Spectral dimension of kappa-deformed spacetime

We investigate the spectral dimension of $\kappa$-spacetime using the $\kappa$-deformed diffusion equation. The deformed equation is constructed for two different choices of Laplacians in $n$-dimensional, $\kappa$-deformed Euclidean spacetime. We use an approach where the deformed Laplacians are expressed in the commutative spacetime itself. Using the perturbative solutions to diffusion equations, we calculate the spectral dimension of $\kappa$-deformed spacetime 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 noncommutative correction to the spectral dimension depends on the topological dimension of the spacetime whereas for the other, it is independent of the topological dimension. We have also analyzed the dimensional flow for the case where the probe particle has a finite extension, unlike a point particle.

Figure 1

Spectral dimension as a function of $\sigma$ with $a=1$ and $n=4$ for $D_{\mu }{D}_{\mu }$ as the Laplacian.

Figure 2

Spectral dimension as a function of $\sigma$ with $a=1$ and $n=4$ for $\square$ as the Laplacian.