where
   | inAsymp : 𝓕.IncomingAsymptotic → 𝓕.States
   | position : 𝓕.PositionStates → 𝓕.States

HepLean/PerturbationTheory/WickContraction/TimeContract.lean

@@ -43,6 +43,14 @@ lemma timeContract_insertAndContract_none (𝓞 : 𝓕.ProtoOperatorAlgebra)
   ext a
   simp
 
+/-- For `φsΛ` a Wick contraction for `φs = φ₀…φₙ`, the time contraction
+  `(φsΛ ↩Λ φ i (some j)).timeContract 𝓞` is equal to the multiple of
+- the time contraction of `φ` with `φⱼ` if `i < i.succAbove j` else
+    `φⱼ` with `φ`.
+- `φsΛ.timeContract 𝓞`.
+This follows from the fact that `(φsΛ ↩Λ φ i (some j))` has one more contracted pair than `φsΛ`,
+corresponding to `φ` contracted with `φⱼ`. The order depends on whether we insert `φ` before
+or after `φⱼ`. -/
 lemma timeConract_insertAndContract_some (𝓞 : 𝓕.ProtoOperatorAlgebra)
     (φ : 𝓕.States) (φs : List 𝓕.States)
     (φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
@@ -69,8 +77,8 @@ lemma timeConract_insertAndContract_some_eq_mul_contractStateAtIndex_lt
     (ht : 𝓕.timeOrderRel φ φs[k.1]) (hik : i < i.succAbove k) :
     (φsΛ ↩Λ φ i (some k)).timeContract 𝓞 =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (φsΛ.uncontracted.filter (fun x => x < k))⟩)
-    • (𝓞.contractStateAtIndex φ [φsΛ]ᵘᶜ
-    ((uncontractedStatesEquiv φs φsΛ) (some k)) * φsΛ.timeContract 𝓞) := by
+    • (𝓞.contractStateAtIndex φ [φsΛ]ᵘᶜ ((uncontractedStatesEquiv φs φsΛ) (some k)) *
+      φsΛ.timeContract 𝓞) := by
   rw [timeConract_insertAndContract_some]
   simp only [Nat.succ_eq_add_one, Fin.getElem_fin, ite_mul, instCommGroup.eq_1,
     ProtoOperatorAlgebra.contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,
@@ -105,7 +113,7 @@ lemma timeConract_insertAndContract_some_eq_mul_contractStateAtIndex_not_lt
     (φsΛ ↩Λ φ i (some k)).timeContract 𝓞 =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (φsΛ.uncontracted.filter (fun x => x ≤ k))⟩)
     • (𝓞.contractStateAtIndex φ [φsΛ]ᵘᶜ
-    ((uncontractedStatesEquiv φs φsΛ) (some k)) * φsΛ.timeContract 𝓞) := by
+      ((uncontractedStatesEquiv φs φsΛ) (some k)) * φsΛ.timeContract 𝓞) := by
   rw [timeConract_insertAndContract_some]
   simp only [Nat.succ_eq_add_one, Fin.getElem_fin, ite_mul, instCommGroup.eq_1,
     ProtoOperatorAlgebra.contractStateAtIndex, uncontractedStatesEquiv, Equiv.optionCongr_apply,