About the foundation of the Kubo Generalized Cumulants theory. A revisited and corrected approach

TL;DR

This paper revisits and corrects the foundational theory of Kubo's generalized cumulants, clarifying their mathematical properties and potential applications in quantum and statistical physics.

Contribution

It provides a rigorous framework for Kubo's generalized cumulants, resolving longstanding theoretical issues and clarifying their relation to moments in a general setting.

Findings

Different definitions of generalized cumulants are possible for the same moments.

A formal expression for cumulants in terms of operators is derived, independent of the moments' nature.

The paper clarifies the limitations of existing formulas relating cumulants and moments.

Abstract

More than fifty years ago, in a couple of seminal works Kubo introduced the important idea of generalized cumulants, extending to stochastic operators this concept, implicitly introduced by Laplace in 1810. Kubo's idea has been applied in several branches of physics, where the result of the average process is a Lioville operator or an effective time evolution operator for the density matrix of spin systems or the reduced density matrix for boson-fermions etc. Despite this success, the theoretical developments in these Kubo works pose problems that were highlighted many years ago by Fox and van Kampen and never solved. These weaknesses and errors, in particular concerning the factorization property of exponentials of cumulants and the explicit expressions that give generalized cumulants in terms of generalized moments and vice-versa, caused some perplexity (and confusion) about the possible application of this procedure, limiting its use, in practice. In the present paper, we give a sound ground to the approach to cumulant operators, working in a general framework that shows the potentiality of the old Kubo's idea, today not yet fully exploited. It results that for the same moment operators, different definitions of generalized cumulants can be adopted. A general Kubo-Meeron closed-form formula giving cumulant operators in terms of moment operators cannot be obtained, but the reverse one, cumulants in terms of operators, is given and, noticeably, formally it {\em does not} depend on the specific nature of the moments, but just on the definition of the generalized cumulants.