19.9 The Casimir Effect on parallel plates
Thus far, we have dealt with the quantization of the free electromagnetic fieldsE~ andB~, Already
at this stage, it is possible to infer measurable predictions, such as the Casimir effect. The Casimir
effect is a purely quantum field theory phenomenon in which the quantization of the electromagnetic
fields in the presence of a electric conductors results in a net force between these conductors.
The most famous example involves two conducting parallel plates, separated by a distancea, as
represented in Fig18(a).
-- a --(a) (b)LLLEp=0Figure 18: The Casimir effect : (a) electromagnetic vacuum fluctuations produce a force
between two parallel plates; (b) the problem in a space box.
The assumptions entering the set-up are as follows.- For a physical conductor, the electric fieldE~pparallel to the plates will vanish for frequencies
much lower than the typical atomic scaleωc. At frequencies much higher than the atomic
scale, the conductor effectively becomes transparent and no constraint is to be imposed on
the electric field. In the frequency region comparable to theatomic scaleωc, the conductivity
is a complicated function of frequency, which we shall simply approximate by a step function
θ(ωc−ω). - The separationashould be taken much larger than interatomic distances, oraωc≫1.
- In order to regularize the problems associated with infinite volume, we study the problem
in a square box in the form of a cube, with periodic boundary conditions, as depicted in