Hexagon model for 3D Lorentzian quantum cosmology

Abstract

We formulate a dynamically triangulated model of three-dimensional Lorentzian quantum gravity whose spatial sections are flat two tori. It is shown that the combinatorics involved in evaluating the one-step propagator ͑the transfer matrix͒ is that of a set of vicious walkers on a two-dimensional lattice with periodic boundary conditions and that the entropy of the model scales exponentially with the volume. We also give explicit expressions for the Teichmüller parameters of the spatial slices in terms of the discrete parameters of the 3D triangulations, and reexpress the discretized action in terms of them. The relative simplicity and explicitness of this model make it ideally suited for an analytic study of the conformal-factor cancellation observed previously in Lorentzian dynamical triangulations and of its relation to alternative, reduced phase space quantizations of 3D gravity. I. MOTIVATIONThe approach of Lorentzian dynamical triangulations 1 ͑LDT͒ leads to a well-defined regularized path integral for 3D quantum gravity, as was shown in ͓3,4͔. The phase structure of this statistical model of causal random geometries has been investigated by Monte Carlo methods in the genus-zero case, where the two-dimensional spatial slices are spheres ͓5,6͔. Perhaps its most striking feature is the emergence in the continuum limit of a well-defined ground state behaving macroscopically like a three-dimensional universe ͓5,7,8͔. This is in contrast with perturbative continuum arguments which suggest that in dу3 Euclideanized gravitational path integrals are generically ill defined because of a divergence due to the conformal mode. Since a Wick rotation from Lorentzian to Euclidean space-time geometries is part of the evaluation of the regularized state sums in LDT, one might expect to encounter a similar problem here, but this is not what happens. Instead, all indications point to a nonperturbative cancellation between the conformal term in the action ͑which still has the same structure as in the continuum͒ and entropy contributions to the state sum ͑that is, ''the measure''͒. It should also be emphasized that this cancellation is not achieved by any ad hoc manipulations of the path integral, for example, by isolating the conformal mode and Wick rotating it in a non-standard way ͑in fact, it is quite impossible to isolate this mode in the non-perturbative setting of LDT͒. Further discussions of the conformal-mode problem and its possible non-perturbative resolution can be found in ͓9,7͔.It is obviously of great interest to understand in a more explicit and analytic fashion how this cancellation occurs and how it gives rise to an effective Hamiltonian whose ground state is the one seen in the numerical simulations of 3D Lorentzian dynamical triangulations. Some progress in this direction has been made recently by mapping the threedimensional LDT model to a two-dimensional Hermitian ABAB-matrix model ͓10,11͔. This latter model has a secondorder phase transition which is absent from the LDT model. This comes about because th...

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