Quantum-Only Metrics in Spherically Symmetric Gravity

Giovanni Modanese

doi:10.3390/quantum2020021

Abstract

The Einstein action for the gravitational field has some properties which make of it, after quantization, a rare prototype of systems with quantum configurations that do not have a classical analogue. Assuming spherical symmetry in order to reduce the effective dimensionality, we have performed a Monte Carlo simulation of the path integral with transition probability e − β | S | . Although this choice does not allow to reproduce the full dynamics, it does lead us to find a large ensemble of metric configurations having action | S | ≪ ħ by several magnitude orders. These vacuum fluctuations are strong deformations of the flat space metric (for which S = 0 exactly). They exhibit a periodic polarization in the scalar curvature R. In the simulation we fix a length scale L and divide it into N sub-intervals. The continuum limit is investigated by increasing N up to ∼ 10 6 ; the average squared action ⟨ S 2 ⟩ is found to scale as 1 / N 2 and thermalization of the algorithm occurs at a very low temperature (classical limit). This is in qualitative agreement with analytical results previously obtained for theories with stabilized conformal factor in the asymptotic safety scenario.

Highlights

Efforts towards the unification of General Relativity and Quantum Mechanics into a coherent theory of Quantum Gravity were started long ago and have been intensifying in the last decades
We have studied the gravitational path integral of spherically symmetric space-times independent in time using numerical Monte Carlo methods
The system is first reduced to the spherically symmetric and time-independent setting, before it is discretized in radial direction

Summary

Introduction

Efforts towards the unification of General Relativity and Quantum Mechanics into a coherent theory of Quantum Gravity were started long ago and have been intensifying in the last decades. We made some early contributions to Quantum Gravity in the covariant formulation by showing that the Wilson loop vanishes to leading order [8] and proposing an alternative expression for the static potential of two sources [9]. This formula was used by Muzinich and Vokos [10] and by Hamber and Liu [11], respectively in perturbation theory and in non-perturbative Regge calculus, to give an estimate of quantum corrections to the Newton potential. The vanishing of the Wilson loop (and of curvature correlations [12]) to leading order is only one of the peculiar aspects of the quantum field theory of gravity, that sets it apart from other successful quantum field theories like QED and QCD

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Conclusion