Many-body systems with SU(1,1) dynamical symmetry: from dynamics to thermodynamics based on the trace formula

TL;DR

This paper develops a trace formula approach for quantum many-body systems with SU(1,1) symmetry, linking group theory, thermodynamics, and dynamics, applicable to both equilibrium and nonequilibrium states.

Contribution

It introduces a trace formula based on Lie group representations that unifies thermodynamic analysis for systems with compact and non-compact symmetries, especially SU(1,1).

Findings

Derived a trace formula consistent with Weyl character formula for compact groups.

Established convergence conditions for traces in non-compact SU(1,1) cases.

Proposed a new method to study thermodynamics of quantum many-body systems.

Abstract

For a quantum (many-body) system with dynamical symmetry described by a given Lie group, we study the trace of exponential operators with complex coefficients in one of the irreducible subspaces in terms of the boson realization. By using this approach, for compact groups, we obtain the result of the trace that is consistent with the well-known Weyl character formula. For non-compact groups (with SU(1,1) as an application), convergent condition of the trace is also obtained. This approach may be a powerful tool to study the thermodynamics of quantum (many-body) systems in equilibrium state or nonequilibrium processes.