Shell structure of the SU (N) generator spectrum: interpretation as spin angular momentum operators

Abstract

We have established that SU(N) symmetry group generators occur in a spectrum with a quantum structure composed of N − 1 configuration shells each containing a definite number of symmetric and antisymmetric pairs of generators specified by quantum numbers l = 1, ..., N − 1 ; m = 0, 1, ..., l. Interpreting the generators as spin angular momentum operators brings the generator spectrum to a form precisely similar to the spectrum of orbital angular momentum states composed of orbital configuration shells containing definite numbers of orbital states specified by orbital and magnetic quantum numbers l = 0, 1, ..., n − 1 ; m = 0, ±1, ..., ±l in the n th-energy level of an atom, thus revealing that the quantum state space of an SU(N) symmetry group corresponds directly to the quantum state space of the n th-energy level of an atom. Within each configuration shell containing specified generators in the SU(N) generator spectrum, we have determined the associated quadratic and Fubini-Veneziano spin angular momentum operators to general order, which we have finally used to obtain the corresponding universal SU(N) quadratic and Fubini-Veneziano spin operators. Basic algebraic relations of the resulting Cartan-Weyl generators have been determined explicitly for general SU(N) symmetry groups. Considering applications to gauge field theories, we easily establish that SU(N) gauge fields have quantum structure corresponding directly to the SU(N) generator spectrum. We have provided elaborate explanations of the important implications of the expanded algebraic properties and quantum structure of the SU(N) generator spectrum to the existing SU(N) gauge field theories.