  We study the quantisation of complex, finite-dimensional, compact, classical
phase spaces C, by explicitly constructing Hilbert-space vector bundles over C.
We find that these vector bundles split as the direct sum of two holomorphic
vector bundles: the holomorphic tangent bundle T(C), plus a complex line bundle
N(C). Quantum states (except the vacuum) appear as tangent vectors to C. The
vacuum state appears as the fibrewise generator of N(C). Holomorphic line
bundles N(C) are classified by the elements of Pic(C), the Picard group of C.
In this way Pic(C) appears as the parameter space for nonequivalent vacua. Our
analysis is modelled on, but not limited to, the case when C is complex
projective space.
