Anomalous dimension in semiclassical gravity

TL;DR

This paper explores the non-commutative geometry of relativistic particles in 3D Einstein gravity, revealing a scale-dependent spectral dimension that becomes non-integer at quantum scales.

Contribution

It introduces a non-commutative heat-kernel framework and demonstrates the scale-dependent spectral dimension in a quantum gravity context.

Findings

Spectral dimension decreases at Planckian scales

Non-commutative heat kernel is constructed using Fourier transform

Spectral dimension reaches below three at small scales

Abstract

The description of the phase space of relativistic particles coupled to three-dimensional Einstein gravity requires momenta which are coordinates on a group manifold rather than on ordinary Minkowski space. The corresponding field theory turns out to be a non-commutative field theory on configuration space and a group field theory on momentum space. Using basic non-commutative Fourier transform tools we introduce the notion of non-commutative heat-kernel associated with the Laplacian on the non-commutative configuration space. We show that the spectral dimension associated to the non-commutative heat kernel varies with the scale reaching a non-integer value smaller than three for Planckian diffusion scales.