Wick's Theorem

Wick's theorem expands a time-ordered product of free-field operators as a normal-ordered product plus the sum of all possible contractions. If the contraction of two operators is denoted by

then the four-operator bosonic case is

Fermionic operators acquire signs from the permutations needed to form the contractions. The theorem reduces free-field correlation functions to two-point functions and is the operator counterpart of the Wick-Isserlis formula. Generalized Wick theorems replace vacuum normal ordering by contractions defined relative to another reference state, including particle-hole and multireference formulations.