The difficulties in the mathematical definition of path integrals are overcome in the theory of continuous quantum measurements

Any real physical process takes place in an external (with respect to the investigated system) medium whose state reflects in one form or another information about the behavior of the system. Therefore, a real model of any process must include a description of the measurement process. This has the consequence that the dynamics of a real system contains both quantum and classical elements. Mathematically, a quantum system subject to continuous measurement can be described by a restricted path integral. The Feynman path integral that is usually employed is an idealization that is not correct in all cases. In a real situation in which the path integral is restricted, the difficulties associated with the mathematical definition of the path integral disappear. These difficulties are in fact a consequence of an unphysical idealization – the neglect of the fact that some information about the behavior of the quantum system remains in classical form in the environment.