Casimir effect between semitransparent mirrors in a Lorentz-violating background

TL;DR

This work analyzes the Casimir effect for a massless scalar field between two semitransparent mirrors in a Lorentz-violating vacuum described by the SME. Using Green's function methods and point-splitting, it yields closed-form expressions for the Casimir energy and shows that the transparency of the mirrors and the LV background combine as a multiplicative factor and an effective rescaling of plate separation, with $E_C(L)=\sqrt{\frac{h^{33}}{h}}\,E_C(\tilde{L})$ and $\tilde{L}=L/\sqrt{-h_{\!33}}$, recovering the standard Lorentz-invariant results in the appropriate limit. In physical realizations, the LV background can be modeled by an anisotropic medium via the Gordon metric, with parameters mapped to $h^{\mu\nu}$ (e.g., in nematic liquid crystals like 5CB), suggesting that precision Casimir measurements could constrain LV coefficients. The paper also provides numerical estimates for realistic materials, illustrating how the Casimir energy can be tuned by mirror transparency and medium anisotropy, thereby bridging high-energy phenomenology and condensed-matter physics.

Abstract

We investigate the Casimir effect for a massless scalar field confined between two parallel semitransparent mirrors in a vacuum modified by spontaneous Lorentz symmetry breaking. Using Green's function techniques and a point-splitting evaluation of the stress-energy tensor, we compute the vacuum expectation value of the energy density $T_{00}$. After a suitable renormalization, the Casimir energy is obtained as the difference between the vacuum configurations with and without the mirrors. We derive closed-form expressions that generalize the conventional result by simultaneously incorporating the mirror transparency and the Lorentz-violating background. Our analysis shows that the effect of transparency and Lorentz violation consists of a multiplicative correction and an effective rescaling of the plate separation, thereby modifying the functional dependence of the energy on the distance. Beyond the formal derivation, we discuss possible physical realizations of this framework, emphasizing anisotropic media such as nematic liquid crystals (e.g., 5CB), where uniaxial dielectric properties could emulate the Lorentz-violating background. Numerical estimates for such systems illustrate the phenomenological impact of our results and open the possibility of constraining Lorentz-violating coefficients through precision Casimir measurements.

Figures (1)

• Figure 1: Left: Casimir energy density (in units of energy per unit area) as a function of plate separation (in nanometers). The black squares show the standard Lorentz-invariant result for perfectly reflecting boundaries. The red triangles indicate the case of nearly perfect mirrors, while the blue circles correspond to a more transparent regime, where the reduced reflectivity weakens the confinement of vacuum fluctuations and thus lowers the Casimir interaction. Right: Density plot of the Casimir energy as a function of the transparency parameters $\lambda$ and $\lambda '$, for a fixed plate separation. The upper-right region corresponds to the limit of perfectly reflecting mirrors, where the energy approaches the standard Casimir result. Along the diagonal ($\lambda = \lambda '$), the mirrors are equally transparent, while deviations from the diagonal highlight asymmetric configurations with different transparency strengths. The energy vanishes smoothly in the transparent limit ($\lambda , \lambda ' \to 0$).