Dynamics of Correlations of Bose and Fermi Particles

TL;DR

This paper develops a mathematical framework for describing the evolution of correlations in quantum many-particle systems of Bose and Fermi particles, linking various hierarchies of equations and solutions.

Contribution

It derives the von Neumann hierarchy for correlation operators and constructs solutions connecting it with quantum BBGKY hierarchies for the first time.

Findings

Derived the von Neumann hierarchy for correlation operators.

Constructed solutions for initial data with chaos property.

Established links between different quantum hierarchies.

Abstract

We discuss the origin of the microscopic description of correlations in quantum many-particle systems obeying Fermi-Dirac and Bose-Einstein statistics. For correlation operators that give the alternative description of the quantum state evolution of Bose and Fermi particles, we deduce the von Neumann hierarchy of nonlinear equations and construct the solution of its initial-value problem in the corresponding spaces of sequences of trace class operators. The links of constructed solution both with the solution of the quantum BBGKY hierarchy and with the nonlinear BBGKY hierarchy for marginal correlation operators are discussed. The solutions of the Cauchy problems of these hierarchies are constructed for initial data satisfying a chaos property.