Casimir effect in DFR space-time

Non-Commutative space-time introduces a fundamental length scale suggested by approaches to quantum gravity. Here we report the analysis of the Casimir effect for parallel plates separated by a distance of $L$ using a Lorentz invariant scalar theory in a non-commutative space-time (DFR space-time), both at zero and finite temperatures. This is done in two ways; one when the additional space-dimensions introduced in DFR space-time are treated as extra dimensions but on par with usual space-dimension and in the second way, the additional dimensions are treated as compact dimensions. Casimir force obtained in the first approach coincides with the result in the extra-dimensional commutative space-time and this is varying as $\frac{1}{L^5}$. In the second approach, we derive the corrections to the Casimir force, which is dependent on the separation between the plate, $L$ and on the size of the extra compactified dimension, $R$. Since correction terms are very small, keeping only the most significant terms of these corrections, we show that for certain values of the R, the corrections due to non-commutativity makes the force between the parallel plates more attractive, and using this, we find lower bound on the value of $R$. We show here that the requirement of the Casimir force and the energy to be real, impose the condition that the weight function used in defining the DFR action has to be a constant. At zero temperature, we find correction terms due to non-commutativity, depend on $L$ and $R$ dependent modified Bessel functions $K_{1}$ and $K_{2}$, with coefficients that vary as $\frac{1}{LR^3}$ and $\frac{1}{L^2R^2}$, respectively . For finite temperature, the Casimir force has correction terms that scale as $\frac{1}{L}$ and $\frac{1}{L^3}$ in high-temperature limit and as $\frac{1}{L^2}$ and $\frac{1}{L^4}$ in the low-temperature limit.