On the UV dimensions of Loop Quantum Gravity

TL;DR

This paper demonstrates that various definitions of spacetime dimension converge in Loop Quantum Gravity's effective regime and shows how UV dimensions can constrain quantum corrections, with a key result of approximately 2.5 dimensions.

Contribution

It unifies different dimensionality concepts in LQG and links UV dimensions to quantum correction ambiguities, providing a new way to constrain quantum gravity models.

Findings

All definitions of spacetime dimension agree in LQG's effective regime.

The UV dimension is approximately 2.5 with quantum corrections.

UV dimensions can constrain ambiguities in quantum gravity modifications.

Abstract

Planck-scale dynamical dimensional reduction is attracting more and more interest in the quantum-gravity literature since it seems to be a model independent effect. However different studies base their results on different concepts of spacetime dimensionality. Most of them rely on the \textit{spectral} dimension, others refer to the \textit{Hausdorff} dimension and, very recently, it has been introduced also the \textit{thermal} dimension. We here show that all these distinct definitions of dimension give the same outcome in the case of the effective regime of Loop Quantum Gravity (LQG). This is achieved by deriving a modified dispersion relation from the hypersurface-deformation algebra with quantum corrections. Moreover we also observe that the number of UV dimensions can be used to constrain the ambiguities in the choice of these LQG-based modifications of the Dirac spacetime algebra. In this regard, introducing the \textit{polymerization} of connections i.e. $K \rightarrow \frac{\sin(\delta K)}{\delta}$, we find that the leading quantum correction gives $d_{UV} = 2.5$. This result may indicate that the running to the expected value of two dimensions is ongoing, but it has not been completed yet. Finding $d_{UV}$ at ultra-short distances would require to go beyond the effective approach we here present.