Casimir effect in Lorentz-violating scalar field theory: a local approach

We study the Casimir effect in the classical geometry of two parallel conductive plates, separated by a distance $L$, for a Lorentz-breaking extension of the scalar field theory. The Lorentz-violating part of the theory is characterized by the term $\lambda \left( u \cdot \partial \phi \right )^{2}$,wheretheparameter$\lambda$andthebackgroundfour-vector$u ^{\mu}$ codify Lorentz symmetry violation. We use Green’s function techniques to study the local behavior of the vacuum stress-energy tensor in the region between the plates. Closed analytical expressions are obtained for the Casimir energy and pressure. We show that the energy density ${\mathcal{E}}_C$ (and hence the pressure) can be expressed in terms of the Lorentz-invariant energy density ${\mathcal{E}}_0$ as follows \begin{align} $\mathcal{E}${C} (L) = \sqrt{\frac{1-\lambda u{n} ^{2}}{1 + \lambda u ^{2}}} $\mathcal{E}${0} (\tilde{L}) , \notag \end{align} where $\tilde{L} = L / \sqrt{1-\lambda u{n} ^{2}}$isarescaledplate-to-plateseparationand$u_{n}$isthecomponentof$\vec{{u}}$ along the normal to the plates. As usual, divergences of the local Casimir energy do not contribute to the pressure.