Dissipative canonical flows in classical and quantum mechanics

A theory of stochastic flows over the algebra of observables of a dynamical system is presented in which the main objective is to ensure that the over all canonical/symplectic structure on the algebra is preserved. We study bo th classical and quantum systems and the importance of physical interpretat ion in the Stratonovich interpretation is stressed. We find the natural for mulation of quantum dissipative systems to be given in terms of quantum sto chastic calculus. This treatment allows for a physically meaningful treatme nt of both constant and nonlinear dissipation. As an application, we quanti ze a mechanical system with the same nonlinear damping mechanism as the van der Pol oscillator. (C) 1999 American Institute of Physics. [S0022-2488(99 )00206-6].