Diffusion in quantum geometry

TLDR: In this article, a general profile of the spectral dimension of the recently introduced multifractional spaces is constructed for the first time, using tools of probability theory and multifractal geometry, and how dimensional flow is controlled by a multiscale fractional diffusion equation, and physically interpreted as a composite stochastic process.

Abstract: The change of the effective dimension of spacetime with the probed scale is a universal phenomenon shared by independent models of quantum gravity. Using tools of probability theory and multifractal geometry, we show how dimensional flow is controlled by a multiscale fractional diffusion equation, and physically interpreted as a composite stochastic process. The simplest example is a fractional telegraph process, describing quantum spacetimes with a spectral dimension equal to 2 in the ultraviolet and monotonically rising to 4 towards the infrared. The general profile of the spectral dimension of the recently introduced multifractional spaces is constructed for the first time. The spectral properties of effective quantum geometries show that the ultraviolet (UV) finiteness of independent theories of quantum gravity is universally associated with a lower spectral dimension of spacetime (typically, dS ∼ 2) at small scales, while dS ∼ 4 in the infrared (IR). Instances are causal dynamical triangulations (CDT) [1], asymptotic safety (QEG) [2, 3], spin foams [4, 5], noncommutative geometry [6], Hoyrava-Lifshitz gravity [7], and other approaches [8]. The change of dimension with the probed scale is known as dimensional reduction or dimensional flow [9]. Understanding its physical meaning is an important piece of the puzzle of quantum gravity, since multiscale behavior is deeply related to the renormalization properties of these theories. Differential geometry and ordinary calculus, as employed in general relativity and field theory, are inadequate to study this and other properties of quantum spacetimes, and stochastic processes and multifractal geometry can offer powerful tools of analysis and novel insight. While there is the tendency to label all multiscale spaces as “fractal,” the accumulated knowledge from these branches of mathematics permit to make sharper statements about the geometric and physical properties of quantum-gravity models. This philosophy inspired the revisiting of a recent problem, the construction of quantum field theories in fractal spacetimes, under a fresh perspective focused on an effective continuum geometry [10], in particular via the formalism of multifractional spacetimes [11]. After a sketch of the classical situation, we will argue that quantum geometry effectively modifies the diffusion equation. A critical appraisal of the latter will allow us to classify quantum geometries in terms of stochastic processes on one hand, and to get a precise back-up to the notion of “fractal spacetime” on the other hand. The aim is to reexamine the spectral dimension starting from its foundation and provide a general, model-independent and analytic description of dimensional flow, confirmed by quantum-gravity examples. This is possible thanks to the presence of universal features in the flow [12]. For a diffusion process to be meaningful, the solution