Probing the quantum nature of spacetime by diffusion

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.

Figure 1

The function $P_{\mathrm{QEG}}(r,\sigma =1)$, Eq. (9), evaluated for a typical RG trajectory of quantum Einstein gravity in $d=3$.

Figure 2

The function $P_{\mathrm{LQG}}(r,\sigma =1)$, Eq. (9), evaluated for the ultraviolet (UV) limit in the LQG-inspired scenario of 6, by setting $d=4$ and $F(p^2)=p^2$.

Figure 3

The function $P_{\mathrm{HL}}(r,t=1,\sigma =1)$, Eq. (10), obtained from the anisotropic diffusion equation of HL gravity [13]. The occurrence of negative values is a generic feature and does not rely on the special choice of parameters in Eq. (6).

Figure 4

The probability density $P(r,\sigma =20)$ in $d=4$, weighted by $r^3$ for normal diffusion (red dotted line), power-law diffusion [Eq. (15) with $\beta =1/2$, green dashed line] and fractional diffusion [Eq. (19) with $\beta =1/2$, thick blue line].

Figure 5

Four-dimensional profiles of the scale-dependent spectral dimension $d_{\mathrm{S}}(\sigma )$ obtained from Eq. (17) with scale-dependent $\delta (k)$ (blue, solid curve) and the multiscale power-law diffusion equation (39) with ${\sigma }_1=0$, ${\beta }_1=1/2$, ${\sigma }_2={10}^{-6}$, ${\beta }_2=1/3$, and ${\sigma }_3={10}^{-2}$, ${\beta }_3=1$ (black, dashed lines) Note that there is no dependence on the constants $c_i$ from Eq. (39).