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In this paper we research all possible finite-dimensional representations andcorresponding values of the Barbero-Immirzi parameter contained in EPRLsimplicity constraints by using Naimark's fundamental theorem of the Lorentzgroup representation theory. It turns out that for each non-zero pure imaginarywith 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 simplicityconstraints, such that the corresponding Lorentz representation is finitedimensional. The converse is also true - for each finite-dimensional Lorentzrepresentation solution of the simplicity constraints $(n, \rho)$, theassociated Barbero-Immirzi parameter is non-zero pure imaginary with rationalmodulus, $\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0$. We solve thesimplicity constraints with respect to the Barbero-Immirzi parameter and thenuse Naimark's fundamental theorem of the Lorentz group representations to findall finite-dimensional representations contained in the solutions.

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