Decoherent histories for a particle coupled to a von Neumann apparatus

摘要

Using the Gell-Mann and Hartle formalism of generalized quantum mechanics ofclosed systems, we study coarse-grained decoherent histories. The system underconsideration is one-dimensional and consists of a particle coupled to a vonNeumann apparatus that measures its position. The particle moves in a quadraticpotential; in particular we consider a driven harmonic oscillator. The realline is divided into intervals of the same length, and coarse-grained historiesare defined by the arithmetic average of the initial and final position of theparticle to be within one of these intervals. The position of the pointercorrelates with this arithmetic average. In addition a constant term is addedto this average, which is a result of the presence of a driving force on theoscillator. We investigate decoherence for such histories via the decoherencefunctional for the particle-apparatus. An exact expression for this functionalhas been derived. If the particle or the pointer of the apparatus is in anexact position state initially, then exact decoherence ensues, and the relativeprobability for such histories is obtained. If the initial state of theparticle, which is centered at some arbitrary point in the $x$-axis, is narrowcompared to the width of the intervals on the $x$-axis, then we obtainapproximate decoherence. The same result follows if the initial state of thepointer is narrow. In these two situations, we evaluate expressions for thedecoherence functional qualitatively and quantitatively. In the latter case weobtain the first two terms in an expansion in powers of the width of theinitial state of the particle or the pointer respectively. Approximate relativeprobabilities can be assigned in this case, and we have obtained an expansionup to second order in powers of the width of the initial state for anexpression for an upper bound of these probabilities.