Spectral Dimension of kappa-deformed space-time

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.