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

TL;DR

This paper introduces a novel numerical method using complex contour deformation to efficiently evaluate Lorentzian simplicial quantum gravity path integrals, overcoming the longstanding sign problem and enabling new physical insights.

Contribution

It develops a holomorphic formulation of Lorentzian simplicial gravity and demonstrates the effectiveness of complex contour deformation techniques for numerical simulations.

Findings

Efficient Monte Carlo simulations for Lorentzian quantum gravity achieved.

A holomorphic formula for Lorentzian simplicial gravity provided.

Proved a complex version of the Gauss-Bonnet theorem.

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.