Quantum theory Perturbative quantum field theory and renormalization Fact

Casimir effect

Builds on Renormalization Zeta function regularization

Statement

Consider the quantized electromagnetic field between two parallel, perfectly conducting plates at distance $a$ , in the limit of large plate area $A$ . Define the vacuum energy as the regularized sum of the zero-point energies $\tfrac12\hbar\omega$ of all modes, minus the same sum without the plates. Then the energy per unit area is $\frac{E}{A} = -\frac{\pi^2 \hbar c}{720\, a^3}.$ So the plates attract each other with the pressure $\pi^2\hbar c/(240\, a^4)$ .

Remarks

The idea of the computation: between the plates, the momentum normal to the plates takes only the values $n\pi/a$ . The sum $\sum_n \tfrac12\hbar\omega_n$ diverges, but its dependence on $a$ is finite after a cutoff, and the limit does not depend on the choice of a smooth cutoff. With zeta function regularization, the computation uses the value $\zeta(-3) = 1/120$ .

Example: a one-dimensional massless field on an interval of length $a$ with vanishing boundary values has modes $\omega_n = n\pi c/a$ . Zeta regularization gives $\tfrac12\sum_{n \ge 1} \hbar n\pi c/a = \frac{\hbar\pi c}{2a}\zeta(-1) = -\frac{\hbar\pi c}{24a}$ , using $\zeta(-1) = -1/12$ .

The result depends on the boundary conditions. Real metals are not perfect conductors at high frequencies, which gives corrections that the Lifshitz theory describes. The force has been measured, and it agrees with the prediction.

Going deeper

For a student

The Casimir effect is the standard example of how vacuum energy differences, rather than the divergent vacuum energy itself, have physical meaning. It is a test case for regularization methods, since zeta regularization, exponential cutoffs and point splitting give the same finite answer. Similar computations give Casimir energies in quantum field theory on compact spaces and in string theory.

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Tag	`29DMC9TR87`
URI	`m2://29DMC9TR87`
Kind	statement, fact
Writing	W0 stub
MSC	81T55
Status	This copy of the site has no build data. Run `m2 status 29DMC9TR87`.
About	Quantum field theory
Uses	Renormalization, Zeta function regularization
Same as	Wikidata Q275393 Wikipedia Casimir effect
Changed	3 November 2026