3d Lorentzian loop quantum gravity and the spinor approach

TL;DR

This paper extends the spinor approach to 3d Lorentzian loop quantum gravity using SU(1,1) recoupling theory, constructing observables and a solvable quantum Hamiltonian constraint, and demonstrating the Lorentzian Ponzano-Regge amplitude as a solution.

Contribution

It introduces a novel generalization of the spinor approach to Lorentzian 3d quantum gravity using SU(1,1) tensor operators and recoupling theory, unifying Euclidean and Lorentzian formalisms.

Findings

Constructed observables using SU(1,1) tensor operators.

Developed a solvable quantum Hamiltonian constraint.

Showed the Lorentzian Ponzano-Regge amplitude solves the constraint.

Abstract

We consider the generalization of the "spinor approach" to the Lorentzian case, in the context of 3d loop quantum gravity with cosmological constant $\Lambda=0$. The key technical tool that allows this generalization is the recoupling theory between unitary infinite-dimensional representations and non-unitary 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, analogue of the one introduced by V. Bonzom and his collaborators in the Euclidean case (with both $\Lambda=0$ and $\Lambda\neq0$). We show that the Lorentzian Ponzano-Regge amplitude is solution of the quantum Hamiltonian constraint by recovering the Biedenharn-Elliott relation (generalized to the case where unitary and non-unitary 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 $\Lambda=0$).