Some problems in large deviations for quantum systems.

We discuss several issues in quantum large deviations. The first problem we discuss is a generalization of Laplace-Vardhan lemma to quantum spin systems or in the language of statistical mechanics a proof of a variational principle for systems with short range and mean field interactions. (Joint work with W. De Roeck, C. Maes and K. Netockny). The second problem we address is a proof a large deviation principle for macroscopic observables in quantum systems. We apply to the quantum case an approach to large deviations which goes back to Lanford and Ruelle and use direct sub additivity arguments. (Joint work with Y. Ogata). Finally we will discuss some open problems.