Casimir effect in DFR space–time

Abstract

Noncommutative 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 [Formula: see text] using a Lorentz invariant scalar theory in a noncommutative 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 [Formula: see text]. In the second approach, we derive the corrections to the Casimir force, which is dependent on the separation between the plate, [Formula: see text] and on the size of the extra compactified dimension, [Formula: see text]. Since correction terms are very small, keeping only the most significant terms of these corrections, we show that for certain values of the [Formula: see text], the corrections due to noncommutativity make the force between the parallel plates more attractive, and using this, we find lower bound on the value of [Formula: see text]. 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 noncommutativity depend on [Formula: see text]- and [Formula: see text]-dependent modified Bessel functions [Formula: see text] and [Formula: see text], with coefficients that vary as [Formula: see text] and [Formula: see text], respectively. For finite temperature, the Casimir force has correction terms that scale as [Formula: see text] and [Formula: see text] in high-temperature limit and as [Formula: see text] and [Formula: see text] in the low-temperature limit.