Radiative Corrections to the Casimir Energy

The lowest radiative correction to the Casimir energy density between two parallel plates is calculated using effective field theory. Since the correlators of the electromagnetic field diverge near the plates, the regularized energy density is also divergent. However, the regularized integral of the energy density is finite and varies with the plate separation $L$ as ${1/L}^7$. This apparently paradoxical situation is analyzed in an equivalent but more transparent theory of a massless scalar field in $1+1$ dimensions confined to a line element of length $L$ and satisfying Dirichlet boundary conditions.