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 invar iant tori and their quantization, we show that nevertheless an importa nt connection between the quantum and classical theories remains to be exploited. This is the relationship between matrix elements of operat ors in the energy diagonal representation and Fourier components of th e corresponding classical dynamical variables that was the basis for H eisenberg's invention of quantum mechanics. We describe a number of pr eviously unknown or little-known aspects of this relationship, with sp ecial emphasis on variational principles and the connection between co mmutation 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 a ccurate than the semiclassical approximation with little additional ef fort compared to the latter.