Normal ordering and a Wick-like reduction theorem for fermions with respect to a multi-determinantal reference state

Abstract

We introduce the notion of normal ordering and Wick-like expansion of a product of fermionic creation/annihilation operators with respect to a multi-determinantal reference state Ψ₀. The new normal ordered products possess the following desirable properties: (a) their expectation values with respect to Ψ₀ vanish, and (b) the normal product of N operators does not depend in a special way on N. The analogues of contractions, unlike in the case of a single determinant reference function, can have n creation and n annihilation operators with n ≥ 1. We prove a Wick-like reordering theorem for a product of creation/annihilation operators, which generates a sum of products in the new normal ordering with any number of contractions. The formula can be generalized to cover products of normal ordered products of operators as well.