Tetrad in $$SL(2,C) \times SU(2) \times U(1)$$ Yang–Mills–Weyl Spacetimes
Alcides Garat
Abstract
A new set of tetrads is introduced within the framework of $$SL(2,C) \times SU(2) \times U(1)$$ Yang–Mills–Weyl field theories in curved spacetimes. A set of field differential equations is analyzed concerning the transformation properties of the tetrad vectors that can be constructed out of the fields satisfying these equations. In previous work it has been found the technique to build these tetrads. Here we are able to prove additional construction properties regarding the new “internal” groups of transformations involved in the formulation. In particular we show how to switch the bosonic “nesting” of tetrads associated to both groups, $$SL(2,C)$$ and $$SU(2)$$ . We also show that the usual two vector fields $${{X}^{\mu }}$$ , $${{Y}^{\mu }}$$ , necessary to gauge the tetrads, can be constructed using currents, that is, Weyl spinors in curved spacetime. Employing our new tetrads we prove that the local group $$SL(2,C)$$ is isomorphic to local groups of tetrad transformations, equivalent to say that the gravitational field is a gauge field. A conjecture is raised in relation with the asymptotic properties of these tetrads. We conjecture that within the set of solutions to the classical field equations we are introducing, there could be one that we might be able to associate to or represent the geometry of a microparticle like the Neutrino or its antiparticle, for instance. We conjecture that we can associate spacetimes to microparticles since all the local symmetries of the standard model can be realized in four-dimensional curved Lorentzian spacetimes. The group isomorphisms between $$U(1)$$ , $$SU(2) \times U(1)$$ or $$SU(3) \times SU(2) \times U(1)$$ on one hand, and local groups of tetrad transformations on the other hand have already been presented in previous manuscripts. In this regard, the asymptotic limit for this set of equations, in particular the Weyl equation on a Minkowskian background in the “far” region, would be the starting point for the standard Quantum Field Theory associated to this particular equation. Standard Quantum Field Theories are then interpreted as devices that deal with perturbative quantum “interactions” between geometries that radiate (create) and absorbe (annihilate) wave modes, but are otherwise never related to the spacetime background geometries that undergo the radiation or absorption processes. Quantum Field Theories just deal with perturbative interacting phenomena in the asymptotic limit to these hypothesized background spacetimes. Isomorphism theorems involving the group structure $$SL(2,C) \times SU(2) \times U(1)$$ are proved. A gauge invariant method to diagonalize the stress-energy tensor is discussed. Beyond the possible association of spacetimes to microparticles, the results found in this work relating gravitation to local internal transformations within the framework of $$SL(2,C) \times SU(2) \times U(1)$$ Yang–Mills–Weyl field theories, are worth being discussed by themselves as proper mathematical and geometrical results. This is a paper about grand Standard Model gauge theories – General Relativity gravity unification and grand group unification in four-dimensional curved Lorentzian spacetimes.