Quantum Liouville Equation

Abstract

The symmetrized product of quantum observables is defined. It is seen as consisting of ordinary multiplication followed by application of the superoperator that orders the operators of coordinate and momentum. This superoperator is defined in the way that allows obstruction free quantization of algebra of observables as well as introduction of operator version of the Poisson bracket. It is shown that this bracket has all properties of the Lie bracket and that it can substitute the commutator in the von Neumann equation leading to quantum Liouville equation.

Probability representation of quantum mechanics and star product quantization

“…The dynamic of quantum particles was described by the Kolmogorov equations for non-negative propagators in the tomography representation in [34]. The symmetrized product of quantum observables is defined in [35].…”