3D Lorentzian loop quantum gravity and the spinor approach

We consider the generalization of the ``spinor approach'' to the Lorentzian case, in the context of three-dimensional loop quantum gravity with cosmological constant $\mathrm{\ensuremath{\Lambda}}=0$. The key technical tool that allows this generalization is the recoupling theory between unitary infinite-dimensional representations and nonunitary finite-dimensional ones, obtained in the process of generalizing the Wigner--Eckart theorem to SU(1,1). We use SU(1,1) tensor operators to build observables and a solvable quantum Hamiltonian constraint, analogous to the one introduced by V. Bonzom and his collaborators in the Euclidean case (with both $\mathrm{\ensuremath{\Lambda}}=0$ and $\mathrm{\ensuremath{\Lambda}}\ensuremath{\ne}0$). We show that the Lorentzian Ponzano-Regge amplitude is the solution of the quantum Hamiltonian constraint by recovering the Biedenharn-Elliott relation [generalized to the case where unitary and nonunitary SU(1,1) representations are coupled to each other]. Our formalism is sufficiently general that both the Lorentzian and the Euclidean case can be recovered (with $\mathrm{\ensuremath{\Lambda}}=0$).

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