This is a short informal commentary to
Leonid Perlov, Michael Bukatin, "Revisiting EPRL: All Finite-Dimensional Solutions by Naimark's Fundamental Theorem", Annales Henri Poincaré, 18 (9), pp. 3035-3048, September 2017.
This paper links together two topics causing rather intense disagreements in quantum gravity research: whether quantum theory should be unitary or non-unitary, and whether Barbero-Immirzi parameter appearing in loop quantum gravity should be real or imaginary.
The mainstream viewpoint states that quantum theory should be unitary, and that Barbero-Immirzi parameter should be real. The opposite viewpoints are influential minority viewpoints (for example, Penrose supports both minority viewpoints here).
This paper shows that these two issues are not independent from each other, but that non-unitary quantum theories and imaginary values of Barbero-Immirzi parameter like to appear together, at least in the context of spin foam loop quantum gravity.
More specifically, in spin foam loop quantum gravity the unitarity or non-unitarity of quantum evolution is expessed as follows: the representations of Lorentz group decorating the edges of spin networks might be unitary or non-unitary. The use of unitary operators in quantum mechanics is tightly related to invariance with respect to shifts along the time axis. In general relativistic framework, there is no preferred time variable and there is no inherent reason to choose unitary operators.
On the other hand, Barbero-Immirzi parameter is tightly connected with the notion of "quantum of area", although they are not quite the same, so the question whether physical area at the fundamental level is real or imaginary is more complicated than just deciding whether Barbero-Immirzi parameter is real or imaginary.
Our paper revisits one of the most popular formulations of loop quantum gravity, namely EPRL [see J.Engle, E.Livine, R.Pereira, C.Rovelli, "LQG vertex with finite Immirzi parameter", https://arxiv.org/abs/0711.0146 and discussion therein].
The simplicity constraints are the equations that make the topological four-dimensional BF model (background free model) the Einstein Quantum Gravity. They contain the Barbero-Immirzi parameter, that is usually set to some constant value before solving the constraints. In our paper we freed Barbero-Immirzi parameter from being a constant and solved the simplicity constraints with respect to it. Among the solutions are two particularly elegant series of Barbero-Immirzi parameters: the rational pure imaginary series yielding all finite-dimensional representations of Lorentz group (those are non-unitary spinor representations, and this belongs to the case of the non-unitary physical space and evolution), and the real series studied earlier (this corresponds to the unitary evolution).
The importance of this result is that the possibility of the non-unitary quantum evolution is obtained as a result of the solution of simplicity constraints (equations which make topological 4D BF model the Einstein Quantum Gravity).