Quantum Systems with Infinite-Dimensional Coordinate Space and the Fourier Transform

In the space of square integrable functions on a Hilbert space with a translation invariant measure, we study unitary groups of operators of shift by vectors of the momentum space. Analyzing the averaging of functionals of Gaussian random processes in the momentum space, we obtain a semigroup of self-adjoint contractions; we establish conditions for the strong continuity of this semigroup and study its generator, which is the operator of multiplication by a quadratic form of a nonpositive trace-class operator in the Hilbert space. We compare the properties of the groups of shift operators in the coordinate and momentum spaces, as well as the properties of semigroups of self-adjoint contractions generated by diffusion in the coordinate and momentum spaces. In addition, we show that one cannot define the Fourier transform as a unitary map that would provide a unitary equivalence of these contraction semigroups.