Physlib

Physlib.QFT.PerturbationTheory.WickAlgebra.WicksTheorem

Wick's theorem

This file constrains the time-dependent version of Wick's theorem for lists of fields containing both fermions and bosons.

Wick's theorem is related to Isserlis' theorem in mathematics.

Wick terms

Wick's theorem

2 declarations

theorem

Φ=Φ′\Phi = \Phi' implies equality of the Wick expansion sums

Let F\mathcal{F} be a field specification. For any two lists of field operators Φ,Φ′∈FieldOp(F)\Phi, \Phi' \in \text{FieldOp}(\mathcal{F}), if Φ=Φ′\Phi = \Phi', then the sum of all Wick terms over the set of Wick contractions is equal for both lists: ∑Λ∈WickContraction(∣Φ∣)wickTerm(Λ,Φ)=∑Λ′∈WickContraction(∣Φ′∣)wickTerm(Λ′,Φ′) \sum_{\Lambda \in \text{WickContraction}(|\Phi|)} \text{wickTerm}(\Lambda, \Phi) = \sum_{\Lambda' \in \text{WickContraction}(|\Phi'|)} \text{wickTerm}(\Lambda', \Phi') where ∣Φ∣|\Phi| denotes the length of the list of field operators and wickTerm(Λ,Φ)\text{wickTerm}(\Lambda, \Phi) is the summand in the Wick expansion corresponding to a specific contraction Λ\Lambda.

theorem

Wick's Theorem: T(∏ϕi)=∑ΛwickTerm(Λ)\mathcal{T}(\prod \phi_i) = \sum_{\Lambda} \text{wickTerm}(\Lambda)

For a given field specification F\mathcal{F} and a list of field operators Φ=[ϕ0,ϕ1,…,ϕn−1]\Phi = [\phi_0, \phi_1, \dots, \phi_{n-1}], the time-ordered product of these operators in the Wick algebra W(F)\mathcal{W}(\mathcal{F}) is equal to the sum over all possible Wick contractions Λ\Lambda of their corresponding Wick terms: T(ϕ0ϕ1…ϕn−1)=∑Λ∈WickContraction(n)wickTerm(Λ,Φ) \mathcal{T}(\phi_0 \phi_1 \dots \phi_{n-1}) = \sum_{\Lambda \in \text{WickContraction}(n)} \text{wickTerm}(\Lambda, \Phi) where T\mathcal{T} is the time-ordering linear map and wickTerm(Λ,Φ)\text{wickTerm}(\Lambda, \Phi) is the summand associated with the contraction Λ\Lambda, including the appropriate fermion sign factors, contractions, and normal-ordered uncontracted fields.