Tetrads in SU(3) × SU(2) × U(1) YangMills geometrodynamics
Abstract
The relationship between gauge and gravity amounts to understanding the underlying new geometrical local structures. These structures are new tetrads specially devised for YangMills theories, Abelian and nonAbelian in fourdimensional Lorentzian curved spacetimes. In the present paper, a new tetrad is introduced for the YangMills SU(3) × SU(2) × 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) × SU(2) × 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 stressenergy 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 nonAbelian with local groups of tetrad transformations. On the local plane one, the Abelian group U(1) of gauge transformations was already proved to be isomorphic to the tetrad local group of transformations LB1, for example. LB1 is SO(1, 1) plus two different kinds of discrete transformations. On the local orthogonal plane two U(1) is isomorphic to LB2 which is just SO(2). That is, we proved that LB1 is isomorphic to SO(2) which is a remarkable result since a noncompact group plus two discrete transformations is isomorphic to a compact group. These new tetrads have displayed manifestly and nontrivially the coupling between YangMills fields and gravity. The new tetrads and the stressenergy 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 YangMills stressenergy 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 fourdimensional curved Lorentzian spacetimes.
 Publication:

International Journal of Geometric Methods in Modern Physics
 Pub Date:
 2018
 DOI:
 10.1142/S0219887818500457
 arXiv:
 arXiv:1207.0912
 Bibcode:
 2018IJGMM..1550045G
 Keywords:

 New tetrads;
 new groups;
 new group isomorphisms;
 gravityYang─Mills classical unification;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 There is a new appendix. The unitary transformations by local SU(2) subgroup elements of a local group coset representative is proved to be a new local group coset representative. This proof is relevant to the study of the memory of the local tetrad SU(3) generated gauge transformations. Therefore, it is also relevant to the group theorems proved in the paper. arXiv admin note: substantial text overlap with arXiv:grqc/0602049