Poisson bracket operator

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.