One-Loop Quantum Corrections to the Casimir Effect for Rough Plates in the Low-Temperature Regime

Abstract

We present a theoretical analysis of the one-loop effective potential of a self-interacting real scalar field in the presence of two parallel conducting plates with geometric roughness. Using WKB methods to evaluate the spectral density of the modified Laplace-Beltrami operator, together with contour integration within a $\zeta$-function regularization scheme, we derive analytical expressions for the quantum corrections to the effective potential induced by perturbative boundary roughness and finite temperature. Furthermore, we compute explicit contributions to the Casimir energy and to the topological mass generation associated with the geometry.