Casimir effect in Lorentz-violating scalar field theory: A local approach
C. A. Escobar, Leonardo Medel, A. Martín-Ruiz
Abstract
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 $\ensuremath{\lambda}{(u\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\partial}\ensuremath{\phi})}^{2}$, where the parameter $\ensuremath{\lambda}$ and the background four-vector ${u}^{\ensuremath{\mu}}$ 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 ${\mathcal{E}}_{C}(L)=\sqrt{\frac{1\ensuremath{-}\ensuremath{\lambda}{u}_{n}^{2}}{1+\ensuremath{\lambda}{u}^{2}}}{\mathcal{E}}_{0}(\stackrel{\texttildelow{}}{L}),$ where $\stackrel{\texttildelow{}}{L}=L/\sqrt{1\ensuremath{-}\ensuremath{\lambda}{u}_{n}^{2}}$ is a rescaled plate-to-plate separation and ${u}_{n}$ is the component of $\stackrel{\ensuremath{\rightarrow}}{u}$ along the normal to the plates. As usual, divergences of the local Casimir energy do not contribute to the pressure.