Editing Casimir effect (section)

== Regularization ==
In order to be able to perform calculations in the general case, it is convenient to introduce a [[regularization (physics)|regulator]] in the summations. This is an artificial device, used to make the sums finite so that they can be more easily manipulated, followed by the taking of a limit so as to remove the regulator.

The [[heat kernel regularization|heat kernel]] or [[Exponential function|exponentially]] regulated sum is
<math display="block">\langle E(t) \rangle=\frac12 \sum_n \hbar |\omega_n |
\exp \bigl(-t |\omega_n |\bigr)\,,</math>

where the limit {{math|''t'' → 0<sup>+</sup>}} is taken in the end. The divergence of the sum is typically manifested as

<math display="block">\langle E(t) \rangle=\frac{C}{t^3} + \textrm{finite}\,</math>

for three-dimensional cavities. The infinite part of the sum is associated with the bulk constant {{mvar|C}} which ''does not'' depend on the shape of the cavity. The interesting part of the sum is the finite part, which is shape-dependent. The [[Gaussian function|Gaussian]] regulator

<math display="block">\langle E(t) \rangle=\frac12 \sum_n \hbar |\omega_n |
\exp \left(-t^2 |\omega_n |^2\right)</math>

is better suited to numerical calculations because of its superior convergence properties, but is more difficult to use in theoretical calculations. Other, suitably smooth, regulators may be used as well. The [[zeta function regulator]]

<math display="block">\langle E(s) \rangle=\frac12 \sum_n \hbar |\omega_n | |\omega_n |^{-s}</math>

is completely unsuited for numerical calculations, but is quite useful in theoretical calculations. In particular, divergences show up as poles in the [[complex plane|complex {{mvar|s}} plane]], with the bulk divergence at {{math|''s'' {{=}} 4}}. This sum may be [[analytic continuation|analytically continued]] past this pole, to obtain a finite part at {{math|''s'' {{=}} 0}}.

Not every cavity configuration necessarily leads to a finite part (the lack of a pole at {{math|''s'' {{=}} 0}}) or shape-independent infinite parts. In this case, it should be understood that additional physics has to be taken into account. In particular, at extremely large frequencies (above the [[plasma frequency]]), metals become transparent to [[photon]]s (such as [[X-ray]]s), and dielectrics show a frequency-dependent cutoff as well. This frequency dependence acts as a natural regulator. There are a variety of bulk effects in [[solid state physics]], mathematically very similar to the Casimir effect, where the [[cutoff frequency]] comes into explicit play to keep expressions finite. (These are discussed in greater detail in ''Landau and Lifshitz'', "Theory of Continuous Media".{{citation needed|date=May 2022}})