Dissipative Quantum Mechanics: The Generalization of the Canonical Quantization and von Neumann Equation

TL;DR

This paper introduces a generalized quantum framework for dissipative systems, extending canonical quantization and the von Neumann equation by incorporating a nonassociative operator W to address quantum ambiguities in dissipative models.

Contribution

It proposes a novel operator algebra with a nonassociative operator W, enabling consistent quantum descriptions of dissipative systems within an extended canonical framework.

Findings

Extended canonical commutation relations with operator W.

Generalized von Neumann equation compatible with dissipative dynamics.

Resolution of quantum ambiguities in dissipative models.

Abstract

The dissipative models in string theory can have more broad range of application: 1) Noncritical strings are dissipative systems in the "coupling constant" phase space. 2) Bosonic string in the affine-metric curved space is dissipative system. But the quantum descriptions of the dissipative systems have well known ambiguities. In order to solve the problems of the quantum description of dissipative systems we suggest to introduce an operator W in addition to usual (associative) operators. The suggested operator algebra does not violate Heisenberg algebra because we extend the canonical commutation relations by introducing an operator W of the nonholonomic quantities in addition to the usual (associative) operators of usual (holonomic) coordinate -momentum functions. To satisfy the generalized commutation relations the operator W must be nonassociative nonLieble (does not satisfied the Jacobi identity) operator. As the result of these properties the total time derivative of the multiplication and commutator of the operators does not satisfies the Leibnitz rule. This lead to compatibility of quantum equations of motion for dissipative systems and canonical commutation relations. The suggested generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems.