Decoherence and classical predictability of phase-space histories.

We consider the decoherence of phase space histories in quantum brownian motion models, consisting of a particle moving under a potential V (x) in contact with a heat bath of temperature T and dissipation constant γ in the Markovian regime. The evolution of the density operator for this system is thus described by a non-unitary master equation. The phase space histories are described by quasiprojectors consisting of gaussian density matrices smeared over large phase space cells. They are characterized by the size [Γ] of the phase space cell together with the size [M ] of the margin (the region at the boundary of Γ in which the Weyl symbol of the projector goes from 1 to 0). By generalizing an earlier result of Hagedorn on the unitary evolution of coherent states, we show that an initial Gaussian density matrix remains approximately Gaussian under non-unitary time evolution, and derive a bound giving the validity of this approximation. This result is then used, following earlier work of Omnès to compute the time evolution of the phase space projectors under the master equation: The evolution of an initial projector P onto a cell Γ is approximately equal to another phase space projector P ′ onto a cell Γ, where Γ