Quantal–Classical Duality and the Semiclassical Trace Formula

Abstract

We consider Hamiltonian systems which can be described both classically and quantum mechanically. Trace formulas establish links between the energy spectra in the quantum description and the spectrum of actions of periodic orbits in the Newtonian description. This is the duality which we investigate in the present paper. The duality holds for chaotic as well as for integrable systems. Billiard systems are a very convenient paradigm and we use them for most of our discussions. However, we also show how to transcribe the results to general Hamiltonian systems. In billiards, it is natural to think of the quantal spectrum (eigenvalues of the Helmholtz equation) and the classical spectrum (lengths of periodic orbits) as two manifestations of the properties of the billiard boundary. The trace formula express this link since it can be thought of as a Fourier transform relation between the classical and the quantum spectral densities. It follows that the two-point statistics of the quantal spectrum is related to the two-point statistics of the classical spectrum via a double Fourier transform. The universal correlations of the quantal spectrum are well known; consequently one can deduce the classical universal correlations. In particular, an explicit expression for the scale of the classical correlations is derived and interpreted. This allows a further extension of the formalism to the case of complex billiard systems, and in particular to the most interesting case of diffusive system. The effects of symmetry and symmetry-breaking are also discussed. The concept of classical correlations allows a better understanding of the so-called diagonal approximation and its breakdown. It also paves the way towards a semiclassical theory that is capable of global description of spectral statistics beyond the breaktime. An illustrative application is the derivation of the disorder-limited breaktime in the case of a disordered chain, thus obtaining a semiclassical theory for localization. We also discuss other applications such as the two-cell systems, periodic chains, and localization theory in more than one dimension. A numerical study of classical correlations in the case of the 3D Sinai billiards is presented. Here it is possible to test some assumptions and conjectures that underlie our formulation. In particular we gain a direct understanding of specific statistical properties of the classical spectrum, as well as their semiclassical manifestation in the quantal spectrum. We also analyze the spectral duality for integrable systems, and show that the Poissonian statistics of both the classical and the quantum spectra can be traced to the same origin.