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 probabilities to alternatives at a fixed moment of time and is not immediately suited to questions of this type. We use the decoherent histories approach to quantum theory to compute the probability of a nonrelativistic particle crossing $x=0\mathrm{}$ during an interval of time. For a system consisting of a single nonrelativistic particle, histories coarse grained according to whether or not they pass through spacetime regions are generally not decoherent, except for very special initial states, and thus probabilities cannot be assigned. Decoherence may, however, be achieved by coupling the particle to an environment consisting of a set of harmonic oscillators in a thermal bath. Probabilities for spacetime coarse grainings are thus calculated by considering restricted density operator propagators of the quantum Brownian motion model. We also show how to achieve decoherence by replicating the system $N$ times and then projecting onto the number density of particles that cross during 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.