Although the WKB approximation for multicomponent systems has been intensively studied in the literature, its geometric and global aspects are much less well understood than in the scalar case. In this paper we give a completely geometric derivation of the transport equation, without using local sections and without assuming complete diagonalizability of the matrix valued principal symbol, or triviality of its eigenbundles. The term (called "no-name term" in some previous literature) appearing in the transport equation in addition to the covariant derivative with respect to a natural projected connection will be a tensor, independent of the choice of any sections. We give a geometric interpretation of this tensor, involving the contraction of the curvature of the eigenbundle and an analog of the second fundamental form with the Poisson tensor in phase space. In the non-degenerate * The research of both authors was partially supported by DOE Contract DE-FG03-93ER25177. We would like to thank Hans Duistermaat, Mikhail Karasev, Robert Littlejohn, and Jim Morehead for helpful discussions.1 case this term may be rewritten in an even simpler geometric form. Finally, we discuss obstructions to the existence of WKB states and give a geometric description of the quantization condition for WKB states for a non-degenerate eigenvalue-function.

B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flattening a given torsion free symplectic connection. In this paper, a classical analog of Fedosov's operations on connections is analyzed and shown to produce the usual exponential mapping of a linear connection on an ordinary manifold. A symplectic version is also analyzed. Finally, some remarks are made on the implications for deformation quantization of Fedosov's index theorem on general symplectic manifolds.

We derive a closed formula for a star-product on complex projective space and on the domain SU (n + 1)/S(U (1) × U (n)) using a completely elementary construction: Starting from the standard star-product of Wick type on C n+1 \ {0} and performing a quantum analogue of Marsden-Weinstein reduction, we can give an easy algebraic description of this starproduct. Moreover, going over to a modified star-product on C n+1 \ {0}, obtained by an equivalence transformation, this description can be even further simplified, allowing the explicit computation of a closed formula for the star-product on CP n which can easily transferred to the domain SU (n + 1)/S

In systems like Yang-Mills or gravity theory, which have a symmetry of gauge type, neither phase space nor configuration space is a manifold but rather an orbifold with singular points corresponding to classical states of non-generically higher symmetry. The consequences of these symmetries for quantum theory are investigated. First, a certain orbifold configuration space is identified. Then, the Schrodinger equation on this orbifold is considered. As a typical case, the Schrodinger equation on (double) cones over Riemannian manifolds is discussed in detail as a problem of selfadjoint extensions. A marked tendency towards concentration of the wave function around the singular points in configuration space is observed, which generically even reflects itself in the existence of additional bound states and can be interpreted as a quantum mechanism of symmetry enhancement.

Based on a closed formula for a star product of Wick type on CP n , which has been discovered in an earlier article of the authors, we explicitly construct a subalgebra of the formal star-algebra (with coefficients contained in the uniformly dense subspace of representative functions with respect to the canonical action of the unitary group) that consists of converging power series in the formal parameter, thereby giving an elementary algebraic proof of a convergence result already obtained by Cahen, Gutt, and Rawnsley. In this subalgebra the formal parameter can be substituted by a real number α: the resulting associative algebras are infinite-dimensional except for the case α = 1/K, K a positive integer, where they turn out to be isomorphic to the finitedimensional algebra of linear operators in the Kth energy eigenspace of an isotropic harmonic oscillator with n + 1 degrees of freedom. Other examples like the 2n-torus and the Poincaré disk are discussed. *