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refactor: More docs for Wick's theorems

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jstoobysmith 2025-02-10 10:40:07 +00:00
parent 4096010e70
commit b30a49d7db
8 changed files with 38 additions and 38 deletions
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8
HepLean/PerturbationTheory/FieldOpAlgebra/StaticWickTerm.lean
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@ -31,8 +31,8 @@ noncomputable section
def staticWickTerm {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 𝓕.FieldOpAlgebra :=
φsΛ.sign • φsΛ.staticContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
/-- For the empty list `[]` of `𝓕.FieldOp`, the `staticWickTerm` of the empty Wick contraction
`empty` of `[]` (its only Wick contraction) is `1`. -/
/-- For the empty list `[]` of `𝓕.FieldOp`, the `staticWickTerm` of the Wick contraction
corresponding to the empty set `∅` (the only Wick contraction of `[]`) is `1`. -/
@[simp]
lemma staticWickTerm_empty_nil :
staticWickTerm (empty (n := ([] : List 𝓕.FieldOp).length)) = 1 := by
@ -116,9 +116,9 @@ lemma staticWickTerm_insert_zero_some (φ : 𝓕.FieldOp) (φs : List 𝓕.Field
For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, the following relation
holds
`φ * φsΛ.staticWickTerm = ∑ k, (φsΛ ↩Λ φ i k).wickTerm`
`φ * φsΛ.staticWickTerm = ∑ k, (φsΛ ↩Λ φ 0 k).staticWickTerm`
where the sum is over all `k` in `Option φsΛ.uncontracted` (so either `none` or `some k`).
where the sum is over all `k` in `Option φsΛ.uncontracted`, so `k` is either `none` or `some k`.
The proof of proceeds as follows:
- `ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum` is used to expand `φ 𝓝([φsΛ]ᵘᶜ)` as