  The concept of elementary particle rests on the idea that it is a physical
system with no excited states, so that all possible states of the particle are
just kinematical modifications of any one of them. In this way instead of
describing the particle attributes it amounts to describe the collection of
consecutive inertial observers who describe the particle in the same
kinematical state. The kinematical state space of an elementary particle is a
homogeneous space of the kinematical group.By considering the largest
homogeneous spaces of both, Galilei and Poincare groups, it is shown how the
spin structure is related to the different degrees of freedom. Finally, the
spacetime symmetry group of a relativistic particle which satisfies Dirac's
equation when quantized, is enlarged to take into account additional symmetries
like spacetime dilations and local rotations. An interaction Lagrangian
invariant under this enlarged group is proposed and the compound system of two
Dirac particles is analyzed.
