Geometry-Induced Vacuum Polarization and Mode Shifts in Maxwell-Klein-Gordon Theory

TL;DR

This paper addresses how geometric confinement on curved surfaces modifies the interacting quantum vacuum in the Maxwell–Klein–Gordon theory. By embedding the scalar field in a thin layer and treating the geometric potential $\Sigma_{\mathrm{geom}}(\mathbf{r})$ as a local mass correction, the authors derive a finite, gauge-invariant correction to vacuum polarization in the long-wavelength limit, encoded in a geometry-induced running of the electromagnetic response. The main result is a local relation $M^2(\mathbf{r}) = m^2 + \Sigma_{\mathrm{geom}}(\mathbf{r})$ that feeds into a one-loop polarization $\delta\Pi_T(\Omega;\mathbf{Q} \to 0)$ proportional to $\Sigma_{\mathrm{geom}}(\mathbf{r})$ times a universal loop factor $\mathcal{I}_0(\omega; m)$, weighted by the electric-energy distribution of cavity modes. Applied to Gaussian bumps, cylinders, and torii, the framework predicts mode-selective spectral shifts, ambient-background renormalization, and explicit angular dependences, offering a route to measurable shifts in high-$Q$ cavities and plasmonic systems. Overall, geometry acts as a controllable renormalization parameter, enabling a form of vacuum engineering in analogue gravity platforms with potential practical impact for precision photonics.

Abstract

Geometric confinement is known to modify single-particle dynamics through effective potentials, yet its imprint on the interacting quantum vacuum remains largely unexplored. In this work, we investigate the Maxwell--Klein--Gordon system constrained to curved surfaces and demonstrate that the geometric potential $Σ_{\mathrm{geom}}(\mathbf{r})$ acts as a local renormalization environment. We show that extrinsic curvature modifies the scalar loop spectrum, entering the vacuum polarization as a position-dependent mass correction $M^2(\mathbf{r}) \to m^2 + Σ_{\mathrm{geom}}(\mathbf{r})$. This induces a finite, gauge-invariant ``geometry-induced running'' of the electromagnetic response. In the long-wavelength regime ($|{\bf Q}|R \ll 1$), we derive a closed-form expression for the relative frequency shift $Δω/ω$, governed by the overlap between the electric energy density and the geometric potential. Applying this formalism to Gaussian bumps, cylindrical shells, and tori, we identify distinct spectral signatures that distinguish these quantum loop corrections from classical geometric optics. Our results suggest that spatial curvature can serve as a tunable knob for ``vacuum engineering,'' offering measurable shifts in high-$Q$ cavities and plasmonic systems.

Figures (6)

• Figure 1: Geometric potential profile and mode selectivity. (a) The radial dependence of the geometric potential $\Sigma_{\mathrm{geom}}$ for a Gaussian bump. Note that the potential vanishes at the isotropic center ($\rho=0$) and peaks at the region of maximum anisotropy ($\rho \approx \sqrt{2}\sigma$). (b) The overlap between the geometric potential and different electromagnetic modes. Higher-order radial modes (red dashed line) sample the potential peak more effectively than the fundamental Gaussian mode (blue solid line), leading to a larger frequency shift. This demonstrates the mechanism of geometry-induced mode splitting.
• Figure 2: Spatial mapping of vacuum polarization on a Gaussian defect. The color map represents the magnitude of the local geometric potential $\Sigma_{\mathrm{geom}}(\mathbf{r})$. Left (Euclidean Ambient): The potential is localized strictly at the "shoulders" of the bump where the curvature anisotropy is maximal, vanishing at the isotropic peak and the flat plane. Right ($\mathbb{C}P^1$ Ambient): The entire surface exhibits a non-zero background color shift due to the constant ambient curvature $-1/r_0^2$, with the shape-induced anisotropy superimposed. This visualizes how the ambient topology acts as a global control parameter for the vacuum energy.
• Figure 3: Uniform geometric potential on a cylindrical shell. 3D visualization of the effective mass correction. Left (Euclidean Ambient): The surface shows a uniform color distribution, reflecting the constant extrinsic curvature $\kappa=1/R$ of the cylinder. Right ($\mathbb{C}P^1$ Ambient): The shell displays a uniform but intensified color shift (larger magnitude). This comparison highlights that while the pattern of the vacuum polarization is dictated by the local shape, its absolute magnitude is renormalized by the ambient geometry.
• Figure 4: Scaling laws for cylindrical geometries. The effective mass squared correction (proportional to the frequency shift) is plotted against the inverse radius $1/R$. The red curves correspond to the Euclidean ambient space, confirming the predicted $\sim 1/R^2$ scaling law driven by extrinsic curvature. The blue curves represent the case of a $\mathbb{C}P^1$ ambient manifold. Note the non-zero intercept as $1/R \to 0$, which provides a spectral signature of the constant ambient curvature background.
• Figure 5: Symmetry breaking in toroidal geometries. The color gradient illustrates the variation of the geometric potential along the poloidal direction. Left (Euclidean Ambient): A distinct gradient is visible from the outer equator (lighter color) to the inner equator (darker color), demonstrating the "geometric trap" effect. Right ($\mathbb{C}P^1$ Ambient): The poloidal modulation persists but is shifted by the global ambient term. This visualization confirms that toroidal topology induces a position-dependent vacuum refractive index.