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feat: More notes

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jstoobysmith 2025-02-05 05:44:40 +00:00
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12
HepLean/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean
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@ -23,18 +23,18 @@ namespace FieldOpFreeAlgebra
open FieldStatistic
/-- The super commutor on the creation and annihlation algebra. For two bosonic operators
or a bosonic and fermionic operator this corresponds to the usual commutator
whilst for two fermionic operators this corresponds to the anti-commutator. -/
/-- For a field specification `𝓕`, the super commutator `superCommuteF` is defined as the linear
map `𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra` such that
`superCommuteF (φ₀ᶜ…φₙᵃ) (φ₀'ᶜ…φₙ'ᶜ)` is equal to
`φ₀ᶜ…φₙᵃ * φ₀'ᶜ…φₙ'ᶜ - 𝓢(φ₀ᶜ…φₙᵃ, φ₀'ᶜ…φₙ'ᶜ) φ₀'ᶜ…φₙ'ᶜ * φ₀ᶜ…φₙᵃ`.
The notation `[a, b]ₛca` is used for this super commutator. -/
noncomputable def superCommuteF : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ]
𝓕.FieldOpFreeAlgebra :=
Basis.constr ofCrAnListFBasis ℂ fun φs =>
Basis.constr ofCrAnListFBasis ℂ fun φs' =>
ofCrAnListF (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF (φs' ++ φs)
/-- The super commutor on the creation and annihlation algebra. For two bosonic operators
or a bosonic and fermionic operator this corresponds to the usual commutator
whilst for two fermionic operators this corresponds to the anti-commutator. -/
@[inherit_doc superCommuteF]
scoped[FieldSpecification.FieldOpFreeAlgebra] notation "[" φs "," φs' "]ₛca" => superCommuteF φs φs'
/-!