Dynamically Triangulating Lorentzian Quantum Gravity

TL;DR

This work develops a non-perturbative, Lorentzian quantum gravity framework via causal dynamical triangulations, employing a well-defined Wick rotation on discretized Lorentzian geometries to enable Euclidean computational treatment. It constructs a transfer-matrix formalism and proves key properties (symmetry, positivity) to support a Hamiltonian description, while proposing ergodic Monte Carlo moves to explore the quantum geometry. In 3D and 4D, the approach yields explicit actions and topological identities, shows the Lorentzian model avoids Euclidean pathologies such as branched polymers and crumpling, and establishes dimension-specific kinematic bounds that constrain the space of admissible geometries. The results provide a robust, regulator-based route toward a continuum quantum gravity theory, with 3D results already indicating extended geometries and 4D presenting both challenges and a clear path for investigation and potential matter coupling.

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.

Figures (5)

• Figure 1: A (3,1)- and a (2,2)-tetrahedron in three dimensions.
• Figure 2: A (4,1)- and a (3,2)-tetrahedron in four dimensions.
• Figure 12: One of a sequence of toroidal sandwich configurations at $\tau +1/2$ which saturate the inequality in (\ref{['inequa']}) in the large-volume limit (opposite sides to be identified). Light links correspond to triangle edges at time $\tau$, dark links to triangle edges at time $\tau +1$.
• Figure 13: Positive (a) and negative (b) Euclidean deficit angles $\delta$.
• Figure 14: Positive space-like (a) and time-like (b) Lorentzian deficit angles $\eta$.