A Simple Quantum Equation for Decoherence through Interaction with the Environment

Abstract

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 Liouvillevon Neumann (LvN) 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.In other words, 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 Schrödinger equation (as well as the LvN equation) in three different ways: “retarded”, “symmetrical”, and “advanced”. One of such three formulations—the retarded one—describes in an elementary way a system that is exchanging (and losing) energy with the environment. In its densitymatrix version, indeed, it can be easily shown that all nondiagonal terms go to zero very rapidly. [A much larger presentation of this work appeared c/o the Laid Archives as the e-print quant-ph/ 9706059, and as the preprint IC/98/74, ICTP; Trieste, 1998].KeywordsQuantum decoherenceInteraction with the environmentQuantum measurement theoryFinite-difference equationsChrononCaldirolaDensity-matrix formalismLiouville—von Neumann equationProceedings