The Gutzwiller Trace Formula for Quantum Systems with Spin

The Gutzwiller trace formula provides a semiclassical approximation for the density of states of a quantum system in terms of classical periodic orbits. In its original form Gutzwiller derived the trace formula for quantum systems without spin. We will discuss the modifications that arise for quantum systems with both translational and spin degrees of freedom and which are either described by Paulior Dirac-Hamiltonians. In addition, spectral densities weighted by expectation values of observables will be considered. It turns out that in all cases the semiclassical approximation yields sums over periodic orbits of the translational motion. Spin contributes via weight factors that take a spin precession along the translational orbits into account. Thirty years ago, after several intermediate steps the Gutzwiller trace formula [1,2] resulted from a detailed semiclassical investigation of the time evolution in quantum mechanics. It opened the way for the application of semiclassical methods to many problems that were so far believed to lie beyond the capability of semiclassics. The most prominent example being semiclassical quantisation rules for classically non-integrable systems, for which Einstein [3] already in 1917 had shown that the usual Bohr-Sommerfeld type quantisation methods fail. Gutzwiller, however, devised a semiclassical expansion for the density of states in terms of a sum over the classical periodic orbits for a huge class of quantum systems, which in particular includes classically chaotic systems. Shortly afterwards, but seemingly independently, also mathematicians became interested in such trace formulae. They devised mathematical proofs for various versions of the trace formula, beginning with the work of Colin de Vèrdiere [4], and Duistermaat and Guillemin [5]. What was lacking so far, however, was a trace formula for quantum systems with a priori non-classical degrees of freedom as, e.g., spin. In such cases it is not immediately clear what the corresponding classical system is whose periodic orbits enter the trace formula, and how the non-classical degrees of freedom have to be taken into account. Even for systems with a classically integrable translational part there do not exist Bohr-Sommerfeld (or EBK) type quantisation rules for the eigenvalues of a Dirac-Hamiltonian, although already in 1932 Pauli [6] began to generalise the WKB method to the Dirac equation. Pauli’s undertaking was only completed in 1963 by Rubinow and Keller [7], and it took again some 30 years before Emmrich and Weinstein [8] proved that due to geometric obstructions for the Dirac equation EBK-