According to quantum electrodynamics, the vacuum, even in the absence of photons, is filled with zero-point energy (ZPE) from all modes

of the electromagnetic field, given by

. The ZPE of the vacuum is infinite unless regularized by frequency cutoffs or other stratagems. In most electromagnetic phenomena, the ZPE can be canceled out or renormalized away. Casimir and Polder in 1948 proposed a physical situation in which the effects of ZPE might be directly observable. They considered two perfectly conducting parallel plates, with area

, separated by a distance

of the order of microns. The electric field inside a perfect conductor equals zero, so this gives boundary values of zero for all modes at the surface. This perturbs the zero-point modes of the electromagnetic field, since modes with horizontal wavelength components greater than

will be excluded from the region between the plates. They will lie in the ultraviolet and beyond and are shown as black waves. Modes outside the plates can belong to a continuum of frequencies. The result is a net attractive force -

. In the above formulas

is the reduced Planck's constant (

),

is the speed of light and

is the frequency of the mode

. Note that this force is a purely quantum effect since it vanishes for

. It is also independent of the electric charge

. Measurement of the Casimir–Polder force involves very delicate experiments, but its validity has been successfully verified to better than 1% accuracy.

In the graphic, the relative scale of

has been greatly exaggerated compared to

, the edge length of each square plate. Conducting plates of area 1

separated by 1 micron (

m) experience an attractive force of 130 dynes (1.30×

N).