Nearly instantaneous alternatives in quantum mechanics

Usual quantum mechanics predicts probabilities for the outcomes of measurements carried out at definite moments of time. However, realistic measurements do not take place in an instant, but are extended over a period of time. The assumption of instantaneous alternatives in usual quantum mechanics is an approximation whose validity can be investigated in the generalized quantum mechanics of closed systems in which probabilities are predicted for spacetime alternatives that extend over time. In this paper we investigate how alternatives extended over time reduce to the usual instantaneous alternatives in the quantum mechanics of a nonrelativistic particle moving in one dimension. Specifically, we consider the coarse-grained alternatives of whether the particle remains to the left, remains to the right, or sometimes crosses the origin in a period of time. We show how the decoherence of this set of spacetime alternatives becomes automatic as the time over which they extend approaches zero and estimate how large this time can be before the interference between the alternatives becomes non-negligible. These results suggest that the time scale over which this kind of coarse graining of quantities such as the center of mass position of a massive body may be extended in time before producing significant interference is much longer than characteristic dynamical time scales. © 1996 The American Physical Society.