Markov Chain Monte Carlo methods for graph refinement in Spinfoam Cosmology

TL;DR

This paper investigates the stability and quantum geometric properties of Lorentzian spinfoam amplitudes under graph refinement, using MCMC methods to compute boundary correlations and entanglement in a quantum cosmological model.

Contribution

It introduces a numerical approach combining MCMC and advanced computational techniques to analyze graph refinement effects in Lorentzian spinfoam cosmology.

Findings

Transition amplitudes remain stable under refinement.

Average boundary geometry is unchanged by refinement.

Quantum fluctuations and correlations are affected by additional degrees of freedom.

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.