Spacetime Probabilities in Nonrelativistic Quantum Mechanics

We demonstrate the existence of spacetime probabilities in nonrelativistic quantum mechanics, that is, quantum mechanical probabilities for a set of alternatives which are associated, not with a surface of constant time, but with spacetime domains with nonzero spatial and temporal width in Newtonian spacetime. We use the criterion that quantum mechanical probabilities can be defined for a set of alternatives if and only if the interference between any two different alternatives vanishes. Although generalized quantum mechanics was formulated on the basis of this criterion, the actual existence of spacetime probabilities has not been known. In this paper we consider a rectangular spacetime domain Ω and introduce a set of spacetime alternatives {Yes, No}: “Yes” is to find a particle in Ω and “No” is the complement to “Yes”. We show that, if the initial amplitude of the particle belongs to a specific class, then the criterion of vanishing interference is met by “Yes” and “No” and spacetime probabilities can be therefore defined for the set {Yes, No}. Owing to the property of the initial amplitude belonging to the class, the resultant probabilities are associated with a clear measurement theoretical meaning.