  The minimal coupling principle is revisited under the quantum perspectives of
the space-time symmetry. This revision is better realized on a Group Approach
to Quantization (GAQ) where group cohomology and extensions of groups play a
preponderant role. We firstly consider the case of the electromagnetic
potential; the Galilei and/or Poincare group is (non-centrally) extended by the
"local" U(1) group. This group can also be seen as a central extension,
parametrized by both the mass and the electric charge, of an
infinite-dimensional group, on which GAQ leads to the dynamics of a particle
moving in the presence of an electromagnetic field. Then we try the
gravitational interaction of a particle by turning into "local" the space-time
translations. However, promoting to "local" the space-time subgroup of the true
symmetry of the quantum free relativistic particle, i.e. the centrally extended
by U(1) Poincare group, results in a new electromagnetic-like force of pure
gravitational origin. This is a consequence of the space-time translations not
being an invariant subgroup of the extended Poincare group and constitutes a
preliminary attempt to a non-trivial mixing of space-time and internal gauge
interactions.
