Spectral dimension of kappa-deformed spacetime

Abstract

We investigate the spectral dimension of κ-space-time using the κ-deformed diffusion equation. The deformed equation is constructed for two different choices of Laplacians in n-dimensional, κ-deformed Euclidean space-time. We use an approach where the deformed Laplacians are expressed in the commutative space-time itself. Using the perturbative solutions to diffusion equations, we calculate the spectral dimension of κ-deformed space-time and show that it decreases as the probe length decreases. By introducing a bound on the deformation parameter, spectral dimension is guaranteed to be positive definite. We find that, for one of the choices of the Laplacian, the non-commutative correction to the spectral dimension depends on the topological dimension of the space-time whereas for the other, it is independent of the topological dimension. We have also analysed the dimensional flow for the case where the probe particle has a finite extension, unlike a point particle.1

Multifractional theories: an unconventional review

“…There are as many multiscale theories as the number of proposals in quantum gravity, plus some more. In fact, dimensional flow (mainly in d s , but in some cases also in d h ) is a universal phenomenon [74][75][76] found in all the main scenarios beyond general relativity: string theory [77], asymptotically-safe gravity (d s ≃ D/2 in D topological dimensions at the UV non-Gaussian fixed point; analytic results) [23,78,79]; CDT (for phase-C geometries, d s ≃ D/2 in the UV [80][81][82][83] or, more recently, d s ≃ 3/2 [84]; numerical results) and the related models of random combs [85,86] and random multigraphs [87,88]; causal sets [89]; noncommutative geometry [90][91][92] and κ-Minkowski spacetime [42,59,[93][94][95][96]; Stelle higher-order gravity (d s = 2 in the UV for any D [30]); nonlocal quantum gravity (d s < 1 in the UV in D = 4) [33].…”