Random quantization of Hamiltonian systems

A quantization of a Hamiltonian system is an ambiguous procedure. Accordingly, we introduce the notion of random quantization, related random variables with values in the set of self-adjoint operators, and random processes with values in the group of unitary operators. The procedures for the averaging of random unitary groups and averaging of random self-adjoint operators are defined. The generalized weak convergence of a sequence of measures and the corresponding generalized convergence in distribution of a sequence of random variables are introduced. The generalized convergence in distribution for some sequences of compositions of random mappings is obtained. In the case of a sequence of compositions of shifts by independent random vectors of Euclidean space, the obtained convergence coincides with the statement of the central limit theorem for a sum of independent random vectors. The results are applied to the dynamics of quantum systems arising in random quantization of a Hamiltonian system.