De Sitter horizon entropy from a simplicial Lorentzian path integral

Abstract

The dimension of the Hilbert space of a quantum gravitational system can be written formally as a path integral partition function over Lorentzian metrics. We implement this in a 2+1 dimensional simplicial minisuperspace model in which the system is a spatial topological disc, and recover by contour deformation through a Euclidean saddle the entropy of the de Sitter static patch, up to discretization artifacts. The model illustrates the importance of integration over both positive and negative lapse to enforce the gravitational constraints, and of restriction to complex metrics for which the fluctuation integrals would converge. Although a strictly Lorentzian path integral is oscillatory, an exponentially large partition function results from unavoidable imaginary contributions to the action. These arise from analytic continuation of the simplicial (Regge) action for configurations with codimension-2 simplices where the metric fails to be Lorentzian. In particular, the dominant contribution comes from configurations with contractible closed timelike curves that encircle the boundary of the disc, in close correspondence with recent continuum results.

24 pages, 12 figures, published version

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• On the lapse contour in the gravitational path integral
• Causal structure and topology change in (2+1)-dimensional simplicial gravity
• Euclidean and complex geometries from real-time computations of gravitational Rényi entropies
• (2+1) Lorentzian quantum cosmology from spin-foams: opportunities and obstacles for semi-classicality
• Thermodynamic stability from Lorentzian path integrals and codimension-two singularities
• Quantum dynamics and thermodynamics of a Minkowski-Minkowski wormhole