Aspectos gerais do efeito Casimir fermiônico e aplicações no espaço-tempo de Minkowski

Abstract

The present work investigates the fermionic Casimir effect under the influence of tempera- ture, with the main objective of determining the vacuum energy associated with a Dirac field defined on a flat spacetime with a circular spatial compactification in the xy plane. Antiperiodic boundary conditions are imposed, allowing for a systematic analysis of the influence of topology and boundary conditions on the quantum spectrum of the system and, consequently, on the properties of the quantum vacuum. The formulation of the problem is developed within the path integral quantization frame- work, which makes it possible to express the zero-point energy in terms of the functional determinant of the Dirac operator. In order to provide a finite and physically consistent definition of the vacuum energy, the generalized zeta function method combined with the asymptotic expansion of the spinor heat kernel is employed. This approach naturally allows for a separation between the zero-temperature Casimir energy contribution and the additional terms associated with thermal corrections. The structure of ultraviolet divergences is analyzed in detail through the asymptotic behavior of the heat kernel, revealing that only a single nonvanishing coefficient contributes to the divergences of the system, being associated with a divergence of Euclidean nature. This feature introduces an intrinsic ambiguity in the zeta function regularization procedure, which cannot be removed solely by standard subtraction schemes. It is therefore shown that the implementation of an additional finite renormalization is required in order to guarantee the uniqueness and physical consistency of the renormalized free energy.