DECOHERENT HISTORIES APPROACH TO THE ARRIVAL TIME PROBLEM

What is the probability of a particle entering a given region of space at any time between t(1) and t(2)? Standard quantum theory assigns pr obabilities to alternatives at a fixed moment of time and is not immed iately suited to questions of this type. We use the decoherent histori es approach to quantum theory to compute the probability of a nonrelat ivistic particle crossing x=0 during an interval of time. For a system consisting of a single nonrelativistic particle, histories coarse gra ined according to whether or not they pass through spacetime regions a re generally not decoherent, except for very special initial states, a nd thus probabilities cannot be assigned. Decoherence may, however, be achieved by coupling the particle to an environment consisting of a s et of harmonic oscillators in a thermal bath. Probabilities for spacet ime coarse grainings are thus calculated by considering restricted den sity operator propagators of the quantum Brownian motion model. We als o show how to achieve decoherence by replicating the system N times an d then projecting onto the number density of particles that cross duri ng a given time interval, and this gives an alternative expression for the crossing probability. The latter approach shows that the relative frequency for histories is approximately decoherent for sufficiently large N, a result related to the Finkelstein-Graham-Hartle theorem.