Prequantization

Abstract

The aim of this chapter is to construct a linear monomorphism from the Poisson algebra of (X, ω) to the space of linear operators on an appropriate Hilbert space, associating to each function f on X a linear operator Pf so that the commutation relations

are satisfied for each pair of functions f and g on X. It should be noted that the mapping f ↦ (ħ/i)ξ_f satisfies (3.1) but it fails to be a monomorphism since its kernel consists of the space of all constant functions on X. Thus, we need a central extension of the Lie algebra of Hamiltonian vector fields on X by the additive group R of real numbers.