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Tetrads in SU(N) Yang-Mills geometrodynamics

Alcides Garat

Abstract

The discovery of the SU(3) symmetry was fundamental as to establishing an ordering principle in particle physics. We already studied how to couple the SU(3) symmetry to the gravitational field in four-dimensional curved Lorentzian spacetimes. 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 the seventies, 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 T3 and hypercharge Y and the standard T3-Y diagrams were extended onto another third axis. Then, instead of the fundamental triplet we have a quartet {u; d; s; c} as the smallest representation of the symmetry group, leading to the introduction of SU(4) as the new group of symmetries. In this paper we will not restrict ourselves exclusively to the symmetry group SU(4) and we will set out to analyze the coupling of the SU(N) symmetry to the gravitational field. To this end new tetrads will be introduced as we did for the SU(3) x SU(2) x U(1) 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 uni?fication in four-dimensional curved Lorentzian spacetimes.

Tetrads in SU(N) Yang-Mills geometrodynamics

Abstract

The discovery of the SU(3) symmetry was fundamental as to establishing an ordering principle in particle physics. We already studied how to couple the SU(3) symmetry to the gravitational field in four-dimensional curved Lorentzian spacetimes. 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 the seventies, 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 T3 and hypercharge Y and the standard T3-Y diagrams were extended onto another third axis. Then, instead of the fundamental triplet we have a quartet {u; d; s; c} as the smallest representation of the symmetry group, leading to the introduction of SU(4) as the new group of symmetries. In this paper we will not restrict ourselves exclusively to the symmetry group SU(4) and we will set out to analyze the coupling of the SU(N) symmetry to the gravitational field. To this end new tetrads will be introduced as we did for the SU(3) x SU(2) x U(1) 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 uni?fication in four-dimensional curved Lorentzian spacetimes.
Paper Structure (6 sections, 2 theorems, 39 equations)

This paper contains 6 sections, 2 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Quotient Space
  3. Tetrads in $SU(N)$
  4. Conclusions
  5. Appendix I
  6. Appendix II

Key Result

Theorem 1

The mapping between the local gauge group of transformations $SU(N)$ and the tensor product of the $N^{2}-1$ local groups of LB1 transformations is isomorphic.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2