Tetrads in SU(N) Yang–Mills geometrodynamics

Abstract

The discovery of the [Formula: see text] symmetry was fundamental as to establishing an ordering principle in particle physics. We already studied how to couple the [Formula: see text] symmetry to the gravitational field in four-dimensional curved Lorentzian space–times. The multiplets of equal quantum numbers are translated through natural elements in Riemannian geometry into local multiplets of equal gravitational field. As quark physics developed since in the 1970s, it was necessary to incorporate new symmetries to the models, that ensued in the incorporation of new quantum numbers like charm, for example, charm is an additive quantum number like isospin [Formula: see text] and hypercharge [Formula: see text] and the standard [Formula: see text] diagrams were extended onto another third axis. Then, instead of the fundamental triplet, we have a quartet [Formula: see text] as the smallest representation of the symmetry group, leading to the introduction of [Formula: see text] as the new group of symmetries. In this paper, we will not restrict ourselves exclusively to the symmetry group [Formula: see text] and we will set out to analyze the coupling of the [Formula: see text] symmetry to the gravitational field. To this end, new tetrads will be introduced as we did for the [Formula: see text] case. These tetrads have outstanding properties that enable these constructions. New theorems will be proved regarding the isomorphic nature of these local symmetry gauge groups and tensor products of groups of local tetrad transformations. This is a paper about grand field unification in four-dimensional curved Lorentzian space–times.