Geometrical Properties of Cumulant Expansions

TL;DR

This paper explores the geometric structure of cumulants in solid-state physics, providing new insights into their transformations, wave operators, and connections to existing methods like Faddeev's equations and the method of increments.

Contribution

It introduces a geometric framework for cumulants, derives their transformation properties, and connects cumulant wave operators to established quantum methods.

Findings

Derived general expressions for cumulant transformations

Represented cumulant wave operators via path integrals in Hilbert space

Connected cumulant Faddeev equations to the method of increments

Abstract

Cumulants represent a natural language for expressing macroscopic properties of a solid. We show that cumulants are subject to a nontrivial geometry. This geometry provides an intuitive understanding of a number of cumulant relations which had been obtained so far by using algebraic considerations. We give general expressions for their infinitesimal and finite transformations and represent a cumulant wave operator through an integration over a path in the Hilbert space. Cases are investigated where this integration can be done exactly. An expression of the ground-state wavefunction in terms of the cumulant wave operator is derived. In the second part of the article we derive the cumulant counterpart of Faddeev`s equations and show its connection to the method of increments.