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feat: Add Wick terms

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jstoobysmith 2025-02-05 08:52:14 +00:00
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HepLean/PerturbationTheory/FieldOpAlgebra/StaticWickTheorem.lean
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@ -3,9 +3,10 @@ Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.PerturbationTheory.FieldOpAlgebra.NormalOrder.WickContractions
import HepLean.PerturbationTheory.WickContraction.Sign.InsertNone
import HepLean.PerturbationTheory.WickContraction.Sign.InsertSome
import HepLean.PerturbationTheory.WickContraction.StaticContract
import HepLean.PerturbationTheory.FieldOpAlgebra.WicksTheorem
import HepLean.Meta.Remark.Basic
/-!
# Static Wick's theorem
@ -29,11 +30,16 @@ lemma static_wick_theorem_nil : ofFieldOpList [] = ∑ (φsΛ : WickContraction
/--
The static Wicks theorem states that
`φ₀…φₙ` is equal to the sum of
`φsΛ.1.sign • φsΛ.1.staticContract * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)`
`φ₀…φₙ` is equal to
`∑ φsΛ, φsΛ.1.sign • φsΛ.1.staticContract * 𝓝(ofFieldOpList [φsΛ.1]ᵘᶜ)`
over all Wick contraction `φsΛ`.
This is compared to the ordinary Wicks theorem in which `staticContract` is replaced with
`timeContract`.
The proof is via induction on `φs`. The base case `φs = []` is handled by `static_wick_theorem_nil`.
The inductive step works as follows:
- The proof considers `φ₀…φₙ` as `φ₀(φ₁…φₙ)` and use the induction hypothesis on `φ₁…φₙ`.
- It also uses `ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum`
-/
theorem static_wick_theorem : (φs : List 𝓕.FieldOp) →
ofFieldOpList φs = ∑ (φsΛ : WickContraction φs.length),