A zeta function Computation of Casimir Energy

A computation of Casimir energy via spectral zeta function is considered in this Chapter. The original
computations deriving the Casimir energy and force consists of first taking limits of the spectral zeta function
and afterwards analytically extending the result. This process of computation presents a weakness in Hendrik
Casimir’s original argument since limit and analytic continuation do not commute. A case of the Laplacian on a
parallelepiped box representing the space as the vacuum between two plates modelled with Dirichlet and
periodic Neumann boundary conditions is constructed to address this anomaly. It involves the derivation of the
regularised zeta function in terms of the Riemann zeta function on the parallelepiped. The values of the Casimir
energy and Casimir force obtained from our derivation agree with those of Hendrik Casimir.