Poisson bracket operator

TL;DR

The paper introduces the Poisson bracket operator as a unified quantum-classical tool for describing operator dynamics, applicable to standard and non-standard systems, and consistent with classical and quantum mechanics principles.

Contribution

It defines a new Poisson bracket operator using quantum analysis, deriving a quantum canonical equation that unifies classical and quantum dynamics.

Findings

The quantum canonical equation reduces to the Heisenberg equation in standard cases.

The operator formalism applies to systems with mixed classical and quantum variables.

The dynamics satisfy Ehrenfest theorem and conserve energy and momentum.

Abstract

We introduce the Poisson bracket operator which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classical limit. Using this, we derive the quantum canonical equation which describes the time evolution of operators. In the standard applications of quantum mechanics, the quantum canonical equation is equivalent to the Heisenberg equation. At the same time, this equation is applicable to c-number canonical variables and then coincides with the canonical equation in classical mechanics. Therefore the Poisson bracket operator enables us to describe classical and quantum behaviors in a unified way. Moreover, the quantum canonical equation is applicable to non-standard system where the Heisenberg equation is not defined. As an example, we consider the application to the system where a c-number and a q-number particles coexist. The derived dynamics satisfies the Ehrenfest theorem and the energy and momentum conservations.