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Casimir Effect of rough plates under a magnetic field in Hořava-Lifshitz theory

Byron Droguett, Claudio Bórquez

TL;DR

This work analyzes the Casimir effect between rough, parallel plates in a 3+1 dimensional Hořava-Lifshitz–like theory under a uniform magnetic field, using a charged-scalar field with Dirichlet boundaries. The geometry is flattened by a coordinate transformation so roughness enters as a perturbative potential, and the spectrum is obtained with perturbation theory and regularized via the $e$-function method. The authors derive both weak- and strong-magnetic-field limits for odd and even anisotropic scaling $z$, showing that energy and force vanish for even $z$ while odd $z$ receive $l$-dependent corrections and surface-roughness enhancements; periodic roughness yields explicit simplified expressions. The results indicate that measurements of the Casimir force could constrain Lorentz-violating parameters in HL-like theories and suggest avenues for incorporating temperature and alternative boundary conditions in future work.

Abstract

We investigate the Casimir effect for parallel plates within the framework of Hořava-Lifshitz theory in $3+1$ dimensions, considering the effects of roughness, anisotropic scaling factor, and an uniform constant magnetic field. Quantum fluctuations are induced by an anisotropic charged-scalar quantum field subject to Dirichlet boundary conditions. To incorporate surface roughness, we apply a coordinate transformation to flatten the plates, treating the remaining roughness terms as potential. The spectrum is derived using perturbation theory and regularized with the $ζ$-function method. As an illustrative example, we consider plates with periodic boundary conditions.

Casimir Effect of rough plates under a magnetic field in Hořava-Lifshitz theory

TL;DR

This work analyzes the Casimir effect between rough, parallel plates in a 3+1 dimensional Hořava-Lifshitz–like theory under a uniform magnetic field, using a charged-scalar field with Dirichlet boundaries. The geometry is flattened by a coordinate transformation so roughness enters as a perturbative potential, and the spectrum is obtained with perturbation theory and regularized via the -function method. The authors derive both weak- and strong-magnetic-field limits for odd and even anisotropic scaling , showing that energy and force vanish for even while odd receive -dependent corrections and surface-roughness enhancements; periodic roughness yields explicit simplified expressions. The results indicate that measurements of the Casimir force could constrain Lorentz-violating parameters in HL-like theories and suggest avenues for incorporating temperature and alternative boundary conditions in future work.

Abstract

We investigate the Casimir effect for parallel plates within the framework of Hořava-Lifshitz theory in dimensions, considering the effects of roughness, anisotropic scaling factor, and an uniform constant magnetic field. Quantum fluctuations are induced by an anisotropic charged-scalar quantum field subject to Dirichlet boundary conditions. To incorporate surface roughness, we apply a coordinate transformation to flatten the plates, treating the remaining roughness terms as potential. The spectrum is derived using perturbation theory and regularized with the -function method. As an illustrative example, we consider plates with periodic boundary conditions.
Paper Structure (8 sections, 41 equations, 4 figures)

This paper contains 8 sections, 41 equations, 4 figures.

Table of Contents

  1. Introducction
  2. Hořava-Lifshitz-like charged-scalar field
  3. The anisotropic operator
  4. The perturbation theory
  5. $\zeta$–function regularization
  6. Weak magnetic field limit
  7. Strong magnetic field limit
  8. Conclusions

Figures (4)

  • Figure 1: Casimir force density as a function of the separation distance $a$. We present the curves for different values of the anisotropic scaling factor: $z=1,3,5$. For the cases $z=3,5$, we set $l=0,01$. The solid curves represent the case without perturbative terms, while the dashed-point curves represent the case with surface roughness $\xi=0.05$ (units of lenght) and no magnetic field. The subfigure shows a close-up of the solid curve in the range $a\sim 10^{-6}$ (units of length) and $\mathcal{F}_{C}\sim 10^{-5}$ (units of lenght$^{-4}$), where it is observed that the dashed curve reflects the small contribution from the magnetic field $qB=0.1$ (units of lenght$^{-2}$)
  • Figure : (a)
  • Figure : (a)
  • Figure : (b)