Docs: Documentation related to Wick contraction (#287)

* refactor: Fix field struct defn.

* rename: FieldStruct to FieldSpecification

* feat: Add examples of field specifications

* docs: Slight improvement of module docs

* refactor: Rename CreateAnnihilate

* docs: Algebras

* Update CrAnStates.lean

HepLean/PerturbationTheory/FieldStatistics/ExchangeSign.lean

@@ -8,6 +8,16 @@ import HepLean.PerturbationTheory.FieldStatistics.Basic
 
 # Exchange sign for field statistics
 
+Suppose we have two fields `φ` and `ψ`, and the term `φψ`, if we swap them
+`ψφ`, we may pick up a sign. This sign is called the exchange sign.
+This sign is `-1` if both fields `ψ` and `φ` are fermionic and `1` otherwise.
+
+In this module we define the exchange sign for general field statistics,
+and prove some properties of it. Importantly:
+- It is symmetric `exchangeSign_symm`.
+- When multiplied with itself it is `1` `exchangeSign_mul_self`.
+- It is a cocycle `exchangeSign_cocycle`.
+
 -/
 
 namespace FieldStatistic
@@ -47,16 +57,17 @@ def exchangeSign : FieldStatistic →* FieldStatistic →* ℂ where
   Defined to be `-1` if both field statistics are `fermionic` and `1` otherwise. -/
 scoped[FieldStatistic] notation "𝓢(" a "," b ")" => exchangeSign a b
 
+/-- The exchange sign is symmetric. -/
+lemma exchangeSign_symm (a b : FieldStatistic) : 𝓢(a, b) = 𝓢(b, a) := by
+  fin_cases a <;> fin_cases b <;> rfl
+
 @[simp]
 lemma exchangeSign_bosonic (a : FieldStatistic) : 𝓢(a, bosonic) = 1 := by
   fin_cases a <;> rfl
 
 @[simp]
 lemma bosonic_exchangeSign (a : FieldStatistic) : 𝓢(bosonic, a) = 1 := by
-  fin_cases a <;> rfl
-
-lemma exchangeSign_symm (a b : FieldStatistic) : 𝓢(a, b) = 𝓢(b, a) := by
-  fin_cases a <;> fin_cases b <;> rfl
+  rw [exchangeSign_symm, exchangeSign_bosonic]
 
 lemma exchangeSign_eq_if (a b : FieldStatistic) :
     𝓢(a, b) = if a = fermionic ∧ b = fermionic then - 1 else 1 := by
@@ -75,6 +86,7 @@ lemma exchangeSign_ofList_cons (a : FieldStatistic)
     𝓢(a, ofList s (φ :: φs)) = 𝓢(a, s φ) * 𝓢(a, ofList s φs) := by
   rw [ofList_cons_eq_mul, map_mul]
 
+/-- The exchange sign is a cocycle. -/
 lemma exchangeSign_cocycle (a b c : FieldStatistic) :
     𝓢(a, b * c) * 𝓢(b, c) = 𝓢(a, b) * 𝓢(a * b, c) := by
   fin_cases a <;> fin_cases b <;> fin_cases c <;> simp