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docs: Notes for normal-ordered Wicks

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jstoobysmith 2025-02-07 09:56:37 +00:00
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8
HepLean/PerturbationTheory/WickContraction/InsertAndContract.lean
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@ -284,6 +284,14 @@ lemma insert_fin_eq_self (φ : 𝓕.FieldOp) {φs : List 𝓕.FieldOp}
use z
rfl
/-- For a list `φs` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
`𝓕.FieldOp`, a `i ≤ φs.length` a sum over
Wick contractions of `φs` with `φ` inserted at `i` is equal to the sum over Wick contractions
`φsΛ` of just `φs` and the sum over optional uncontracted elements of the `φsΛ`.
I.e. `∑ (φsΛ : WickContraction (φs.insertIdx i φ).length), f φsΛ` is equal to
`∑ (φsΛ : WickContraction φs.length), ∑ (k : Option φsΛ.uncontracted), f (φsΛ ↩Λ φ i k) `.
where `(φs.insertIdx i φ)` is `φs` with `φ` inserted at position `i`. -/
lemma insertLift_sum (φ : 𝓕.FieldOp) {φs : List 𝓕.FieldOp}
(i : Fin φs.length.succ) [AddCommMonoid M] (f : WickContraction (φs.insertIdx i φ).length → M) :
∑ c, f c =