Extraordinary properties of functional integrals and groups of diffeomorphisms

A review of the work of the authors is presented, in which corollaries of the quasi-invariance of functional integrals on the Wiener measure with respect to the action of a group of diffeomorphisms are studied, and the behavior of functional integrals with nonlinear nonlocal change of variables of integration is investigated as well. Using these substitutions, the functional integrals over discontinuous paths can be determined. The simplest models of the (Euclidean) quantum field theory are offered, in which the presence of hidden internal symmetries or the allowance for discontinuous paths in functional integrals leads to a number of paradoxical properties contradicting the conventional view.