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 space-time dimensionality. Most of them rely on the spectral dimension; others refer to the Hausdorff dimension; and, very recently, the thermal dimension has also been introduced. 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 space-time algebra. In this regard, introducing the polymerization of connections, that is, K -> 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 ultrashort distances would require going beyond the effective approach we here present.