Decoherence in the classical limit of histories of a particle coupled to a von Neumann apparatus

Abstract: Using the Gell-Mann and Hartle formalism of generalized quantum mechanics of closed systems, we study the classical limit of coarse-grained spacetime histories and their decoherence. The system under consideration is one-dimensional and consists of a particle coupled to a von Neumann apparatus that performs a measurement of the position of the particle during the finite time interval during which the histories of this system take place. We consider two cases: a free particle and a harmonic oscillator. The real line is divided into intervals of the same length, and coarse-grained histories are defined by the time average of the position of the particle on a given Feynman path to be within one of these intervals. The position of the pointer in each Feynman path correlates with this time average. The class operators for this system have been evaluated, and the decoherence functional shows that these coarse-grained histories do not decohere, not even when initially either the particle or the pointer is in an eigenstate of position. Decoherence is obtained only when the classical limit is taken. Qualitative arguments for decoherence in the classical limit are presented for the case of a general particle potential.