Quantum States from Tangent Vectors

Details Quantum States from Tangent Vectors Quantum States from Tangent Vectors

2004-07-20

arxiv.org/abs/quant-ph/0407159v2

Mod.Phys.Lett. A19 (2004) 2339-2352

Physics

Quantum Physics

Refs. updated

Scientific paper

10.1142/S0217732304015634

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.

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