New Classical Limits of Quantum Theories

Quantum fluctuations of some systems vanish not only in the limit $\hbar\to 0$, but also as some other parameters (such as $1\over N$, the inverse of the number of `colors' of a Yang-Mills theory) vanish. These lead to new classical limits that are often much better approximations to the quantum theory. We describe two examples: the familiar Hartree--Fock-Thomas-Fermi methods of atomic physics as well as the limit of large spatial dimension. Then we present an approach of the Hecke operators on modular forms inspired by these ideas of quantum mechanics. It explains in a simple way why the spectra of these operators tend to the spectrum of random matrices for large weight for the modular forms.