Casimir force

Two parallel ideal conducting plates at zero temperature have separation-dependent electromagnetic energy per area $-{\pi }^2\hslash c/(720d^3)$ and pressure $-{\pi }^2\hslash c/(240d^4)$. The negative sign denotes attraction in this geometry. Finite temperature, material response and boundary geometry change the ideal result; an absolute divergent vacuum energy is not the measured quantity.

For a massless real scalar in one spatial dimension, applying $e^{-\omega /(\pi \Lambda )}$ to both the Dirichlet mode sum and the unbounded-line integral gives the same leading bulk energy $\pi L{\Lambda }^2/2$. Their finite difference is $-\pi /(24L)$ in units $\hslash =c=1$, producing the attractive force $-\pi /(24L^2)$. This scalar boundary force is distinct from the retarded atom-surface Casimir-Polder interaction.