Markov Chain Monte Carlo methods for graph refinement in Spinfoam Cosmology

Abstract

We study the behaviour of the Lorentzian Engle-Pereira-Rovelli-Livine spinfoam amplitude with homogeneous boundary data, under a graph refinement going from five to twenty boundary tetrahedra. This can be interpreted as a wave function of the universe, for which we compute boundary geometrical operators, correlation functions and entanglement entropy. The numerical calculation is made possible by adapting the Metropolis-Hastings algorithm, along with recently developed computational methods appropriate for the deep quantum regime. We confirm that the transition amplitudes are stable against such refinement. We find that the average boundary geometry does not change, but the new degrees of freedom correct the quantum fluctuations of the boundary and the correlations between spatial patches. The expectation values are compatible with their geometrical interpretation and the correlations between neighbouring patches decay when computed across different spinfoam vertices.

34 pages, 18 figures

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• Cosmology in Lorentzian Regge calculus: causality violations, massless scalar field and discrete dynamics
• A Monte Carlo algorithm for spin foam intertwiners
• Effective geometry of Bell-network states on a dipole graph
• (2+1) Lorentzian quantum cosmology from spin-foams: opportunities and obstacles for semi-classicality