Zero point energy on extra dimensions : Noncommutative torus

In this paper we calculate the zero point energy density experienced by observers on ${\mathcal{M}}^{4}$ due to a massless scalar field defined throughout ${\mathcal{M}}^{4}\ifmmode\times\else\texttimes\fi{}{\mathbf{T}}_{F}^{2}$, where ${\mathbf{T}}_{F}^{2}$ are fuzzy extra dimensions. Using the Green's function approach we calculate the energy density for the commutative torus and the fuzzy torus. We also calculate the energy density for the fuzzy torus using the Hamiltonian approach. Agreement is shown between the Green's function and Hamiltonian approaches.