Dynamical Symmetry Group Based on Dirac Equation and Its Generalization to Elementary Particles

Abstract

The dynamical or exact symmetry group of the nonrelativistic Kepler problem (a symmetry group in four dimensions) is generalized to the Dirac equation and further to elementary particles. The former is a ten-parameter group of rank two isomorphic to a group in five dimensions, the latter a 16-parameter group of rank four isomorphic to a group in six dimensions. Both groups contain the real Lorentz group and couple the space-time quantum numbers with the internal quantum numbers. The 16-parameter group has a 15-parameter simple subgroup and contains two three-dimensional rotation groups, one for ordinary spin and one for isotopic spin. The concept of "inhomogeneous dynamical group" is introduced. The inhomogeneous group contains two new additive quantum numbers to describe the hypercharge and the baryon number and leads to a mass spectrum. The third component of isospin and the new additive quantum numbers commute with all the six generators of the Lorentz group. A further generalization leads to a group where all three isospin generators commute with the Lorentz group.