The EPRL intertwiners and corrected partition function

TL;DR

This paper investigates the properties of the EPRL intertwiners, proving injectivity and non-isometry of the EPRL map, and derives a new complete formula for the partition function that extends beyond traditional SU(2) spin-foam models.

Contribution

It provides a rigorous analysis of the EPRL map's properties and derives a corrected, comprehensive partition function formula surpassing previous SU(2) spin-foam models.

Findings

EPRL map is injective for n-valent vertices from SO(3) to SO(3)xSO(3)

EPRL map is not isometric, affecting amplitude formulations

Derived a new, complete partition function formula extending beyond SU(2) models

Abstract

Do the SU(2) intertwiners parametrize the space of the EPRL solutions to the simplicity constraint? What is a complete form of the partition function written in terms of this parametrization? We prove that the EPRL map is injective for n-valent vertex in case when it is a map from SO(3) into SO(3)xSO(3) representations. We find, however, that the EPRL map is not isometric. In the consequence, in order to be written in a SU(2) amplitude form, the formula for the partition function has to be rederived. We do it and obtain a new, complete formula for the partition function. The result goes beyond the SU(2) spin-foam models framework.