Geodesic
Random quantization of Hamiltonian systems
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 498 (2021), pp. 31-36.

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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.
Keywords: random linear operator, random operator-valued function, operator-valued random process, law of large numbers, central limit theorem, Markovian process, Kolmogorov equation. 
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     title = {Random quantization of {Hamiltonian} systems},
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J. E. Gough; Yu. N. Orlov; V. Zh. Sakbaev; O. G. Smolyanov. Random quantization of Hamiltonian systems. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 498 (2021), pp. 31-36. http://geodesic.mathdoc.fr/item/DANMA_2021_498_a5/

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