A Phenomenological Running Fractional Laplacian and the Spectral Dimension Flow of Quantum Spacetime

Abstract

We study a scale-dependent fractional d’Alembertian $(-\Box)^{\alpha(\ell)}$ as a phenomenological model for the dimensional reduction observed in quantum gravity. Within Calcagni’s fractional operator framework, the spectral dimension $d_s = 4/\alpha$ flows from $d_s = 4$ in the infrared to a sub-$4$ value in the ultraviolet, controlled by a single running exponent $\alpha(\ell)$. We fit a three-parameter sigmoid ansatz to the published spectral dimension data of Ambjørn, Jurkiewicz, and Loll (2005), finding $\alpha_{\rm UV} = 2.42 \pm 0.17$ ($d_s^{\rm UV} = 1.65 \pm 0.12$), in agreement with the CDT result $d_s^{\rm UV} = 1.80 \pm 0.25$ at the $0.6\sigma$ level.

We test robustness against four alternative functional forms (exponential, stretched exponential, rational, Gaussian); the fit value $\alpha_{\rm UV}$ varies in the range $1.74$ to $5.0$, so the data are consistent with but do not require Calcagni’s super-renormalizable regime $\alpha > 2$. We numerically verify finiteness of the one-loop tadpole and self-energy integrals at the fit value, in agreement with power counting.

§ 8 compares against the Lauscher–Reuter (2005) asymptotic-safety prediction $d_s^{\rm UV} = 2$ and clarifies the relationship between the running fractional exponent of a scalar kinetic operator and the running cosmological constant of asymptotic safety. The framework remains phenomenological. Unitarity for $\alpha > 1$ requires the Anselmi–Piva fakeon prescription as developed by Calcagni and Rachwał (2022).

DOI

https://doi.org/10.5281/zenodo.19557695