Effective gauge group of pure loop quantum gravity is SO(3): New estimate of the Immirzi parameter

TL;DR

This paper proposes that the effective gauge group in pure four-dimensional loop quantum gravity is SO(3), leading to a revised estimate of the Immirzi parameter around 0.170, and discusses implications for black hole entropy and spin representations.

Contribution

It demonstrates that the effective gauge group is SO(3), modifies the spectra of geometric operators, and provides a new estimate for the Immirzi parameter based on entropy matching.

Findings

Effective gauge group is SO(3), not SU(2).

New Immirzi parameter estimate is approximately 0.170.

Logarithmic correction coefficient is robust across spin configurations.

Abstract

We argue that the effective gauge group for {\it pure} four-dimensional loop quantum gravity(LQG) is SO(3) (or $SO(3,C)$) instead of SU(2) (or $SL(2,C)$). As a result, links with half-integer spins in spin network states are not realized for {\it pure} LQG, implying a modification of the spectra of area and volume operators. Our observations imply a new value of $\gamma \approx 0.170$ for the Immirzi parameter which is obtained from matching the Bekenstein-Hawking entropy to the number of states from LQG calculations. Moreover, even if the dominant contribution to the entropy is not assumed to come from configurations with the minimum spins, the results of both pure LQG and the supersymmetric extension of LQG can be made compatible when only integer spins are realized for the former, while the latter also contains half-integer spins, together with an Immirzi parameter for the supersymmetric case which is twice the value of the SO(3) theory. We also verify that the $-{1/2}$ coefficient of logarithmic correction to the Bekenstein-Hawking entropy formula is robust, independent of whether only integer, or also half-integer spins, are realized.