Invariant tori and Heisenberg matrix mechanics: a new window on the quantum-classical correspondence

After a brief review of the extensive work done on the theory of invariant tori and their quantization, we show that nevertheless an important connection between the quantum and classical theories remains to be exploited. This is the relationship between matrix elements of operators in the energy diagonal representation and Fourier components of the corresponding classical dynamical variables that was the basis for Heisenberg's invention of quantum mechanics. We describe a number of previously unknown or little-known aspects of this relationship, with special emphasis on variational principles and the connection between commutation relations and quantization of action variables. As a single illustration of the utility of these ideas we show that it is possible to obtain approximate solutions to the quantum scheme that are more accurate than the semiclassical approximation with little additional effort compared to the latter. (orig.)