Central extensions and physics

Abstract

In this paper two themes are considered; first of all we consider the question under what circumstances a central extension of the Lie algebra of a given Lie group determines a central extension of this Lie group (and how many different ones). The answer will be that if we give the algebra extension in the form of a left invariant closed 2-form ω on the Lie group, then there exists an associated group extension iff the group of periods of ω is a discrete subgroup of IR and ω admits a momentum mapping for the left action of the group on itself.The second theme concerns the process of pre-quantization; we show that the construction needed to answer the previous question is exactly the same as the construction of prequantum bundles in geometric quantization. Moreover we show that the formalism of prequantization over a symplectic manifold and the formalism of quantum mechanics (where the projective Hilbert spaces replaces the (symplectic) phase space) can be identified (modulo some ≪details≫ concerning infinite dimensions).