Casimir Effect in Lorentz Invariant Non-commutative Space-Time

Quantum gravity has been studied using various approaches, and all of these approaches introduce a fundamental length scale in the theory. Non-Commutative space-time is an approach which incorporates this fundamental minimum length scale naturally. Though the length scale at which the Casimir effect is measured and the scale at which quantum gravity effects are expected are very different, it is worth studying the possible modification of the Casimir effect due to space-time non-commutativity. The Casimir effect is the phenomenon wherein a physical force between macroscopic boundaries confining space, such as the ones introduced by placing two parallel plates, arise due to the vacuum fluctuation of the quantized field. It is shown that vacuum fluctuations of the quantized electromagnetic field lead to either attraction or repulsion force between the plates depending on the geometry of the plates. The effects of the existence of a minimal length scale and the presence of extra dimensions on the Casimir effect has been studied in recent time. Thus it is of intrinsic interest to study the Casimir effect in Doplicher-Fredhegan-Robertson (DFR) space-time, a non-commutative space-time that naturally introduces a minimum length scale and has extra dimensions. Here we study the Casimir effect by analyzing the vacuum fluctuation of the scalar field in Lorentz invariant non-commutative space-time, DFR space-time. This is calculated by studying the scalar field when there are two parallel plates, separated by a distance and modeled by two \(\delta \) -functions. We calculate modifications to Casimir force and Casimir energy for both at zero and finite temperature. This is done in two ways; first, by treating the extra spatial dimensions introduced in the DFR space-time in the same manner as usual spatial dimensions of commutative space-time, and in the second, the extra dimension is treated as a compact dimension.