Hermitian vector fields and special phase functions

Abstract

We start by analysing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a spacetime connection (preserving the time fibring and the spacelike metric) and an electromagnetic field. In the second case, we consider a spacetime equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase functions and show that the Lie algebra of Hermitian vector fields turns out to be naturally isomorphic to the Lie algebra of special phase functions. Eventually, we compare the Galilei and Einstein cases.

References in corpus (2)

• On symmetries in Galilei classical mechanics
• Comparison between Geometric Quantisation and Covariant Quantum Mechanics

Cited by in corpus (7)

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• Frölicher-smooth geometries, quantum jet bundles and BRST symmetry
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• Geometric structures of the classical general relativistic phase space
• Semi--vector spaces and units of measurement
• On the characterization of infinitesimal symmetries of the relativistic phase space
• On the geometry of the energy operator in quantum mechanics