Timelike surfaces in Lorentz covariant loop gravity and spin foam models

TL;DR

This paper develops a Lorentz covariant canonical formulation of general relativity for timelike foliations, quantizes it using loop methods, and finds agreement with spin foam models regarding area spectra.

Contribution

It introduces a Lorentz covariant canonical approach for timelike foliations and demonstrates its consistency with spin foam predictions.

Findings

Derived the spectrum of the area operator for timelike surfaces.

Established a correspondence between loop quantum gravity and spin foam models.

Achieved perfect agreement in area spectra between the two approaches.

Abstract

We construct a canonical formulation of general relativity for the case of a timelike foliation of spacetime. The formulation possesses explicit covariance with respect to Lorentz transformations in the tangent space. Applying the loop approach to quantize the theory we derive the spectrum of the area operator of a two-dimensional surface. Its different branches are naturally associated to spacelike and timelike surfaces. The results are compared with the predictions of Lorentzian spin foam models. A restriction of the representations labeling spin networks leads to perfect agreement between the states as well as the area spectra in the two approaches.