Mathematics of open quantum systems

任何量子力学系统都必须被认为是与环境不断相互作用的。这是由于两个事实：第一，不可能将一个系统与其周围环境完全隔离(通常不能控制，并且以不可忽视的方式影响系统)。其次，对量子力学系统的任何观察都是通过与环境(测量设备)的相互作用来实现的。因此，分析环境对系统的影响的开放量子系统理论在数学物理学中受到极大关注也就不足为奇了。此外，开放量子系统的严格理论目前正在经历重大进展。主要驱动力是合适的现代数学工具的发展。这些工具位于演化群谱理论、算符代数理论、散射理论、量子共振理论和量子场论的交叉点上。在量子计算、量子信息理论和量子测量理论中都有现代的应用。据说，最近开发的一种新的方法使我们能够对回归平衡的过程(平衡的稳定性)进行更精确、可量化的描述，并解释了远离平衡的渐近状态的出现和性质。这一物理学理论的一个重要成果是从系统的微观哈密顿量子统计描述出发，推导出热力学和电学的宏观定律。这些定律被表示为温度、熵、密度、热通量和电荷通量等十个量之间的关系。相信本项目提出的研究将进一步发展、延伸和巩固开放量子系统的数学理论。这是通过考虑新的物理模型类别和开发新的数学技术来实现的。这些结果引起了各种各样的数学家和物理学家的兴趣。

Any quantum mechanical system must be regarded as being in constant interactions with an environment. This is due to two facts: Firstly, it is impossible to isolate completely a system from its surroundings (which cannot be controlled in general, and which influence the system in a non-negligible way). Secondly, any observation of a quantum mechanical system is brought about by an interaction with an environment (measuring device). It is therefore not surprising that the theory of open quantum systems, which analyzes environment-induced effects on systems, receives much attention in mathematical physics.     The rigorous theory of open quantum systems is currently experiencing significant progress. The main driving force is the development of suitable modern mathematical tools. Those tools are situated at the intersection of spectral theory of evolution groups, the theory of operator algebras, scattering theory, the theory of quantum resonances, and quantum field theory. Modern applications are found in quantum computing, quantum information theory and the theory of quantum measurement.     A recently developed, new approach to the topic has allowed us to obtain a  precise, quantifyable description of the process of return to equilibrium (stability of equilibria), and it has explained the emergence and the properties of asymptotic states far from equilibrium. An important achievement of this theory for physics is the derivation of macroscopic laws of thermodynamics and electricity, starting from a microscopic, hamiltonian quantum statistical description of the system. These laws are expressed as relations bewteen quantities such as temperature, entropy, density, and fluxes of heat and of charge.     The research proposed in this project will further develop, extend and solidify the mathematical theory of open quantum systems. This is done by considering new classes of physical models and by developing new mathematical techniques. The results are of interest to a wide variety of mathematicians and physicists.