Star exponentials and Wigner functions for time-dependent harmonic oscillators

TL;DR

This work develops a deformation-quantization treatment of the time-dependent harmonic oscillator with time-varying mass and frequency, focusing on the Wigner distribution and the star exponential. It shows that the star exponential can be computed from the quantum propagator and, via a suitable scaling and a nonlinear time variable, reduces to a simple harmonic oscillator with a quantized Lewis-Riesenfeld invariant, yielding time-dependent phases consistent with the Schrödinger picture. The diagonal Wigner functions arise from a Fourier-Dirichlet expansion and link invariant eigenvalues to phase-space structure. The results cover Caldirola-Kanai and time-dependent frequency oscillators, aligning with prior methods and suggesting extensions to damping and field-theoretic contexts.

Abstract

In this paper, we address the Wigner distribution and the star exponential function for a time-dependent harmonic oscillator for which the mass and the frequency terms are considered explicitly depending on time. To such an end, we explore the connection between the star exponential, naturally emerging within the context of deformation quantization, and the propagators constructed through the path integral formalism. In particular, the Fourier-Dirichlet expansion of the star exponential implies a distinctive quantization of the Lewis-Riesenfeld invariant. Further, by introducing a judicious time variable, we recovered a time-dependent phase function associated with the Lewis-Riesenfeld construction of the standard Schrödinger picture. In particular, we applied our results to the cases of the Caldirola-Kanai and the time-dependent frequency harmonic oscillators, recovering relevant results previously reported in the literature.