On some feature and application of the Faddeev formulation of gravity

TL;DR

This paper explores the Faddeev formulation of gravity, highlighting its unique feature of allowing discontinuous fields and analyzing the quantization of surface areas, leading to a specific area spectrum related to the Barbero-Immirzi parameter.

Contribution

It introduces the application of the Faddeev formulation to discrete spacetime and derives the area spectrum proportional to the Barbero-Immirzi parameter, including its value for ten-dimensional spacetime.

Findings

Area spectrum proportional to the Barbero-Immirzi parameter

Discontinuous fields enable independent elementary spacetime pieces

Barbero-Immirzi parameter value for 10-dimensional spacetime is approximately 0.39

Abstract

In the Faddeev formulation of gravity, the metric is regarded as composite field, bilinear of $d = 10$ 4-vector fields. A unique feature is that this formulation admits the discontinuous fields. On the discrete level, when spacetime is decomposed into the elementary 4-simplices, this means that the 4-simplices may not coincide on their common faces, that is, be independent. We apply this to the particular problem of quantization of the surface regarded as that composed of virtually independent elementary pieces (2-simplices). We find the area spectrum being proportional to the Barbero-Immirzi parameter $\gamma$ in the Faddeev gravity and described as a sum of spectra of separate areas. According to the known in the literature approach, we find that $\gamma$ exists ensuring Bekenstein-Hawking relation for the statistical black hole entropy for arbitrary $d$, in particular, $\gamma = 0.39...$ for genuine $d = 10$.