Complex, Lorentzian, and Euclidean simplicial quantum gravity: numerical methods and physical prospects

Abstract

Evaluating gravitational path integrals in the Lorentzian has been a long-standing challenge due to the numerical sign problem. We show that this challenge can be overcome in simplicial quantum gravity. By deforming the integration contour into the complex, the sign fluctuations can be suppressed, for instance using the holomorphic gradient flow algorithm. Working through simple models, we show that this algorithm enables efficient Monte Carlo simulations for Lorentzian simplicial quantum gravity. In order to allow complex deformations of the integration contour, we provide a manifestly holomorphic formula for Lorentzian simplicial gravity. This leads to a complex version of simplicial gravity that generalizes the Euclidean and Lorentzian cases. Outside the context of numerical computation, complex simplicial gravity is also relevant to studies of singularity resolving processes with complex semi-classical solutions. Along the way, we prove a complex version of the Gauss-Bonnet theorem, which may be of independent interest.

Light ray fluctuations in simplicial quantum gravity

“…is the vertex share of the Lorentzian area. In the continuum limit −δ v /A v approaches R for θ defined in (12) [21,8]. 2 This explains why −k v A v δv Av = −kδ v forms the lattice version of the Einstein-Hilbert term ik d D x √ −gR.…”

“…In simplicial quantum gravity [9,10,11,12,13,14], the sum is over simplicial spacetime configuration which describe curved spacetime by combining flat simplicies (Figure 3). While extensive works have been carried out in the past in the Euclidean signature, the present study of light ray fluctuations is based on a Lorentzian version of the theory [8] that sums Lorentzian simplicial spacetime configurations [21,22].…”

“…In this work we study light ray fluctuations across a quantum region of spacetime in Lorentzian simplicial quantum gravity [8]. The study of gravitational path integrals defined in terms of simplicial spacetime configurations á la Regge [9] has a long history [10,11,12,13,14].…”

“…The study of gravitational path integrals defined in terms of simplicial spacetime configurations á la Regge [9] has a long history [10,11,12,13,14]. While previous works focused on the Euclidean theory, there has been a growing interest in the Lorentzian theory in recent years [15,16,17,18,19,8,20]. As in causal dynamical triangulation, the light ray paths on a piecewise flat simplicial spacetime configuration is obtainable, in particular using Lorentzian trigonometry which we illustrate in Section 3.1.…”

Backpropagating Hybrid Monte Carlo algorithm for fast Lefschetz thimble calculations

“…[16].). It has attracted attention also in the context of quantum cosmology since it provides a unique prescription to make the original oscillatory path integral of Lorentzian quantum gravity well defined [17,18]. 3 Historically, this idea was born out of the previous proposal based on tempering [23,24,25], which amounts to sampling configurations on many integration contours in parallel, swapping the configurations after a fixed interval with certain probability.…”