Derivative expansion for the Casimir effect at zero and finite temperature in $d+1$ dimensions

We apply the derivative expansion approach to the Casimir effect for a real scalar field in $d$ spatial dimensions, to calculate the next to leading order term in that expansion, namely, the first correction to the proximity force approximation. The field satisfies either Dirichlet or Neumann boundary conditions on two static mirrors, one of them flat and the other gently curved. We show that, for Dirichlet boundary conditions, the next to leading order term in the Casimir energy is of quadratic order in derivatives, regardless of the number of dimensions. Therefore it is local, and determined by a single coefficient. We show that the same holds true, if $d \neq 2$, for a field which satisfies Neumann conditions. When $d=2$, the next to leading order term becomes nonlocal in coordinate space, a manifestation of the existence of a gapless excitation (which do exist also for $d> 2$, but produce sub-leading terms). We also consider a derivative expansion approach including thermal fluctuations of the scalar field. We show that, for Dirichlet mirrors, the next to leading order term in the free energy is also local for any temperature $T$. Besides, it interpolates between the proper limits: when $T \to 0$ it tends to the one we had calculated for the Casimir energy in $d$ dimensions, while for $T \to \infty$ it corresponds to the one for a theory in $d-1$ dimensions, because of the expected dimensional reduction at high temperatures. For Neumann mirrors in $d=3$, we find a nonlocal next to leading order term for any $T>0$.