Determining SU (N) symmetry group generators

Abstract

SU(N) symmetry groups are useful in formulating gauge theories of elementary particle interactions in quantum field theory. Gauge bosons and particle states are associated with the symmetry group generators. The accuracy of the physical structure and predictions of the gauge theory thus depends on the accuracy of the group generators. In this article, we present an accurate mathematical method for determining all generators of an SU(N) symmetry group for any N ≥ 2. Group generators are characterized as diagonal or non-diagonal symmetric and antisymmetric partners. There are N(N − 1) traceless non-diagonal symmetric and antisymmetric generators, (N −1) traceless diagonal antisymmetric generators and (N − 1) non-traceless diagonal symmetric generators. An SU(N) symmetry group is therefore specified by a total of N 2 − 1 standard traceless non-diagonal and diagonal symmetric and antisymmetric generators and (N − 1) non-traceless diagonal symmetric generators. The procedure is particularly effective in enumerating the correct generators of SU(N) groups used in formulating various models of gauge theories of elementary particle interactions driven by fundamental forces of nature. As simple illustrations, we have applied the procedure to determine the generators of the SU(2), SU(3) , SU(4) , SU(5) , SU(6) and SU(7) symmetry groups. While the SU(2) and SU(3) symmetry group generators have been used as the building blocks of the largely successful Standard Model (SM) of quantum field theory, the determination of the correct forms of the complete set of generators of the SU(5) and SU(6) groups, comprising the standard 52 − 1 = 24 or 62 − 1 = 35 traceless non-diagonal and diagonal symmetric and antisymmetric generators and the other emerging 5 − 1 = 4 or 6 − 1 = 5 non-traceless diagonal symmetric generators, respectively, reveals that the physical structure and predictions of the SU(5) and SU(6) models of the Grand Unified Theory needs a radical review, particularly with respect to the incorrect forms of the standard third, fourth and fifth traceless diagonal antisymmetric generators used in the current forms of the models.