Mathematics of open quantum systems

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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.