The Wigner representation of classical mechanics, quantization and classical limit

Starting from the Liouvillian formulation of classical physics, it is possible by means of a Fourier transform to introduce the Wigner representation and to derive an operator structure to classical mechanics. The importance of this new representation lies in the fact that it turns out to be the suitable route to establish a general method of quantization directly from the equations of motion without alluding to the existence of Hamiltonian and Lagrangian functions. Following this approach we quantize only the motion of a Brownian particle with non-linear friction in the Markovian approximation—the thermal bath may be quantum or classical—, thus when the bath is classically described we obtain a master equation which reduces to the Caldeira–Leggett equation (vide, A.O. Caldeira and A.J. Leggett, Physica A 121 (1983) 587) for the linear friction case, and when the reservoir is quantum we get an equation reducing to the one found by Caldeira et al. (Phys. Rev. A 40 (1989) 3438). By neglecting the environmental influence, we show that the system can be approximately described by equations of motion in terms of wave function, such as the Schrödinger–Langevin equation and equations of the Caldirola–Kanai type. Finally, to make the present study self-consistent we evaluate the classical limit of these dynamical equations employing a new classical limiting method ℏ→0.