Quantum gravity on a square graph

TL;DR

This paper explores the quantisation of quantum metrics on a Lorentzian square graph, deriving correlation functions and analyzing quantum fluctuations, providing insights into quantum geometry and its relation to scalar field theories.

Contribution

It introduces a functional-integral approach to quantising edge lengths of a square graph and compares quantum metric fluctuations with scalar field theories on similar backgrounds.

Findings

Correlation functions with fixed relative uncertainty of 1/√8

Expected geometry is a rectangle with uniform edge lengths

Quantum metric fluctuations form a finite theory interpolating between scalar and fully fluctuating models

Abstract

We consider functional-integral quantisation of the moduli of all quantum metrics defined as square-lengths $a$ on the edges of a Lorentzian square graph. We determine correlation functions and find a fixed relative uncertainty $\Delta a/<a>= 1/\sqrt{8}$ for the edge square-lengths relative to their expected value $<a>$. The expected value of the geometry is a rectangle where parallel edges have the same square-length. We compare with the simpler theory of a quantum scalar field on such a rectangular background. We also look at quantum metric fluctuations relative to a rectangular background, a theory which is finite and interpolates between the scalar theory and the fully fluctuating theory.