A first look at transition amplitudes in (2+1)-dimensional causal dynamical triangulations

TL;DR

This paper investigates transition amplitudes in (2+1)-dimensional causal dynamical triangulations, revealing insights into quantum spacetime geometry, artifacts of lattice regularization, and potential dominance of Lorentzian de Sitter space.

Contribution

It provides the first numerical simulations of transition amplitudes in (2+1)D CDT with fixed boundary geometries, and compares these to theoretical expectations, highlighting artifacts and Lorentzian dominance.

Findings

Transition amplitudes align with effective gravitational action.

Stalks are numerical artifacts of lattice regularization.

Evidence suggests Lorentzian de Sitter space dominates ground state.

Abstract

We study a lattice regularization of the gravitational path integral--causal dynamical triangulations--for (2+1)-dimensional Einstein gravity with positive cosmological constant in the presence of past and future spacelike boundaries of fixed intrinsic geometries. For spatial topology of a 2-sphere, we determine the form of the Einstein-Hilbert action supplemented by the Gibbons-Hawking-York boundary terms within the Regge calculus of causal triangulations. Employing this action we numerically simulate a variety of transition amplitudes from the past boundary to the future boundary. To the extent that we have so far investigated them, these transition amplitudes appear consistent with the gravitational effective action previously found to characterize the ground state of quantum spacetime geometry within the Euclidean de Sitter-like phase. Certain of these transition amplitudes convincingly demonstrate that the so-called stalks present in this phase are numerical artifacts of the lattice regularization, seemingly indicate that the quantization technique of causal dynamical triangulations differs in detail from that of the no-boundary proposal of Hartle and Hawking, and possibly represent the first numerical simulations of portions of temporally unbounded quantum spacetime geometry within the causal dynamical triangulations approach. We also uncover tantalizing evidence suggesting that Lorentzian not Euclidean de Sitter spacetime dominates the ground state on sufficiently large scales.