Hermitian vector fields and special phase functions

TL;DR

This paper explores the structure of Hermitian vector fields on line bundles over different spacetimes, establishing an isomorphism with a Lie algebra of special phase functions and comparing Galilei and Einstein frameworks.

Contribution

It identifies a natural Lie algebra of special phase functions and proves its isomorphism with the Lie algebra of Hermitian vector fields in both Galilei and Einstein spacetimes.

Findings

Lie algebra of Hermitian vector fields is isomorphic to special phase functions

Comparison between Galilei and Einstein spacetime cases

Framework unifies geometric structures in quantum mechanics contexts

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.