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

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.

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