Quantization of Damped Mechanical Systems Based on the Doubling of Degrees of Freedom

Quantizing dissipative systems like damped harmonic oscillators remains an ongoing challenge in theoretical physics due to their non-conservative nature. This study addresses this issue by employing the method of doubling the degrees of freedom, transforming an open, dissipative system into a closed, conservative one through the introduction of a time-reversed mirror system. The aim is to derive a quantum model consistent with classical mechanics using the Hamilton-Jacobi formalism and to formulate an equivalent Schrödinger equation. Through canonical transformations and analytical solutions, we demonstrate that damping effectively vanishes in the quantum model. These results offer new insights into the quantization of non-conservative systems and contribute to the advancement of quantum open-system theory.