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

In this paper, 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 {em 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, without having recourse to any statistical approach). Namely, the mere introduction (not of a "time-lattice", but simply) of the ``chronon" allows us 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 em 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.