Geometric aspects of the quantization of a rigid body M. Modugno C. Tejero Prieto R. Vitolo Published in B. Kruglikov, V. Lychagin, E. Straume: Differential Equations – Geometry, Symmetries and Integrability, Proceedings of the 2008 Abel Symposium, Springer, 275-285. Abstract In this paper we review our results on the quantization of a rigid body. The fact that the configuration space is not simply connected yields two inequivalent quantizations. One of the quantizations allows us to recover classically double-valued wave functions as single val- ued sections of a non-trivial complex line bundle. This reopens the problem of a physical interpretation of these wave functions. 1 Introduction The idea of writing quantum mechanics in a coordinate-free way circulated among physicists and mathematicians as a natural consequence of the gen- eral relativity principle. One of the main features of quantum mechanics is that it must contain, according to Dirac’s ideas, a correspondence with classical mechanics. Having symplectic mechanics at hand, it was natural to formulate a correspondence principle between classical symplectic mechanics and quantum mechanics that associates a self adjoint operator on a Hilbert space with every quantizable classical observable [13, 21]. This is the heart of what has been called the Geometric Quantization (GQ for short). The above theory proved to be useful in some physically simple situations, but showed to have a number of drawbacks, discussed in detail in Section 2. The aim of this paper is to discuss some features of a recent geometric approach to quantum theory, the Covariant Quantum Mechanics (CQM for short). The CQM (introduced by Jadczyk and Modugno [10] and further developed in [1, 11, 12, 15, 16, 17, 19, 23, 25]) has two distinguished features with respect to GQ: on one hand, it is simpler, because it deals only with 1