Binary operations on algebras of observables are studied in the quantum as well as in the classical case. It is shown that certain natural compatibility conditions with the associative product imply properties which are usually additionally required.
We present a generalization of the classical central limit theorem to the case of non-commuting random variables which are bm-independent and indexed by a partially ordered set. As the set of indices I we consider discrete lattices in symmetric positive cones, with the order given by the cones. We show that the limit measures have moments which satisfy recurrences generalizing the recurrence for the Catalan numbers.
One of the most interesting phenomena exhibited by ultracold quantum gases is the appearance of vortices when the gas is put in rotation. The talk will bring a survey of some recent progress in understanding this phenomenon starting from the many-body ground state of a Bose gas with short range interactions. Mathematically this amounts to describing solutions of a linear Schrödinger equation with a very large number of variables in terms of a nonlinear equation with few variables and analyzing the...