[Paper Review] Semiclassical Electromagnetic Casimir Self-Energies

Martin Schaden|ArXiv.org|Apr 17, 2006

Quantum Electrodynamics and Casimir Effect4 references3 citations

TL;DR

This paper applies semiclassical methods based on periodic orbits to compute electromagnetic Casimir self-energies for spherical and cylindrical cavities, demonstrating that the semiclassical approach accurately reproduces the finite, repulsive Casimir energy of a spherical cavity (within 1%) but predicts a vanishing Casimir energy for a cylindrical cavity due to destructive interference of optical paths. The discrepancy arises because the cylindrical case depends sensitively on ultraviolet details of the boundary, unlike the robust spherical case.

ABSTRACT

The electromagnetic Casimir energies of a spherical and a cylindrical cavity are analyzed semiclassically. The field theoretical self-stress of a spherical cavity with ideal metallic boundary conditions is reproduced to better than 1%. The subtractions in this case are unambiguous and the good agreement is interpreted as evidence that finite contributions from the exterior of the cavity are small. The semiclassical electromagnetic Casimir energy of a cylindrical cavity on the other hand vanishes to any order in the real reflection coefficients. The Casimir energy of a cylindrical cavity with a perfect metallic and infinitesimally thin boundary on the other hand is finite and negative [17]. Contrary to the spherical case and in agreement with Barton's perturbative analysis [31], the subtractions in the spectral density for the cylinder are not universal when only the interior modes of are taken into account [43]. The Casimir energy of a cylindrical cavity therefore depends sensitively on the physical nature of the boundary in the ultraviolet whereas the Casimir energy of a spherical one does not. The extension of the semiclassical approach to more realistic systems is sketched.

Motivation & Objective

• To investigate the accuracy and robustness of semiclassical methods in computing electromagnetic Casimir self-energies for classically integrable systems.
• To compare semiclassical results with exact field-theoretic calculations for spherical and cylindrical cavities.
• To examine the role of boundary conditions and exterior modes in determining the finiteness and physical meaning of Casimir energies.
• To assess whether the Casimir energy depends on global geometric properties or is sensitive to local ultraviolet details of the boundary.
• To extend the semiclassical approach to more realistic systems, including dielectrics and rough surfaces, via frequency-dependent and stochastic reflection coefficients.

Proposed method

• The study uses a dual representation of Casimir energy in terms of periodic orbits, derived from Gutzwiller's trace formula, to compute the oscillating part of the spectral density.
• The semiclassical approximation is applied to the spectral density of integrable systems, with subtractions made to render the Casimir energy finite and physically meaningful.
• The method relies on optical phases accumulated along classical periodic orbits within the cavity, with reflection coefficients encoding boundary properties.
• For cylindrical cavities, the approach shows that contributions from periodic orbits cancel exactly to all orders in real reflection coefficients, leading to a vanishing Casimir energy.
• Finite-temperature corrections are incorporated by introducing a periodic extra dimension of length ℏc/(kT), modeling thermal fluctuations in the path integral.
• Surface roughness is modeled by modifying the dispersion relation to include a stochastic damping term proportional to the variance in periodic orbit length.

Experimental results

Research questions

• RQ1 Can the semiclassical approach accurately reproduce the known finite Casimir self-energy of a spherical cavity with ideal metallic boundaries?
• RQ2 Why does the semiclassical Casimir energy of a cylindrical cavity vanish to all orders in the reflection coefficient, despite the exact field-theoretic result being finite and negative?
• RQ3 To what extent does the Casimir energy depend on the ultraviolet details of the boundary, particularly in non-integrable or chaotic systems?
• RQ4 Is the Casimir energy robust under changes in boundary conditions if it depends only on global geometric properties rather than local surface details?
• RQ5 Can the semiclassical method be extended to model realistic systems such as dielectrics and rough surfaces with frequency-dependent and stochastic reflection coefficients?

Key findings

• The semiclassical method reproduces Boyer’s result for the spherical cavity with better than 1% accuracy, confirming the repulsive nature of the Casimir self-energy.
• The Casimir energy of a cylindrical cavity vanishes to all orders in the real reflection coefficients, consistent with perturbative analyses and geometric optics.
• The subtractions required to render the Casimir energy finite are not universal for the cylinder, indicating sensitivity to ultraviolet details of the boundary.
• In contrast to the cylinder, the spherical cavity’s Casimir energy is robust and insensitive to exterior contributions, suggesting a more universal physical interpretation.
• The method can be extended to dielectrics by using complex, frequency-dependent reflection coefficients, reproducing Lifshitz theory in the slab limit.
• Surface roughness suppresses contributions from wavelengths much smaller than the roughness scale ε, modeled via a modified dispersion relation with an imaginary quadratic term in energy.