A SIMPLE QUANTUM EQUATION FOR DECOHERENCE through interaction with the environment

Within the density matrix formalism, it is shown that a simple way to get decoherence is through the introduction of a “quantum” of time (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 rather simple quantum equation to describe the decoherence effects due to interaction with the environment. Namely, the mere introduction (not of a “time-lattice”, but simply) of a “chronon” allows us to go on from differential to finite-difference equations; and in particular to write down the Scroedinger 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. [A much larger presentation of this work appeared c/o the Lanl Archives as the e-print quant-ph/9706059, and as the preprint IC/98/74, ICTP; Trieste, 1998]. keywords: quantum decoherence, interaction with the environment, quantum measurement theory, finite-difference equations, chronon, Caldirola, density-matrix formalism, Liouville–von Neumann equation, proceedings