O papel das condições de fronteira na energia de vácuo: uma análise via métodos global e local

Abstract

This work investigates the Casimir effect for a real scalar field in (3+1) dimensions, considering different boundary conditions - Dirichlet, Neumann, mixed, and quasi-periodic. Two distinct methodologies for calculating the vacuum energy are analyzed: the global approach, based on the sum over the field modes, and the local approach, founded on the expectation value of the energy-momentum tensor. We demonstrate that, in the case of infinite parallel plates, the two approaches are not equivalent due to the non-commutativity between the mode summation and spatial integration. In this scenario, the energy obtained by the local method exhibits divergences near the surfaces, while the global approach, after zeta function regularization and subsequent renormalization, yields finite results consistent with the literature. In contrast, for the quasi-periodic case where there are no physical boundaries, the two methodologies coincide exactly, indicating that the discrepancy stems from the translational symmetry breaking caused by the boundary conditions. Using the Wightman function and the energy-momentum tensor formalism, we derive unified expressions that describe both scenarios. For the massive case, we demonstrate the exponential decay of the vacuum energy in the large-mass limit, also recovering known results for the massless field. The Casimir force per unit area is calculated for all configurations, showing attractive or repulsive behavior depending on the boundary conditions and the phase parameter in the quasi-periodic case. The results underscore the relevance of boundary conditions in determining the vacuum energy and suggest directions for future extensions, such as the analysis of other quantum

fields (electromagnetic and fermionic), the inclusion of thermal effects, the study of non- trivial geometries and topologies, and also investigations into non-minimal coupling with

curvature. This work contributes to the fundamental understanding of quantum vacuum properties in confined and compactified spaces, establishing connections between global and local formalisms in quantum field theory calculations.