Connecting dissipation and noncommutativity: A Bateman system case study

TL;DR

This paper investigates quantum effects on Bateman oscillators in noncommutative space, revealing a duality where noncommutativity can induce or cancel damping effects, with implications for understanding dissipation in quantum systems.

Contribution

It introduces a novel quantization approach for Bateman oscillators in noncommutative space, showing how noncommutativity can renormalize damping and create a duality with dissipative systems.

Findings

Renormalized damping can be non-zero even if bare damping is zero.

Noncommutative parameter can be tuned to cancel damping effects.

Duality between dissipative commutative and non-dissipative noncommutative theories.

Abstract

Quantum effects on a pair of Bateman oscillators embedded in an ambient noncommutative space (Moyal plane) is analyzed using both path integral and canonical quantization schemes within the framework of Hilbert-Schmidt operator formulation. We adopt a method which is distinct from the one which employs 't Hooft's scheme of quantization, carried out earlier in the literature where the ambient space was taken to be commutative. Our quantization shows that we end up finally again with a Bateman system except that the damping factor undergoes renormalization. The corresponding expression shows that the renormalized damping factor can be non-zero even if "bare" one is zero to begin with. Conversely, the noncommuatative parameter $\theta$, taken to be a free one now, can be fine-tuned to get a vanishing renormalized damping factor. This indicates a duality between dissipative commutative theory and non-dissipative noncommutative theory.