Non-perturbative Lorentzian Quantum Gravity, Causality and Topology Change

TL;DR

We develop a non-perturbative, Lorentzian 2D quantum gravity by summing over causal geometries in a discrete path integral, which is exactly solvable and yields a Lorentzian continuum propagator admitting a proper-time gauge quantization via the Hamiltonian $\hat{H}= -\partial^2/\partial L^2 + \Lambda$. With cylinder topology and no topology changes, the Lorentzian theory differs from Liouville/matrix-model gravity, exhibiting a finite disc amplitude $W_\Lambda(L)=e^{-\sqrt{\Lambda}L}$ and intrinsic Hausdorff dimension $d_H=2$. Allowing spatial topology changes (baby universes) drives the continuum limit into the Euclidean Liouville universality class, yielding scaling exponents $\varepsilon=1/2$ and $\eta=3/2$, a disc amplitude $W_\Lambda(X)$ matching Euclidean results, and $d_H=4$. The work highlights how causality constraints shape 2D quantum gravity, and how topology changes can reconcile Lorentzian and Euclidean formulations, with potential implications for higher-dimensional quantum gravity.

Abstract

We formulate a non-perturbative lattice model of two-dimensional Lorentzian quantum gravity by performing the path integral over geometries with a causal structure. The model can be solved exactly at the discretized level. Its continuum limit coincides with the theory obtained by quantizing 2d continuum gravity in proper-time gauge, but it disagrees with 2d gravity defined via matrix models or Liouville theory. By allowing topology change of the compact spatial slices (i.e. baby universe creation), one obtains agreement with the matrix models and Liouville theory.

Figures (5)

• Figure 1: The propagation of a spatial slice from step $t$ to step $t+1$. The ends of the strip should be joined to form a band with topology $S^1 \times [0,1]$.
• Figure 2: The circle of convergence in the complex $g$ plane (radius 1/2), and the critical lines, ending in $g=\pm 1/2$.
• Figure 3: A "baby universe" branches off locally in one time-step.
• Figure 4: An alternative representation of the process in fig. \ref{['branching']}: a "baby universe" is created by a global pinching.
• Figure 5: Marking a vertex in the bulk of $W_\Lambda(X)$. The mark has a distance $T$ from the boundary loop, which itself has one marked vertex.