The Group Law for The New Internal-Spacetime Mapping Between The Group of Internal Yang-Mills Gauge Transformations and The Groups (õLB1)3 and (õLB2)3 of Spacetime Tetrad Transformations

In previous works it has been demonstrated that all the standard model local gauge groups are isomorphic to local groups of special tetrad transformations. The skeleton-gauge-vector tetrad vector structure enables to prove all of these isomorphism theorems. These new tetrads have been specially constructed for Yang–Mills theories, Abelian and non-Abelian in four-dimensional Lorentzian spacetimes. In the present paper a new tetrad is employed for the Yang–Mills SU(2) × U(1) formulation. These new tetrads establish a connection between local groups of gauge transformations and local groups of spacetime tetrad transformations. We will prove that these Yang–Mills tetrads under the local Yang-Mills gauge transformations not only transform a local group into another local group but also satisfy the group law.

PACS numbers: 12.10.-g, 04.40.Nr, 04.20.Cv, 11.15.-q, 02.40.Ky, 02.20.Qs, MSC2010, 51H25, 53c50, 20F65, 70s15, 70G65, 70G45.