Research proof index

For a field specification 𝓕, an element φ of 𝓕.CrAnFieldOp, a list φs of 𝓕.CrAnFieldOp, the following relation holds [φ, 𝓝(φ₀…φₙ)]ₛ = ∑ i, 𝓢(φ, φ₀…φᵢ₋₁) • [φ, φᵢ]ₛ * 𝓝(φ₀…φᵢ₋₁φᵢ₊₁…φₙ). The proof of this result ultimately goes as follows - The definition of normalOrder is used to rewrite 𝓝(φ₀…φₙ) as a scalar multiple of a `ofCrAnList...

Within a proto-operator algebra we have that φ * 𝓝ᶠ(φ₀φ₁…φₙ) = 𝓝ᶠ(φφ₀φ₁…φₙ) + [anpart φ, 𝓝ᶠ(φ₀φ₁…φₙ)]ₛF.

Within a proto-operator algebra, N(φφ₀φ₁…φₙ) = s • N(φ₀…φₖ₋₁φφₖ…φₙ), where s is the exchange sign for φ and φ₀…φₖ₋₁.

For a field specification 𝓕, and φ and ψ elements of 𝓕.FieldOp, then timeContract φ ψ is in the center of 𝓕.WickAlgebra.

For a field specification 𝓕, and φ and ψ elements of 𝓕.FieldOp, if φ and ψ are time-ordered then timeContract φ ψ = [anPart φ, ofFieldOp ψ]ₛ.

For a list φs of 𝓕.FieldOp, then 𝓣(φs) is equal to the sum of - ∑ φsΛ, φsΛ.wickTerm where the sum is over all Wick contraction φsΛ which have no contractions of equal time. - ∑ φsΛ, φsΛ.sign • φsΛ.timeContract * (∑ φssucΛ, φssucΛ.wickTerm), where the first sum is over all Wick contraction φsΛ which only have equal time contractions and the...