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arXiv · 0810.3893

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

B. Belchev·M. A. Walton
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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.

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B. Belchev, M. A. Walton. 2008-10-21. On Wigner functions and a damped star product in dissipative phase-space quantum mechanics. https://doi.org/10.1016/j.aop.2008.10.009

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