Cylindrical Wigner measures

TL;DR

This paper develops a unified framework for analyzing the semiclassical behavior of bosonic quantum systems using cylindrical Wigner measures, applicable to both finite and infinite degrees of freedom.

Contribution

It introduces a comprehensive approach to semiclassical analysis of bosonic systems via cylindrical Wigner measures, bridging finite and infinite-dimensional cases.

Findings

Cylindrical Wigner measures coincide with classical measures in infinite-dimensional systems.

The approach applies to relativistic quantum field theories and many-body systems.

Characterization of cylindrical Wigner measures and their properties derived from quantum states.

Abstract

In this paper we study the semiclassical behavior of quantum states acting on the C*-algebra of canonical commutation relations, from a general perspective. The aim is to provide a unified and flexible approach to the semiclassical analysis of bosonic systems. We also give a detailed overview of possible applications of this approach to mathematical problems of both axiomatic relativistic quantum field theories and nonrelativistic many body systems. If the theory has infinitely many degrees of freedom, the set of Wigner measures, i.e. the classical counterpart of the set of quantum states, coincides with the set of all cylindrical measures acting on the algebraic dual of the space of test functions for the field, and this reveals a very rich semiclassical structure compared to the finite-dimensional case. We characterize the cylindrical Wigner measures and the \emph{a priori} properties they inherit from the corresponding quantum states.