Effective Trace Framework for Self-Similar Casimir Systems

TL;DR

This paper develops a unified effective framework for analyzing vacuum traces in self-similar Casimir systems, bridging rigorous mathematical bounds with phenomenological models across multiple physical regimes.

Contribution

It introduces a cohesive approach combining thermal and vacuum traces for fractal geometries, differentiating backreaction effects from local trace anomalies, and analyzing finite prefractal realizations.

Findings

Unified effective framework for self-similar Casimir systems.

Differentiation of backreaction from trace anomalies.

Analytical conditions for transitioning to predictive electromagnetic theory.

Abstract

The interaction of quantum fields with fractal and self-similar geometries encompasses multiple distinct physical regimes, including spectral geometry on intrinsic fractals, macroscopic self-similar Casimir configurations, and bounded Euclidean cavities with fractal boundaries. While the thermal equations of state and spectral asymptotics for these systems are well established, a cohesive treatment of the vacuum trace frequently conflates rigorous mathematical bounds with phenomenological models. In this manuscript, we systematically decouple these regimes and advance a unified effective framework combining the rigorous thermal trace of fractal radiation with a zero-temperature integrated vacuum trace for plate-like self-similar geometries. We demonstrate that for systems governed by a scale-dependent Casimir coefficient $C(d_s, \ln(d/\ell_*))$, the anisotropic stress-energy tensor produces an integrated vacuum trace proportional to its logarithmic running, $\partial_{\ln d}C$. We strictly differentiate this effective macroscopic backreaction from first-principles local trace anomalies on genuine fractal boundaries. Finally, we analyze finite-level ($n$) prefractal realizations, establishing the analytical prerequisites necessary to transition this effective formalism into a quantitatively predictive electromagnetic theory amenable to experimental verification.