Quantum mechanics of measurements distributed in time. A path-integral formulation

Consider measurements that provide information about the position of a nonrelativistic, one-dimensional, quantum-mechanical system. An outstanding question in quantum mechanics asks how to analyze measurements distributed in time—i.e., measurements that provide information about the position at more than one time. I develop a formulation in terms of a path integral and show that it applies to a large class of measurements distributed in time. For measurements in this class, the path-integral formulation provides the joint statistics of a sequence of measurements. Specialized to the case of instantaneous position measurements, the path-integral formulation breaks down into the conventional machinery of nonrelativistic quantum mechanics: a system quantum state evolving in time according to two rules—between measurements, unitary evolution, and at each measurement, ‘‘collapse of the wave function’’ (‘‘reduction of the state vector’’). For measurements distributed in time, the path-integral formulation has no similar decomposition; the notion of a system quantum state evolving in time has no place.