Editing Casimir effect (section)

== Derivation of Casimir effect assuming zeta-regularization ==
{{hatnote|See '''[[v:Quantum mechanics/Casimir effect in one dimension|Wikiversity]]''' for an elementary calculation in one dimension.}}

In the original calculation done by Casimir, he considered the space between a pair of conducting metal plates at distance {{mvar|a}} apart. In this case, the standing waves are particularly easy to calculate, because the transverse component of the electric field and the normal component of the magnetic field must vanish on the surface of a conductor. Assuming the plates lie parallel to the {{mvar|xy}}-plane, the standing waves are

<math display="block">\psi_n(x,y,z;t)=e^{-i\omega_nt} e^{ik_xx+ik_yy} \sin(k_n z) \,,</math>

where {{mvar|ψ}} stands for the electric component of the electromagnetic field, and, for brevity, the polarization and the magnetic components are ignored here. Here, {{mvar|k<sub>x</sub>}} and {{mvar|k<sub>y</sub>}} are the [[wavenumber]]s in directions parallel to the plates, and

<math display="block">k_n=\frac{n\pi}{a}</math>

is the wavenumber perpendicular to the plates. Here, {{mvar|n}} is an integer, resulting from the requirement that {{mvar|ψ}} vanish on the metal plates. The frequency of this wave is

<math display="block">\omega_n=c \sqrt{{k_x}^2 + {k_y}^2 + \frac{n^2\pi^2}{a^2}} \,,</math>

where {{mvar|c}} is the [[speed of light]]. The vacuum energy is then the sum over all possible excitation modes. Since the area of the plates is large, we may sum by integrating over two of the dimensions in {{mvar|k}}-space. The assumption of [[Van Hove singularity#Theory|periodic boundary conditions]] yields,

<math display="block">\langle E \rangle=\frac{\hbar}{2} \cdot 2
\int \frac{A \,dk_x \,dk_y}{(2\pi)^2} \sum_{n=1}^\infty \omega_n \,,</math>

where {{mvar|A}} is the area of the metal plates, and a factor of 2 is introduced for the two possible polarizations of the wave. This expression is clearly infinite, and to proceed with the calculation, it is convenient to introduce a [[regularization (physics)|regulator]] (discussed in greater detail below). The regulator will serve to make the expression finite, and in the end will be removed. The [[Zeta function regularization|zeta-regulated]] version of the energy per unit-area of the plate is

<math display="block">\frac{\langle E(s) \rangle}{A}=\hbar
\int \frac{dk_x \,dk_y}{(2\pi)^2} \sum_{n=1}^\infty \omega_n
\left| \omega_n \right|^{-s} \,.</math><!-- Nice, how can it be fixed? -->

In the end, the limit {{math|''s'' → 0}} is to be taken. Here {{mvar|s}} is just a [[complex number]], not to be confused with the shape discussed previously. This integral sum is finite for {{mvar|s}} [[real number|real]] and larger than 3. The sum has a [[pole (complex analysis)|pole]] at {{math|''s'' {{=}} 3}}, but may be [[analytic continuation|analytically continued]] to {{math|''s'' {{=}} 0}}, where the expression is finite. The above expression simplifies to:

<math display="block">\frac{\langle E(s) \rangle}{A}=
\frac{\hbar c^{1-s}}{4\pi^2} \sum_n \int_0^\infty 2\pi q \,dq
\left | q^2 + \frac{\pi^2 n^2}{a^2} \right|^\frac{1-s}{2} \,,</math>

where [[Polar coordinate system|polar coordinates]] {{math|''q''<sup>2</sup> {{=}} ''k<sub>x</sub>''<sup>2</sup> + ''k<sub>y</sub>''<sup>2</sup>}} were introduced to turn the [[Multiple integral|double integral]] into a single integral. The {{mvar|q}} in front is the Jacobian, and the {{math|2''π''}} comes from the angular integration. The integral converges if {{math|Re(''s'') > 3}}, resulting in

<math display="block">\frac{\langle E(s) \rangle}{A}=
-\frac {\hbar c^{1-s} \pi^{2-s}}{2a^{3-s}} \frac{1}{3-s}
\sum_n \left| n \right| ^{3-s}=
-\frac {\hbar c^{1-s} \pi^{2-s}}{2a^{3-s}(3-s)}\sum_n \frac{1}{\left| n\right| ^{s-3}} \,.</math>

The sum diverges at {{mvar|s}} in the neighborhood of zero, but if the damping of large-frequency excitations corresponding to analytic continuation of the [[Riemann zeta function]] to {{math|''s'' {{=}} 0}} is assumed to make sense physically in some way, then one has

<math display="block">\frac{\langle E \rangle}{A}=
\lim_{s\to 0} \frac{\langle E(s) \rangle}{A}=
-\frac {\hbar c \pi^2}{6a^3} \zeta (-3) \,.</math>

But {{math|''ζ''(−3) {{=}} {{sfrac|1|120}}}} and so one obtains

<math display="block">\frac{\langle E \rangle}{A}=
-\frac {\hbar c \pi^2}{720 a^3}\,.</math>

The analytic continuation has evidently lost an additive positive infinity, somehow exactly accounting for the zero-point energy (not included above) outside the slot between the plates, but which changes upon plate movement within a closed system. The Casimir force per unit area {{math|{{sfrac|''F''<sub>c</sub>|''A''}}}} for idealized, perfectly conducting plates with vacuum between them is

<math display="block">\frac{F_\mathrm{c}}{A}=-\frac{d}{da} \frac{\langle E \rangle}{A} = -\frac {\hbar c \pi^2} {240 a^4}</math>

where
* {{mvar|ħ}} is the [[reduced Planck constant]],
* {{mvar|c}} is the [[speed of light]],
* {{mvar|a}} is the [[distance]] between the two plates

The force is negative, indicating that the force is attractive: by moving the two plates closer together, the energy is lowered. The presence of {{mvar|ħ}} shows that the Casimir force per unit area {{math|{{sfrac|''F''<sub>c</sub>|''A''}}}} is very small, and that furthermore, the force is inherently of quantum-mechanical origin.

By [[Integral|integrating]] the equation above it is possible to calculate the energy required to separate to infinity the two plates as:

<math display="block">\begin{align}
U_E(a) &= \int F(a) \,da = \int - \hbar c \pi^2 \frac {A} {240 a^4} \,da \\[4pt]
&=  \hbar c \pi^2 \frac {A} {720 a^3} 
\end{align}</math>

where
* {{mvar|ħ}} is the [[reduced Planck constant]],
* {{mvar|c}} is the [[speed of light]],
* {{mvar|A}} is the [[area]] of one of the plates,
* {{mvar|a}} is the [[distance]] between the two plates

In Casimir's original derivation,<ref name=":1">{{Cite journal|last=Casimir|first=H. B. G.|date=1948|title=On the attraction between two perfectly conducting plates|url=https://www.dwc.knaw.nl/DL/publications/PU00018547.pdf |archive-url=https://web.archive.org/web/20130418012405/http://www.dwc.knaw.nl/DL/publications/PU00018547.pdf |archive-date=2013-04-18 |url-status=live|journal=Proc. Kon. Ned. Akad. Wet.|volume=51|pages=793}}</ref> a moveable conductive plate is positioned at a short distance {{mvar|a}} from one of two widely separated plates (distance {{mvar|L}} apart). The zero-point energy on ''both'' sides of the plate is considered. Instead of the above ''ad hoc'' analytic continuation assumption, non-convergent sums and integrals are computed using [[Euler–Maclaurin summation]] with a regularizing function (e.g., exponential regularization) not so anomalous as {{math|{{abs|''ω<sub>n</sub>''}}<sup>−''s''</sup>}} in the above.<ref>{{cite journal |last1=Ruggiero |first1=Zimerman |last2=Villani |title=Application of Analytic Regularization to the Casimir Forces |journal=Revista Brasileira de Física |volume=7 |number=3 |year=1977 |url=http://www.sbfisica.org.br/bjp/download/v07/v07a43.pdf |archive-url=https://web.archive.org/web/20150403165408/http://www.sbfisica.org.br/bjp/download/v07/v07a43.pdf |archive-date=2015-04-03 |url-status=live}}</ref>

=== More recent theory ===
Casimir's analysis of idealized metal plates was generalized to arbitrary dielectric and realistic metal plates by [[Evgeny Lifshitz]] and his students.<ref name=":2">{{cite journal |last1=Dzyaloshinskii |first1=I E |last2=Lifshitz |first2=E M |last3=Pitaevskii |first3=Lev P |title=General Theory of van der Waals' Forces |journal=Soviet Physics Uspekhi |volume=4 |pages=153 |year=1961 |doi=10.1070/PU1961v004n02ABEH003330 |issue=2 |bibcode=1961SvPhU...4..153D }}</ref><ref>{{cite journal |last1=Dzyaloshinskii |first1=I E |last2=Kats |first2=E I |title=Casimir forces in modulated systems |arxiv=cond-mat/0408348 |journal=Journal of Physics: Condensed Matter |volume=16 |pages=5659 |year=2004 |doi=10.1088/0953-8984/16/32/003 |issue=32 |bibcode=2004JPCM...16.5659D |s2cid=250897415 }}</ref> Using this approach, complications of the bounding surfaces, such as the modifications to the Casimir force due to finite conductivity, can be calculated numerically using the tabulated complex dielectric functions of the bounding materials. Lifshitz's theory for two metal plates reduces to Casimir's idealized {{math|{{sfrac|1|''a''<sup>4</sup>}}}} force law for large separations {{mvar|a}} much greater than the [[skin depth]] of the metal, and conversely reduces to the {{math|{{sfrac|1|''a''<sup>3</sup>}}}} force law of the [[London dispersion force]] (with a coefficient called a [[Hamaker constant]]) for small {{mvar|a}}, with a more complicated dependence on {{mvar|a}} for intermediate separations determined by the [[Dispersion (optics)|dispersion]] of the materials.<ref>V. A. Parsegian, ''Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists'' (Cambridge Univ. Press, 2006).</ref>

Lifshitz's result was subsequently generalized to arbitrary multilayer planar geometries as well as to anisotropic and magnetic materials, but for several decades the calculation of Casimir forces for non-planar geometries remained limited to a few idealized cases admitting analytical solutions.<ref name=Rodriguez11-review /> For example, the force in the experimental sphere–plate geometry was computed with an approximation (due to Derjaguin) that the sphere radius {{mvar|R}} is much larger than the separation {{mvar|a}}, in which case the nearby surfaces are nearly parallel and the parallel-plate result can be adapted to obtain an approximate {{math|{{sfrac|''R''|''a''<sup>3</sup>}}}} force (neglecting both skin-depth and [[Orders of approximation|higher-order]] curvature effects).<ref name=Rodriguez11-review /><ref>B. V. Derjaguin, I. I. Abrikosova, and E. M. Lifshitz, ''Quarterly Reviews, Chemical Society'', vol. 10, 295–329 (1956).</ref> However, in the 2010s a number of authors developed and demonstrated a variety of numerical techniques, in many cases adapted from classical [[computational electromagnetics]], that are capable of accurately calculating Casimir forces for arbitrary geometries and materials, from simple finite-size effects of finite plates to more complicated phenomena arising for patterned surfaces or objects of various shapes.<ref name=Rodriguez11-review>{{cite journal |first1=A. W. |last1=Rodriguez |first2=F. |last2=Capasso |title=The Casimir effect in microstructured geometries |journal=Nature Photonics |volume=5 |pages=211–221 |year=2011 |doi=10.1038/nphoton.2011.39 |last3=Johnson |first3=Steven G. |issue=4 |bibcode=2011NaPho...5..211R }} Review article.</ref><ref name=Reid2013>{{cite journal |first1=M. T. H. |last1=Reid |first2=J. |last2=White |first3=S. G. |last3=Johnson |title=Computation of Casimir interactions between arbitrary three-dimensional objects with arbitrary material properties |journal=Physical Review A |volume=84 |pages=010503(R) |year=2011 |doi=10.1103/PhysRevA.84.010503 |issue=1 |arxiv=1010.5539 |bibcode=2011PhRvA..84a0503R |s2cid=197461628 }}</ref>