A simple quantum (finite-difference) equation for dissipation and decoherence

Within the density matrix formalism, it is shown that a simple way to get decoherence is through the introduction of a quantum of time (or rather of a chronon): thus replacing the differential Liouville-von Neumann equation with a finite-difference version of it. In this way, one is given the possibility of using a very simple quantum equation to describe the decoherence effects due to dissipation, and of partially solving the measurement-problem in quantum mechanics (avoiding any recourse to the wave function collapse). Namely, the mere introduction (not of a time-lattice, but simply) of the chronon allows to go on from differential to finite-difference equations; and in particular to write down the Schroedinger equation (as well as the Liouville-von Neumann equation) in three different ways: retarded, symmetrical, and advanced. One of such three formulations - the retarded one - describes in an elementary way a system which is exchanging (and losing) energy with the environment. In its density-matrix version, indeed, it can be easily shown that all non-diagonal terms go to zero very rapidly.