Generalized Cumulant Expansion Method

TLDR

The moment‑generating function of stochastic variables defines cumulants, and this concept can be extended to q‑numbers, generalized exponentials, and non‑standard averaging procedures. The study demonstrates that cumulants and their generating functions can be generalized extensively using the formal properties of exponentials. The authors prove foundational theorems and illustrate that the generalized cumulant expansion unifies numerous existing quantum and statistical mechanics techniques.

Abstract

The moment generating function of a set of stochastic variables defines the cumulants or the semi-invariants and the cumulant function. It is possible, simply by formal properties of exponential functions, to generaiize to a great extent the concepts of cumulants and cumulant function. The stochastic variables to be considered need not be ordinary c -numbers but they may be q -numbers such as used in quantum mechanics. The exponential function which defines a moment generating function may be any kind of generalized exponential, for example an ordered exponential with a certain prescription for ordering q -number variables. The definition of average may be greatly generalized as far as the condition is fulfilled that the average of unity is unity. After statements of a few basic theorems these generalizations are discussed here with certain examples of application. This generalized cumulant expansion provides us with a point of view from which many existent methods in quantum mechanics and statistical mechanics can be unified.