Revisiting EPRL: All Finite-Dimensional Solutions by Naimark's Fundamental Theorem

TL;DR

This paper classifies all finite-dimensional solutions of the EPRL simplicity constraints by linking them to specific pure imaginary rational Barbero-Immirzi parameters using Naimark's theorem.

Contribution

It provides a complete characterization of finite-dimensional Lorentz group representations satisfying EPRL constraints, revealing the precise form of the Barbero-Immirzi parameter involved.

Findings

Finite-dimensional solutions exist for pure imaginary rational Barbero-Immirzi parameters.

Each finite-dimensional Lorentz representation corresponds to a specific Barbero-Immirzi parameter.

The classification is complete and bidirectional, linking solutions and parameters.

Abstract

In this paper we research all possible finite-dimensional representations and corresponding values of the Barbero-Immirzi parameter contained in EPRL simplicity constraints by using Naimark's fundamental theorem of the Lorentz group representation theory. It turns out that for each non-zero pure imaginary with rational modulus value of the Barbero-Immirzi parameter $\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0$, there is a solution of the simplicity constraints, such that the corresponding Lorentz representation is finite dimensional. The converse is also true - for each finite-dimensional Lorentz representation solution of the simplicity constraints $(n, \rho)$, the associated Barbero-Immirzi parameter is non-zero pure imaginary with rational modulus, $\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0$. We solve the simplicity constraints with respect to the Barbero-Immirzi parameter and then use Naimark's fundamental theorem of the Lorentz group representations to find all finite-dimensional representations contained in the solutions.