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.

中文翻译：

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哈密​​顿系统的随机量化

摘要

哈密​​顿系统的量化是一个模棱两可的过程。因此，我们引入了随机量化的概念、具有自伴随算子集合中值的相关随机变量以及具有酉算子组中的值的随机过程。定义了随机酉群的平均和随机自伴随算子的平均的过程。介绍了测度序列的广义弱收敛性和相应的随机变量序列分布的广义收敛性。获得了一些随机映射组合序列分布的广义收敛性。在由欧几里德空间的独立随机向量组成的移位序列的情况下，获得的收敛性与独立随机向量之和的中心极限定理的陈述一致。结果被应用于在哈密顿系统的随机量化中产生的量子系统的动力学。