Semiclassics for Quantum Systems with Internal Degrees of Freedom: the Semiclassical Correspondence Generalized

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. 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 for 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 and we prove a generalization of Egorov's Theorem and a semiclassical limit formula of Szegö-type. We employ a quantization scheme for the internal degrees of freedom which enables us to map their quantum character to a classical model and thus obtain a classical description for both the translational and internal degrees of freedom."