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Cumulant expansions of operator groups of quantum many-particle systems

V. I. Gerasimenko, I. V. Gapyak

TL;DR

The paper develops a nonperturbative framework for quantum many-particle dynamics by constructing cluster (cumulant) expansions of operator groups tied to the von Neumann and Heisenberg equations. It provides generating operators for nonperturbative solutions to the BBGKY hierarchy and its dual for reduced observables, with rigorous existence and convergence results under explicit conditions. The approach unifies state- and observable-centered descriptions through dual cumulant structures and shows how perturbative representations arise as special cases, while offering pathways to quantum kinetic limits. The work lays a rigorous foundation for analyzing correlations and dynamics in systems with finite or infinite particle numbers and suggests extensions to bosonic/fermionic statistics.

Abstract

The article presents a method of cluster expansions for groups of operators associated with the von Neumann equations for states and the Heisenberg equations for observables, aiming to construct generating operators for nonperturbative solutions to the Cauchy problem for hierarchies of evolution equations of many-particle quantum systems.

Cumulant expansions of operator groups of quantum many-particle systems

TL;DR

The paper develops a nonperturbative framework for quantum many-particle dynamics by constructing cluster (cumulant) expansions of operator groups tied to the von Neumann and Heisenberg equations. It provides generating operators for nonperturbative solutions to the BBGKY hierarchy and its dual for reduced observables, with rigorous existence and convergence results under explicit conditions. The approach unifies state- and observable-centered descriptions through dual cumulant structures and shows how perturbative representations arise as special cases, while offering pathways to quantum kinetic limits. The work lays a rigorous foundation for analyzing correlations and dynamics in systems with finite or infinite particle numbers and suggests extensions to bosonic/fermionic statistics.

Abstract

The article presents a method of cluster expansions for groups of operators associated with the von Neumann equations for states and the Heisenberg equations for observables, aiming to construct generating operators for nonperturbative solutions to the Cauchy problem for hierarchies of evolution equations of many-particle quantum systems.

Paper Structure

This paper contains 8 sections, 2 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Cumulants of operator groups of quantum systems
  3. Hierarchies of evolution equations of quantum systems
  4. Nonperturbative solution of the BBGKY hierarchy of quantum systems
  5. Representation of reduced density operators by the perturbation theory
  6. Nonperturbative solution of the hierarchy of evolution equations for reduced observables
  7. Representation of reduced observables by perturbation theory
  8. Conclusion

Key Result

Theorem 1

If $\alpha>e$, the nonperturbative solution of the Cauchy problem of the BBGKY hierarchy (h),(hi) is represented by series expansions (se), whose generating operators are the cumulants of the corresponding order of the groups of operators of the von Neumann equation (grGs). For initial states $F(0)\

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2