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Tetrads in SU(3) X SU(2) X U(1) Yang-Mills geometrodynamics

Alcides Garat

Abstract

The relationship between gauge and gravity amounts to understanding underlying new geometrical local structures. These structures are new tetrads specially devised for Yang-Mills theories, Abelian and Non-Abelian in four-dimensional Lorentzian curved spacetimes. In the present manuscript a new tetrad is introduced for the Yang- Mills SU(3) x SU(2) x U(1) formulation. These new tetrads establish a link between local groups of gauge transformations and local groups of spacetime transformations that we previously called LB1 and LB2. New theorems are proved regarding isomorphisms between local internal SU(3) x SU(2) x U(1) groups and local tensor products of spacetime LB1 and LB2 groups of transformations. These new tetrads define at every point in spacetime two orthogonal planes that we called blades or planes one and two. These are the local planes of covariant diagonalization of the stress-energy tensor. These tetrads are gauge dependent. Tetrad local gauge transformations leave the tetrads inside the local original planes without leaving them. These local tetrad gauge transformations enable the possibility to connect local gauge groups Abelian or non-Abelian with local groups of tetrad transformations. These new tetrads have displayed manifestly and non-trivially the coupling between Yang-Mills fields and gravity. The new tetrads and the stress-energy tensor allow for the introduction of three new local gauge invariant objects. Using these new gauge invariant objects and in addition a new general local duality transformation, a new algorithm for the gauge invariant diagonalization of the Yang-Mills stress-energy tensor is developed as an application. This is a paper about grand Standard Model gauge theories - General Relativity gravity unification and grand group unification in four-dimensional curved Lorentzian spacetimes.

Tetrads in SU(3) X SU(2) X U(1) Yang-Mills geometrodynamics

Abstract

The relationship between gauge and gravity amounts to understanding underlying new geometrical local structures. These structures are new tetrads specially devised for Yang-Mills theories, Abelian and Non-Abelian in four-dimensional Lorentzian curved spacetimes. In the present manuscript a new tetrad is introduced for the Yang- Mills SU(3) x SU(2) x U(1) formulation. These new tetrads establish a link between local groups of gauge transformations and local groups of spacetime transformations that we previously called LB1 and LB2. New theorems are proved regarding isomorphisms between local internal SU(3) x SU(2) x U(1) groups and local tensor products of spacetime LB1 and LB2 groups of transformations. These new tetrads define at every point in spacetime two orthogonal planes that we called blades or planes one and two. These are the local planes of covariant diagonalization of the stress-energy tensor. These tetrads are gauge dependent. Tetrad local gauge transformations leave the tetrads inside the local original planes without leaving them. These local tetrad gauge transformations enable the possibility to connect local gauge groups Abelian or non-Abelian with local groups of tetrad transformations. These new tetrads have displayed manifestly and non-trivially the coupling between Yang-Mills fields and gravity. The new tetrads and the stress-energy tensor allow for the introduction of three new local gauge invariant objects. Using these new gauge invariant objects and in addition a new general local duality transformation, a new algorithm for the gauge invariant diagonalization of the Yang-Mills stress-energy tensor is developed as an application. This is a paper about grand Standard Model gauge theories - General Relativity gravity unification and grand group unification in four-dimensional curved Lorentzian spacetimes.

Paper Structure

This paper contains 13 sections, 2 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Quotient Space
  3. Tetrads in $SU(3)$
  4. Applications
  5. Gauge invariants
  6. Diagonalization of the stress-energy tensor
  7. Conclusions
  8. Appendix I
  9. Appendix II
  10. Appendix III
  11. Abelian $U(1)$
  12. Non-Abelian $SU(2) \times U(1)$
  13. Non-Abelian $SU(3) \times SU(2) \times U(1)$

Key Result

Theorem 1

The mapping between the local gauge group of transformations $SU(3)$ and the tensor product of the eight local groups of LB1 transformations is isomorphic.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2