Semiclassics for quantum systems with internal degrees of freedom

Abstract

We study semiclassical properties of quantum systems with internal degrees of freedom. While translational degrees of freedom are described as coordinates on the cotangent bundle of a configuration manifold, the internal ones find their classical description on more general symplectic manifolds, such as coadjoint orbits of compact Lie groups or Kaehler manifolds. The quantum space for the translational degrees of freedom, square integrable functions on the configuration space, has to be ``tensored' with the representation space for the internal ones.Consequently, quantum observables are operators that take values in the endomorphisms of the representation space corresponding to the internal degrees of freedom. One part of this work is concerned with the generalization of semiclassical techniques to the non-scalar setting. In particular, we study the semiclassical time evolution of observables. For this purpose we construct semiclassical projections whose range is almost invariant under the quantum mechanical time evolution, and we prove a generalization of Egorov's Theorem. Furthermore, the existence of semiclassical projections allows us to state a semiclassical limit formula of Szegoe-type. We employ a quantization scheme for the internal degrees of freedom which enables us to map their quantum character, still remanescent in the above results, to a classical model and thus obtain a classical description for both the translational and internal degrees of freedom. Using this framework we can reformulate the results achieved before; in addition we prove a quantum ergodicity theorem and study the time evolution of coherent states. The quantization method for the internal degrees, in addition, provides a method to perform a semiclassical limit also for the internal degrees of freedom, which can be combined with the translational semiclassical limit. Also in this case the time evolution of coherent states is considered.