We propose an alternative approach to gauge theoretical treatment of quantum mechanics by lifting quantum state functions to the holomorphic tangent bundle T(C). PACS 2010: 02.40.-k, 03.65.-w
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 spacesC, 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 toC. 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 whenC is complex projective space CP n.