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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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HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Lemmas.lean
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@ -347,7 +347,7 @@ noncomputable def contractStateAtIndex (φ : 𝓕.FieldOp) (φs : List 𝓕.Fiel
/--
For a field specification `𝓕`, a `φ` in `𝓕.FieldOp` and a list `φs` of `𝓕.FieldOp`
the following relation holds in the algebra `𝓕.FieldOpAlgebra`,
`φ * 𝓝(φ₀φ₁…φₙ) = 𝓝(φφ₀φ₁…φₙ) + ∑ i, (𝓢(φ,φ₀φ₁…φᵢ₋₁) • [anPartF φ, φᵢ]ₛ) * 𝓝(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)`.
`φ * 𝓝(φ₀φ₁…φₙ) = 𝓝(φφ₀φ₁…φₙ) + ∑ i, (𝓢(φ,φ₀φ₁…φᵢ₋₁) • [anPart φ, φᵢ]ₛ) * 𝓝(φ₀φ₁…φᵢ₋₁φᵢ₊₁…φₙ)`.
The proof of ultimetly goes as follows:
- `ofFieldOp_eq_crPart_add_anPart` is used to split `φ` into its creation and annihilation parts.