Groups of Operators for Evolution Equations of Quantum Many-Particle Systems

TL;DR

This paper investigates the properties of operator groups governing the evolution of quantum many-particle systems, introducing cumulant-based expansions for infinite-particle dynamics.

Contribution

It develops a cumulant-based framework for analyzing groups of operators in quantum many-particle evolution equations, unifying hierarchies like von Neumann and BBGKY.

Findings

Cumulants form the basis of expansions for infinite-particle evolution operators.

The work provides a new mathematical approach to quantum many-particle dynamics.

Establishes properties of operator groups for different hierarchies in quantum systems.

Abstract

The aim of this work is to study the properties of groups of operators for evolution equations of quantum many-particle systems, namely, the von Neumann hierarchy for correlation operators, the BBGKY hierarchy for marginal density operators and the dual BBGKY hierarchy for marginal observables. We show that the concept of cumulants (semi-invariants) of groups of operators for the von Neumann equations forms the basis of the expansions for one-parametric families of operators for evolution equations of infinitely many particles.