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 nonzero 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 nonzero 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.