Abstract

There is an extensive literature on the dynamic law of large numbers for systems of quantum particles, that is, on the derivation of an equation describing the limiting individual behavior of particles in a large ensemble of identical interacting particles. The resulting equations are generally referred to as nonlinear Schrödinger equations or Hartree equations, or Gross–Pitaevskii equations. In this paper, we extend some of these convergence results to a stochastic framework. Specifically, we work with the Belavkin stochastic filtering of many-particle quantum systems. The resulting limiting equation is an equation of a new type, which can be regarded as a complex-valued infinite-dimensional nonlinear diffusion of McKean–Vlasov type. This result is the key ingredient for the theory of quantum mean-field games developed by the author in a previous paper.

中文翻译：

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用于多粒子系统的量子随机滤波和控制的大数定律

摘要

有大量关于量子粒子系统的大数动力学定律的文献，即关于描述相同相互作用粒子的大型集合中粒子的极限个体行为的方程的推导。得到的方程通常称为非线性薛定谔方程或 Hartree 方程，或 Gross-Pitaevskii 方程。在本文中，我们将其中一些收敛结果扩展到随机框架。具体来说，我们使用多粒子量子系统的 Belavkin 随机滤波。得到的极限方程是一种新型方程，可以看作是一个复值无限维非线性扩散的 McKean-Vlasov 型。