In this paper we prove injectivity of the EPRL map for |\gamma|<1, filling the gap of our previous paper.

The kernel and the injectivity of the EPRL map

We calculate the black-hole entropy in loop quantum gravity as a function of the horizon area and provide the exact formula for the leading and sub-leading terms. By comparison with the Bekenstein–Hawking formula, we uniquely fix the value of the 'quantum of area' in the theory.

https://iopscience.iop.org/article/10.1088/0264-9381/21/22/015/pdf

Black hole entropy in Loop Quantum Gravity

Do the SU(2) intertwiners parametrize the space of the Engle, Pereira, Rovelli, Livine (EPRL) solutions to the simplicity constraint? What is the complete form of the partition function written in terms of this parametrization? We prove that the EPRL map is injective in the general n-valent vertex case for the Barbero–Immirzi parameter less than 1. We find, however, that the EPRL map is not isometric. In the consequence, a partition function can be defined either using the EPRL intertwiners Hilbert product or the SU(2) intertwiners Hilbert product. We use the EPRL one and derive a new, complete formula for the partition function. Next, we view it in terms of the SU(2) intertwiners. The result, however, goes beyond the SU(2) spin-foam models' framework and the original EPRL proposal.

https://iopscience.iop.org/article/10.1088/0264-9381/27/16/165020/pdf

The kernel and the injectivity of the EPRL map

References

Black hole entropy in Loop Quantum Gravity

The EPRL intertwiners and corrected partition function

Related Papers (2)

On the injectivity of the nodal map

Characterizing injectivity of classes of maps via classes of matrices