Deformation quantization of linear dissipative systems

A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of a non-stationary Poisson bracket, i.e. the corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization procedure involving a non-stationary star-product * t and an " extended " operator of time derivative D t = ∂ t + · · · , differentiating the * t-product. As in the usual case, the * t-algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional Tr t. Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.