On representations of a chiral alternative to vierbein

TL;DR

This paper explores a novel chiral alternative to the vierbein in quantum gravity, representing it via Dirac gamma matrices and relating it to Killing-Yano tensors, aiming to deepen understanding of spacetime structure.

Contribution

It constructs specific representations of the chiral vierbein alternative using Dirac gamma matrices and extends these to curved spacetime, proposing a new physical interpretation.

Findings

Representation of the chiral alternative in terms of gamma matrices

Extension to curved spacetime using Penrose-Newman formalism

Conjecture linking these objects to Killing-Yano tensors

Abstract

In an attempt to facilitate the construction of a quantum theory of gravity, 't Hooft has considered a chiral alternative to the vierbein field in general theory of relativity. These objects, $ f^a{}_{\mu\nu}$, behave like the "cube root" of the metric tensor. We try to construct specific representations of these tensors in terms of Dirac $\gamma$ matrices in Euclidean and Minkowski space and promote these to curved space through Penrose-Newman formalism. We conjecture that these new objects, with physical significance, are the analog of Killing-Yano tensors.