Nonperturbative 3D Lorentzian quantum gravity

Abstract

We have recently introduced a discrete model of Lorentzian quantum gravity, given as a regularized nonperturbative state sum over simplicial Lorentzian space-times, each possessing a unique Wick rotation to the Euclidean signature. We investigate here the phase structure of the Wick-rotated path integral in three dimensions with the aid of computer simulations. After fine tuning the cosmological constant to its critical value, we find a whole range of the gravitational coupling constant k 0 for which the functional integral is dominated by nondegenerate three-dimensional space-times. We therefore have a situation in which a well-defined ground state of extended geometry is generated dynamically from a nonperturbative state sum of fluctuating geometries. Remarkably, its macroscopic scaling properties resemble those of a semiclassical spherical universe. Measurements so far indicate that k 0 defines an overall scale in this extended phase, without affecting the physics of the continuum limit. These findings provide further evidence that discrete Lorentzian gravity is a promising candidate for a nontrivial theory of quantum gravity.

A proper-time cure for the conformal sickness in quantum gravity

“…One can indeed identify simplicial geometries (whose spatial volumes oscillate rapidly in proper time) with a large and negative Euclidean action. Nevertheless, it has been established by numerical simulations that for the 3d model there is a large range of the gravitational coupling constant where such modes do not play a role [16,17]. This entails a win of "entropy over energy", that is, well-behaved geometries outnumber completely the potentially dangerous ones associated with conformal excitations.…”

“…As we have mentioned earlier, this scenario seems to be realized in the non-perturbative approach based on piece-wise linear Lorentzian geometries [10,11,12], which is one of the few well-defined regularized path integrals available that do not rely on any fixed background geometry. Numerical investigations of the corresponding continuum theory in d = 3 have shown that for sufficiently large bare Newton constant there is a phase whose ground state has a stable and extended geometry, without the large fluctuations indicative of conformal excitations [16].…”

Hexagon model for 3D Lorentzian quantum cosmology

“…5 We recognize here a simplified feature of the hexagon model, compared with the most general dynamically triangulated 3D gravity model ͓4,5͔, even if we restricted its integer-t slices to be flat tori. Namely, although the toroidal two-geometries forming the spatial boundaries of a space-time sandwich ⌬tϭ1 by no means fix the threegeometry in between the two slices, they determine essentially uniquely the value of the sandwich action S(⌬tϭ1).…”

“…A main task in solving the model is therefore the computation of the number of distinct interpolating 3-geometries between two adjacent flat two-tori. 5 The wedge flip is an obvious candidate for a Monte Carlo move in numerical simulations of the hexagon model; it will have to be augmented by moves that can change the 's and other physical variables. Before we can give a precise definition of this combinatorial problem, we still need to specify how we are going to parametrize the coloring of the rhombic intersection pattern.…”

Unexpected spin-off from quantum gravity

“…In "pure" gravity (i.e. without matter coupling) this is sufficient to produce geometries whose effective (or Hausdorff) large-scale dimension d H -in the sense of ensemble averages -equals the dimension d of their microscopic triangular building blocks, for d = 2 [8], d = 3 [9] and d = 4 [10], which was not the case for the corresponding Euclidean models, and is an indication that the geometries are much better behaved.…”