Hermitian vector fields and covariant quantum mechanics of a spin particle

TL;DR

This paper classifies quantum vector fields for a spin particle in covariant quantum mechanics, establishing an isomorphism with special phase functions, thus providing a covariant method to derive quantum operators and observables.

Contribution

It introduces a classification of quantum vector fields as Hermitian vector fields and links them to classical phase space functions, offering a new covariant quantization approach.

Findings

Quantum vector fields form a Lie algebra isomorphic to special phase functions.

Provides a covariant procedure to generate quantum operators from classical functions.

Establishes a structural link between quantum vector fields and classical phase space functions.

Abstract

In the context of Covariant Quantum Mechanics for a spin particle, we classify the ``quantum vector fields'', i.e. the projectable Hermitian vector fields of a complex bundle of complex dimension 2 over spacetime. Indeed, we prove that the Lie algebra of quantum vector fields is naturally isomorphic to a certain Lie algebra of functions of the classical phase space, called ``special phase functions''. This result provides a covariant procedure to achieve the quantum operators generated by the quantum vector fields and the corresponding observables described by the special phase functions.