Abstract
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.
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REFERENCES
F. A. Berezin, Theor. Math. Phys. 6, 141–155 (1971).
Yu. N. Orlov, Foundations of Quantization of Degenerate Dynamical Systems (Mosk. Fiz.-Tekh. Inst., Moscow, 2004) [in Russian].
S. Bonaccorsi, F. Cottini, and D. Mugnolo, Appl. Math. Optim. (2021). https://doi.org/10.1007/s00245-020-09732-w
Ya. A. Butko, Stochastics Dyn. (2018). https://doi.org/10.1142/S0219493718500211
I. D. Remizov, J. Math. Phys. 60 (7), 071505 (2019).
I. V. Volovich, Theor. Math. Phys. 164, 1128–1135 (2010).
Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov, Izv. Math. 80 (6), 1131–1158 (2016).
P. Chernoff, J. Funct. Anal. 2 (2), 238–242 (1968).
K. B. Sinha and S. Srivastava, Theory of Semigroups and Applications (Springer, Singapore, 2017).
Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov, Proc. Steklov Inst. Math. 306, 196–211 (2019).
Funding
John Gough acknowledges funding by the French National Research Agency under the grant Q-COAST ANR-19-CE48-0003.
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Translated by I. Ruzanova
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Gough, J., Orlov, Y.N., Sakbaev, V.Z. et al. Random Quantization of Hamiltonian Systems. Dokl. Math. 103, 122–126 (2021). https://doi.org/10.1134/S106456242103008X
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DOI: https://doi.org/10.1134/S106456242103008X