Many interesting C * -algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution C * -algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, including geometric quantization of symplectic manifolds and the C * -algebra of a Lie groupoid.

A number of papers over the past eight years have claimed to solve the fractional Schrödinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schrödinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported groundstate, which is based on a piecewise approach, is definitely not a solution of the fractional Schrödinger equation for general fractional parameters α. On a more positive note, we present a solution to the fractional Schrödinger equation for the one-dimensional harmonic oscillator with α = 1.

ABSTRACT. Using very weak criteria for what may constitute a noncommutative geometry, I show that a pseudo-Riemannian manifold can only be smoothly deformed into noncommutative geometries if certain geometric obstructions vanish. These obstructions can be expressed as a system of partial differential equations relating the metric and the Poisson structure that describes the noncommutativity. I illustrate this by computing the obstructions for well known examples of noncommutative geometries and quantum groups. These rigid conditions may cast doubt on the idea of noncommutatively deformed space-time.

Abstract. The quantization of vector bundles is defined. Examples are constructed for the well controlled case of equivariant vector bundles over compact coadjoint orbits. (A coadjoint orbit is a symplectic manifold with a transitive, semisimple symmetry group.) In preparation for the main result, the quantization of coadjoint orbits is discussed in detail.This subject should not be confused with the quantization of the total space of a vector bundle such as the cotangent bundle.

The quantum Euclidean spheres, S N−1 q , are (noncommutative) homogeneous spaces of quantum orthogonal groups, SO q (N). The * -algebra A(S N−1 q ) of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, unipotent matrix. We explicitly construct complete sets of generators for the K-theory (by nontrivial self-adjoint idempotents and unitaries) and the K-homology (by nontrivial Fredholm modules) of the spheres S N−1 q . We also construct the corresponding Chern characters in cyclic homology and cohomology and compute the pairing of K-theory with K-homology. On odd spheres (i. e., for N even) we exhibit unbounded Fredholm modules by means of a natural unbounded operator D which, while failing to have compact resolvent, has bounded commutators with all elements in the algebra A(S N−1 q ).