Quantum gravity in the triangular gauge

TL;DR

This paper introduces a complete gauge fixing of Lorentz symmetry in quantum gravity using a triangular gauge, simplifying the phase space and enabling new Hilbert space representations that facilitate the study of non-degenerate quantum geometries.

Contribution

It presents a novel gauge fixing approach that reduces the gauge symmetry, opening up alternative quantum gravity formulations beyond Loop Quantum Gravity.

Findings

Classical solution of Gauss constraints in the triangular gauge.

New Hilbert space representations free of non-Abelian gauge complications.

Enhanced ability to analyze non-degenerate quantum geometries.

Abstract

Vielbeins are necessary when coupling General Relativity (GR) to fermionic matter. This enhances the gauge group of GR to include local Lorentz transformations. In view of a reduced phase space formulation of quantum gravity, in this work we completely gauge fix that Lorentz gauge symmetry by using a so-called triangular gauge. Having solved the Gauss constraints already classically opens access to new Hilbert space representations which are free of the complications that otherwise arise due to a non Abelian gauge symmetry. In that sense, a connection formulation as being pursued in Loop Quantum Gravity is no longer the only practicable option and other less dimension dependent representations e.g. based on triads and even metrics suggest themselves. These formulations make it easier to identify states representing non-degenerate quantum geometries and thus to investigate the hypersurface deformation algebra which implicitly assumes non-degeneracy.