New Dynamical Group for the Relativistic Quantum Mechanics of Elementary Particles

Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincaré quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group ${\mathcal{G}}_5$ is suggested as the symmetry group of dynamics. ${\mathcal{G}}_5$ contains both the nonrelativistic Galilei group and the Poincaré group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of ${\mathcal{G}}_5$ by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\tilde{\mathcal{G}}}_5$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\tilde{\mathcal{G}}}_5$ correspond to infinite towers of states with increasing spin.