Dynamically Triangulating Lorentzian Quantum Gravity

TL;DR

This paper introduces a non-perturbative Lorentzian quantum gravity model using simplicial regularization, providing a well-defined Wick rotation and analyzing its geometric properties in 3 and 4 dimensions.

Contribution

It offers a complete description of a Lorentzian dynamical triangulation approach with a well-defined Hamiltonian and Monte Carlo methods, avoiding Euclidean phase pathologies.

Findings

The model has a well-defined transfer matrix and Hamiltonian.

Pathological phases in Euclidean models are absent in the Lorentzian case.

The approach is suitable for numerical simulations of quantum gravity.

Abstract

Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d less than 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation. We present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4. This includes a derivation of Lorentzian simplicial manifold constraints, the gravitational actions and their Wick rotation. We define a transfer matrix for the system and show that it leads to a well-defined self-adjoint Hamiltonian. In view of numerical simulations, we also suggest sets of Lorentzian Monte Carlo moves. We demonstrate that certain pathological phases found previously in Euclidean models of dynamical triangulations cannot be realized in the Lorentzian case.