Abstract
The Feynman path integral places the classical action in the very core of quantum evolution. Instead of contrasting the multiplicity of quantum paths with the scarcity of classical trajectories, we can avail ourselves of the variational principle by which they are identified. This leads to wide ranging semiclassical approximations for the evolution operator. The Fourier transform of the trace of this operator then supplies the density of quantum energy levels as a sum over the classical periodic orbits: The Gutzwiller trace formula. The periodic orbits also explain the correlations between levels, following universal classes of random matrices, if the classical system is chaotic. Conversely, discrepancies between the semiclassical spectra and the universal random matrix spectra uncouvered correlations between periodic orbits that had never been suspectected in the long history of classical mechanics.
Keywords:
path integral; classical trajectory; variational principle; periodic orbit; level density; random matrices
