Free probability theory, random matrices and quantum chaos

In this talk, we will discuss recent advances in the theory of quantum chaos in quantum many-body systems using a specific branch of mathematics of non-commutative probability spaces, namely, the free probability theory. We shall argue that time evolution generated by many-body quantum systems that show full random matrix behavior produces asymptotically free random variables. This provides a new and generic definition of dynamical quantum chaos, and also helps to bridge the different notions of chaos that have been studied previously in the literature.