Quantization of Non-Hamiltonian Systems

TL;DR

This paper introduces a generalized Weyl quantization method that extends to non-Hamiltonian and dissipative systems, enabling their consistent quantization and analysis.

Contribution

It proposes a new quantization framework for non-Hamiltonian systems, broadening the scope of Weyl quantization beyond traditional Hamiltonian mechanics.

Findings

Provides a generalized quantization method applicable to dissipative systems

Derives standard Weyl quantization as a special case

Demonstrates the approach with examples like damped harmonic oscillator

Abstract

In this paper a generalization of Weyl quantization which maps a dynamical operator in a function space to a dynamical superoperator in an operator space is suggested. Quantization of dynamical operator, which cannot be represented as Poisson bracket with some function, is considered. The usual Weyl quantization of observables can be derived as a specific case of suggested quantization if dynamical operator is an operator of multiplication on a function. This approach allows to define consistent Weyl quantization of non-Hamiltonian and dissipative systems. Examples of the harmonic oscillator with friction and a system which evolves by Fokker-Planck-type equation are considered.