Dirac's Method for the Two-Dimensional Damped Harmonic Oscillator in the Extended Phase Space

TL;DR

This paper applies Dirac's method to quantize a two-dimensional damped harmonic oscillator in extended phase space, revealing its singular nature and providing a clear reduced phase space quantum description.

Contribution

It offers a novel application of Dirac's canonical quantization to the damped oscillator in extended phase space, clarifying its constraint structure and quantization process.

Findings

The system is shown to be singular and first-class constrained.

Classical Hamiltonian is proportional to a first-class constraint.

Quantum description reduces to the original quantum Hamiltonian.

Abstract

The system of two-dimensional damped harmonic oscillator is revisited in the extended phase space. It is an old problem already addressed by many authors that we present here in some fresh points of view and carry on smoothly a whole discussion. We show that the system is singular. The classical Hamiltonian is proportional to the first-class constraint. We pursue with the Dirac's canonical quantization procedure by fixing the gauge and provide a reduced phase space description of the system. As result the quantum system is simply modeled by the original quantum Hamiltonian.