Complex time transformations peculiarities for wave function collapse description using quantum path integrals

A quantum path integral was transformed into the real form using a complex representation of the time. Such procedure gives the possibility to specify measures for the sets of the virtual paths in continual integrals determining amplitudes of quantum states transitions. The transition amplitude is a real function of the complex time modulus. Negative time values correspond to the reverse sequence of events. The quantum evolution description in form of the virtual paths mechanical motion does not depend on the sign of the time, due to the reversibility of the classical mechanics laws. This allows to consider the negative half of the imaginary axis of the time for the path integral measure determination. In this case this integral has the form of Wiener's integral having the well-known measure. As the wave function collapse is irreversible effect, the causal chain of events cannot be changed. Thus, to describe the collapse the transformation of quantum path integrals have to be performed in upper half plane of the complex time. It is shown that the Wiener measure for the real continual integral can be continued analytically on this actual range of the complex time. This allows to use the quantum path integral for any actual range of the complex time.

wave function collapse, path integral, Wiener measure, complex time, Wick rotation