  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.
