Graded Lie algebra of Hermitian tangent valued forms

TL;DR

This paper classifies Hermitian tangent valued forms on complex line bundles, linking them to tangent forms and base space forms, with implications for covariant quantization in gauge theories.

Contribution

It provides a direct classification of Hermitian tangent valued forms and their representation via special phase functions, simplifying previous complex approaches.

Findings

Local classification obtained in coordinates

Global classification requires a Hermitian connection

Results facilitate covariant quantization of observables

Abstract

In the theory of so called "Covariant Quantum Mechanics" a basic role is played by Hermitian vector fields on a complex line bundle in the frameworks of Galilei and Einstein spacetimes. In fact, it has been proved that the Lie algebra of Hermitian vector fields is naturally isomorphic to a Lie algebra of "special functions" of the phase space. Indeed, this is the source of the covariant quantisation of the above special functions. In the original version of the theory, this result was formulated and proved in a rather involved way; now, we have achieved a more direct and simple approach to the classification of Hermitian vector fields and to their representation via special phase functions. In view of a possible covariant quantisation of a larger class of "observables" it is natural to consider the Hermitian tangent valued forms. Thus, this paper is devoted to a self--contained analysis of the graded Lie algebra of Hermitian tangent valued forms of a complex line bundle and to their classification in terms of tangent valued forms and forms of the base space. The local classification is obtained in coordinates. For the global classification we need a Hermitian connection: indeed, this is just the connection required in gauge theories.