Tensor network approach to 2d Lorentzian quantum Regge calculus

TL;DR

This paper applies tensor renormalization group methods to a 2D Lorentzian quantum Regge calculus model, successfully reproducing exact results and suggesting the model can describe smooth geometries, highlighting TRG's potential in quantum gravity.

Contribution

It introduces a tensor network formulation of 2D Lorentzian QRC and demonstrates TRG's effectiveness in analyzing its properties and suppressing problematic configurations.

Findings

Reproduces exact space-time area values using TRG.

Finds potential suppression of pinched geometries in the Lorentzian model.

Suggests TRG as a promising tool for numerical quantum gravity studies.

Abstract

We demonstrate a tensor renormalization group (TRG) calculation for a two-dimensional Lorentzian model of quantum Regge calculus (QRC). This model is expressed in terms of a tensor network by discretizing the continuous edge lengths of simplicial manifolds and identifying them as tensor indices. The expectation value of space-time area, which is obtained through the higher-order TRG method, nicely reproduces the exact value. The Lorentzian model does not have the spike configuration that was an obstacle in the Euclidean QRC, but it still has a length-divergent configuration called a pinched geometry. We find a possibility that the pinched geometry is suppressed by checking the average edge length squared in the limit where the number of simplices is large. This implies that the Lorentzian model may describe smooth geometries, although the investigation of the higher moments is required to make the statement more conclusive. Our results also indicate that TRG is a promising approach to numerical study of simplicial quantum gravity.