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

Abstract

We argue that the effective gauge group for 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 pure LQG, implying a modification of the spectra of area and volume operators. Our observations imply a new value of γ ≈ 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.

Flux-area operator and black hole entropy

“…1. The first assumption would be equivalent to replacing SUð2Þ by SOð3Þ in LQG (see [14,20]). The second assumption had been believed in for a while, but turned out to be incorrect [3].…”

“…Within each of these subsets of area eigenvalues the right proportionality between entropy and area can be found after fixing an appropriate value for even or odd so if one can physically justify the elimination of one of these two sectors in the spectrum of the area operator one would get the desired result without any coarse graining. Given that the spacing between consecutive eigenvalues depends on the choice of it is actually possible to have the type of area quantization proposed in [9] with a distance between consecutive eigenvalues of an integer multiple of 4 log2 or connect with some proposals involving the choice of SOð3Þ as the internal symmetry group [14,20]. Another potential problem that should be faced at some point is the recovery of a thermal spectrum for the Hawking radiation with such an equally-spaced spectrum [21].…”

Super-group field cosmology

“…Even though colour has already been incorporated in group field theory [35,36], it has not so far been incorporated in any model of group field cosmology. Furthermore, the supersymmetric loop quantum gravity has been studied only at the second quantized level [37][38][39][40], but has never been studied at the level of third quantized loop quantum gravity or loop quantum cosmology. In fact, no third quantized theory, including group field theory, has ever been supersymmetrized.…”