Rigorous second-order cumulant expansion

Abstract

In several occasions (typically in the quantum-mechanical cases) the random variables of a many-body stochastic process are q-number variables rather than c-numbers. The canonical ensemble part i t ion function, which can be thought of as the moment generating function for the interaction part of the hamil tonian H of the system, is usually computed through a cumulang cluster expansion (1). Suck an expansion however is always performed under the assumption that the interaction part of H might be considered small enough to be kept only up to the first perturbative order. The motivation is twofold. On one hand the interaction energy is indeed often muck smaller than the single-particle energy. On the other hand the interaction part of H is very frequently restricted to a linear combination of pair interaction terms, and the lat ter circumstance allows a very convenient and elegant procedure whereby the problem is reduced to a two-particle problem. Briefly, the argument goes as follows (2). Let