The Cluster's Wick's Theorem

A page about Wick's theorem — the combinatorial tool for computing time-ordered products of operators.

Wick's theorem

Wick's theorem states that the time-ordered product of operators can be written as the normal-ordered product plus all possible contractions. For two operators: T(A B) = :AB: + contraction(A, B). For n operators: T(A_1 ... A_n) = :A_1 ... A_n: + sum of all single contractions + sum of all double contractions + ... + all full contractions. The contraction of two operators is their vacuum expectation value: contraction(A, B) = <0|T(A B)|0> = D_F(x - y) for scalar fields. In the cluster, Wick's theorem states that the edit time-ordered product can be written as the edit normal-ordered product plus all possible edit contractions.

The normal ordering

Normal ordering :O: places all annihilation operators to the right of all creation operators. The vacuum expectation value of a normally ordered product is zero: <0|:O:|0> = 0. In the cluster, the edit normal ordering places all edit annihilation operators to the right of all edit creation operators. The edit vacuum expectation value of a normally ordered edit product is zero.

The contractions

The contraction of two operators A and B is defined as contraction(A, B) = T(A B) - :A B:. For a scalar field, contraction(phi(x), phi(y)) = <0|T(phi(x) phi(y))|0> = D_F(x - y) = integral d^4 k / (2 pi)^4 exp(ik . (x - y)) / (k^2 - m^2 + i epsilon). In the cluster, the edit contraction is the edit Feynman propagator.

The S-matrix expansion

Wick's theorem is essential for computing S-matrix elements in perturbation theory. The interaction picture S-matrix is S = T exp(-i integral d^4 x H_int(x)). Expanding the exponential and applying Wick's theorem gives the Feynman diagrams. In the cluster, Wick's theorem is essential for computing edit S-matrix elements. The edit interaction picture S-matrix is expanded and Wick's theorem gives the edit Feynman diagrams.

This theorem

This page is about Wick's theorem. T(A B) = :AB: + contraction. The contraction is <0|T(A B)|0>. The normal ordering has zero vacuum expectation value. Wick's theorem is used to expand the S-matrix. The Feynman diagrams emerge from Wick's theorem. The theorem is real.