Extended Hilbert Phase Space and Dissipative Quantum Systems

We suggest an extension of the Hilbert Phase Space formalism, which appears to be naturally suited for application to the dissipative (open) quantum systems, such as those described by the non-stationary (time-dependent) Hamiltonians $H(x,p,t)$. A notion of quantum differential is introduced, highlighting the difference between the quantum and classical propagators. The equation of quantum dynamics of the generalised Wigner function in the extended Hilbert phase space is derived, as well as its classical limit, which serves as the generalisation of the classical Liouville equation for the domain of non-stationary Hamiltonian dynamical systems. This classical limit is then studied at some length and it is shown that in the extended phase space the energy plays the role of the coordinate and time that of the conjugated momentum and not the other way around, as might be suggested by the covariant treatment of these quantities arising from the 4-coordinate and 4-momentum in the relativistic context. Furthermore, the canonical form of the equation obtained suggests that that which is perceived as motion in the ordinary phase space is an equilibrium configuration in the extended phase space.