On Wigner functions and a damped star product in dissipative phase-space quantum mechanics

TL;DR

This paper compares a new damped star product model for dissipative quantum mechanics with existing phase-space methods, extending it to Wigner functions and revealing that the modified equations of motion align with classical dynamics.

Contribution

It introduces a novel damped star product framework for dissipative quantum systems and extends it to Wigner functions, connecting quantum dissipation with classical equations of motion.

Findings

The damped star product relates to a complex Hamiltonian.

Wigner functions satisfy classical equations of motion under this framework.

The approach effectively models dissipation in phase-space quantum mechanics.

Abstract

Dito and Turrubiates recently introduced an interesting model of the dissipative quantum mechanics of a damped harmonic oscillator in phase space. Its key ingredient is a non-Hermitian deformation of the Moyal star product with the damping constant as deformation parameter. We compare the Dito-Turrubiates scheme with phase-space quantum mechanics (or deformation quantization) based on other star products, and extend it to incorporate Wigner functions. The deformed (or damped) star product is related to a complex Hamiltonian, and so necessitates a modified equation of motion involving complex conjugation. We find that with this change the Wigner function satisfies the classical equation of motion. This seems appropriate since non-dissipative systems with quadratic Hamiltonians share this property.