Some remarks on Relativistic Diffusion and the Spectral Dimension Criterion

TL;DR

This paper compares relativistic diffusion descriptions using RSEAC and Telegraph's Equation, showing they agree at high energies with spectral dimension 2, but differ at low energies where TE correctly predicts the spacetime dimension of 4.

Contribution

It demonstrates the equivalence of RSEAC and TE at high energies and clarifies their differences in the infrared regime for spectral dimension calculations.

Findings

Both RSEAC and TE yield spectral dimension 2 at high energies.

TE correctly predicts spectral dimension 4 in the IR regime.

RSEAC predicts spectral dimension 3 in the IR regime.

Abstract

The spectral dimension $d_s$ for high energies is calculated using the Relativistic Schr\"{o}dinger Equation Analytically Continued (RSEAC) instead of the so-called Telegraph's Equation (TE), in both ultraviolet (UV) and infrared (IR) regimens. Regarding the TE, the recent literature presents difficulties related to its stochastic derivation and interpretation, advocating the use of the RSEAC to properly describe the relativistic diffusion phenomena. Taking into account that the Lorentz symmetry is broken in UV regime at Lifshitz point, we show that there exists a degeneracy in very high energies, meaning that both the RSEAC and the TE correctly describe the diffusion processes at these energy scales, at least under the spectral dimension criterion. In fact, both the equations yield the same result, namely, $d_s = 2$, a dimensional reduction that is compatible with several theories of quantum gravity. This result is reached even when one takes into account a cosmological model - the De Sitter one - for a flat Universe. On the other hand, in the IR regimen, such degeneracy is lifted in favor of the approach via TE, due to the fact that only this equation provides the correct value for $d_s$, which is equal to the actual number of spacetime dimensions, i. e., $d_s = 4$, while RSEAC furnishes $d_s=3$, so that a diffusing particle described by this latter experiences a three-dimensional spacetime.