On the UV dimensions of Loop Quantum Gravity

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 $\mathit{\text{spectral}}$ dimension, others refer to the $\mathit{\text{Hausdorff}}$ dimension and, very recently, it has been introduced also the $\mathit{\text{thermal}}$ 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 $\mathit{\text{polymerization}}$ of connections i.e. $K\to \frac{\sin (\delta K)}{\delta }$, 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.