  We argue that tangent vectors to classical phase space give rise to quantum
states of the corresponding quantum mechanics. This is established for the case
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.
