Non-perturbative 3d Lorentzian Quantum Gravity

TL;DR

The paper investigates a non-perturbative, discrete model of 3D Lorentzian quantum gravity built from causal triangulations with a Wick-rotated Euclidean action. Through Monte Carlo simulations, it maps the phase diagram, identifying a central extended phase where a ground-state 3D geometry spontaneously emerges and scales consistently with a semi-classical spherical universe. The gravitational coupling $k_0$ primarily sets an overall length scale in this phase and does not seem to renormalize, suggesting a continuum limit without fine-tuning of $k_0$. These results provide evidence that Lorentzian dynamical triangulations yield a well-defined continuum theory in 3D and bolster the case for applying the framework to 4D with potential matter coupling.

Abstract

We have recently introduced a discrete model of Lorentzian quantum gravity, given as a regularized non-perturbative state sum over simplicial Lorentzian space-times, each possessing a unique Wick rotation to 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 non-degenerate three-dimensional space-times. We therefore have a situation in which a well-defined ground state of extended geometry is generated dynamically from a non-perturbative state sum of fluctuating geometries. Remarkably, its macroscopic scaling properties resemble those of a semi-classical 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 {\it Lorentzian} gravity is a promising candidate for a non-trivial theory of quantum gravity.

Figures (11)

• Figure 1: The three types of tetrahedral building blocks used in discrete 3d Lorentzian quantum gravity, and their location with respect to the spatial slices of constant integer $t$.
• Figure 5: How the tetrahedral building blocks give rise to dual bi-coloured graphs (drawn as red and blue horizontal lines) at half-integer $t$.
• Figure 6: Graphical illustration of the Monte Carlo moves in terms of dual bi-coloured graphs. In moves 1 and 2&3, dual links from two adjacent sandwiches are affected. Moves 4&5 take place within a given $[t,t\!+\! 1]$-sandwich.
• Figure 7: The order parameter $\tau=N_{22}/N_3$ for configurations with $T\!=\! 64$, and $N_3\!=\! 16,000$ (long dark curve) and 64,000 (short light curve), plotted as a function of $k_0$. The curve is a linear interpolation between data points. (Error bars smaller than width of curve.)
• Figure 8: The order parameter $\tau=N_{22}/N_3$ for configurations with $T \!=\! 1$ and $N_3\!=\! 16,000$, but with free boundary conditions, plotted as a function of $k_0$. The curve is a linear interpolation between data points. (Error bars smaller than width of curve.)