Probing the quantum nature of spacetime by diffusion

Abstract

Many approaches to quantum gravity have resorted to diffusion processes to characterize the spectral properties of the resulting quantum spacetimes. We critically discuss these quantum-improved diffusion equations and point out that a crucial property, namely positivity of their solutions, is not preserved automatically. We then construct a novel set of diffusion equations with positive semidefinite probability densities, applicable to asymptotically safe gravity, Hořava-Lifshitz gravity and loop quantum gravity. These recover all previous results on the spectral dimension and shed further light on the structure of the quantum spacetimes by assessing the underlying stochastic processes. Pointing out that manifestly different diffusion processes lead to the same spectral dimension, we propose the probability distribution of the diffusion process as a refined probe of quantum spacetime.

Diffusion in multiscale spacetimes

“…In the presence of one or more fundamental quantum scales, the new operator is in fact a sum of operators of different orders. To the best of our knowledge, there are only two concrete examples of a quantum-gravity diffusion equation with (multi)fractional diffusion operator: multifractional spacetimes [27] (but as an optional construction) and, perhaps more interestingly for the habitués of the field, maybe also quantum Einstein gravity [36]. There, the deformation of ∂ σ is realized because the cutoff scale of the theory is not identified with the physical momentum as usual but, as a powerful alternative, with diffusion time; the anomalous scaling is due to the renormalization group flow realizing asymptotic safety in the UV, while the presence of several scales (and, hence, of several diffusion operators in the same equation) is guaranteed by the type of action.…”

“…The interested reader can find the proof in the references cited above. The equivalence between fractional and iterated Brownian motion can play a role in the interpretation of quantum geometry at the UV fixed point, including QEG [36]. For this reason, it is important to stress the physical meaning of transport equations such as (71) and (84).…”

“…Therefore, as a first approximation, it is sufficient to study diffusion equations with two diffusion operators ∂ β 1 σ and ∂ β 2 σ , 10 or with two Laplacians ∂ 2γ 1 |x| and ∂ 2γ 2 |x| . Certain realizations of QEG can be regarded as a two-scale system, where there is an intermediate genuine plateau [36].…”

“…Therefore, it can be regarded either as an independent proposal for a fundamental theory or an effective framework wherein to better understand the multiscale geometry of the other approaches (examples are [21,35,36]). For this reason, we believe it is important to exploit the tools available in multifractional spacetimes as much as possible.…”

On a renormalization group scheme for causal dynamical triangulations

“…As I demonstrate in section 3.3, the rate of decay for σ > σ cl sets the scale of the large scale curvature, which is also the scale dS that characterizes the central accumulation. 5 If one possessed a model of the spectral dimension's behavior for σ < σ cl , of which there are several [15,18,19,21,22,23,29,32,34,37,38], then one could presumably also extract a scale qm from the rate of increase. Once again, as expected, based on numerical measurements alone, these three scales are characterized by dimensionless numbers.…”