On Geometrization of Spinors in a Complex Spacetime

TL;DR

This paper proposes a geometric framework for spinor fields in complex spacetime, revealing multiple spinor sets with SU(3) and U(1) symmetries, linking geometry with quantum field properties.

Contribution

It introduces a novel geometric approach to spinors using parametric coordinates on null manifolds, connecting spinor behavior with complex internal hyperspaces.

Findings

Identifies eight sets of parametric coordinate systems behaving as spinors.

Shows these spinors form two SU(3) triplets with internal rotations.

Demonstrates coupling of seven spinors with a U(1) field.

Abstract

While general relativity provides a complete geometric theory of gravity, it fails to explain the other three forces of nature, i.e., electromagnetism and weak and strong interactions. We require the quantum field theory (QFT) to explain them. Therefore, in this article, we try to geometrize the spinor fields. We define a parametric coordinate system in the tangent space of a null manifold and show that these parametric coordinates behave as spinors. By introducing a complex internal hyperspace on a tangent space of a null manifold, we show that we can get eight sets of such parametric coordinate systems that can behave as eight spinor fields. These spinor fields contain two triplets that can rotate among themselves under SU(3). Seven of these spinor fields also couple with a U(1) field with different strengths. We also show that while these spinors can be assigned a tensor weight $1/2$ or $-1/2$, in a $L^p$ space where the coordinates, instead of adding up in quadrature, add up in $p$th power, we can get a parametric space that contains similar spinors of tensor weight $1/p$.