A simple quantum (finite-difference) equation for dissipation anddecoherence

摘要

Within the density matrix formalism, it is shown that a simple wayto get decoherence is through the introduction of a "quantum" of time (or ratherof a chronon): thus replacing the differential Liouville-von Neumann equation witha finite-difference version of it. In this way, one is given the possibility of using avery simple quantum equation to describe the decoherence effects due to dissipation,and of partially solving the measurement-problem in quantum mechanics (avoidingany recourse to the wave function collapse). Namely, the mere introduction (notof a "time-lattice", but simply) of the "chronon" allows us to go on from differen-tial to finite-difference equations; and in particular to write down the Schroedingerequation (as well as the Liouville-von Neumann equation) in three different ways:"retarded", "symmetrical", and "advanced". One of such three formulationstheretarded one—describes in an elementary way a system which is exchanging (andlosing) energy with the environment. In its density-matrix version, indeed, it canbe easily shown that all non-diagonal terms go to zero very rapidly.