Casimir effect
Casimir forces on parallel plates
Casimir forces on parallel plates

In quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field. The typical example is of two uncharged metallic plates in a vacuum, like capacitors placed a few micrometers apart, without any external electromagnetic field. In a classical description, the lack of an external field also means that there is no field between the plates, and no force would be measured between them.[1] When this field is instead studied using quantum electrodynamics, it is seen that the plates do affect the virtual photons which constitute the field, and generate a net force[2]—either an attraction or a repulsion depending on the specific arrangement of the two plates. Although the Casimir effect can be expressed in terms of virtual particles interacting with the objects, it is best described and more easily calculated in terms of the zero-point energy of a quantized field in the intervening space between the objects. This force has been measured, and is a striking example of an effect purely due to second quantization.[3][4] However, the treatment of boundary conditions in these calculations has led to some controversy. In fact "Casimir's original goal was to compute the van der Waals force between polarizable molecules" of the metallic plates. Thus it can be interpreted without any reference to the zero-point energy (vacuum energy) or virtual particles of quantum fields.[5]

Dutch physicists Hendrik B. G. Casimir and Dirk Polder proposed the existence of the force and formulated an experiment to detect it in 1948 while participating in research at Philips Research Labs. The classic form of the experiment, described above, successfully demonstrated the force to within 15% of the value predicted by the theory.[6]

Because the strength of the force falls off rapidly with distance, it is only measurable when the distance between the objects is extremely small. On a submicron scale, this force becomes so strong that it becomes the dominant force between uncharged conductors. In fact, at separations of 10 nm—about 100 times the typical size of an atom—the Casimir effect produces the equivalent of 1 atmosphere of pressure (101.325 kPa), the precise value depending on surface geometry and other factors.[7]

In modern theoretical physics, the Casimir effect plays an important role in the chiral bag model of the nucleon; and in applied physics, it is significant in some aspects of emerging microtechnologies and nanotechnologies.[8]

## Overview

The Casimir effect can be understood by the idea that the presence of conducting metals and dielectrics alters the vacuum expectation value of the energy of the second quantized electromagnetic field.[9] Since the value of this energy depends on the shapes and positions of the conductors and dielectrics, the Casimir effect makes itself manifest as a force between such objects.

## Vacuum energy

The causes of the Casimir effect are described by quantum field theory, which states that all of the various fundamental fields, such as the electromagnetic field, must be quantized at each and every point in space. In a simplified view, a "field" in physics may be envisioned as if space were filled with interconnected vibrating balls and springs, and the strength of the field can be visualized as the displacement of a ball from its rest position. Vibrations in this field propagate and are governed by the appropriate wave equation for the particular field in question. The second quantization of quantum field theory requires that each such ball-spring combination be quantized, that is, that the strength of the field be quantized at each point in space. At the most basic level, the field at each point in space is a simple harmonic oscillator, and its quantization places a quantum harmonic oscillator at each point. Excitations of the field correspond to the elementary particles of particle physics. However, even the vacuum has a vastly complex structure, so all calculations of quantum field theory must be made in relation to this model of the vacuum.

The vacuum has, implicitly, all of the properties that a particle may have: spin, or polarization in the case of light, energy, and so on. On average, most of these properties cancel out: the vacuum is, after all, "empty" in this sense. One important exception is the vacuum energy or the vacuum expectation value of the energy. The quantization of a simple harmonic oscillator states that the lowest possible energy or zero-point energy that such an oscillator may have is

${E} = \begin{matrix} \frac{1}{2} \end{matrix} \hbar \omega \ .$

Summing over all possible oscillators at all points in space gives an infinite quantity. To remove this infinity, one may argue that only differences in energy are physically measurable; this argument is the underpinning of the theory of renormalization. In all practical calculations, this is how the infinity is always handled. In a deeper sense, however, renormalization is unsatisfying, and the removal of this infinity presents a challenge in the search for a Theory of Everything. Currently there is no compelling explanation for how this infinity should be treated as essentially zero; a non-zero value is essentially the cosmological constant and any large value causes trouble in cosmology.

## Effects

Casimir's observation was that the second-quantized quantum electromagnetic field, in the presence of bulk bodies such as metals or dielectrics, must obey the same boundary conditions that the classical electromagnetic field must obey. In particular, this affects the calculation of the vacuum energy in the presence of a conductor or dielectric.

Consider, for example, the calculation of the vacuum expectation value of the electromagnetic field inside a metal cavity, such as, for example, a radar cavity or a microwave waveguide. In this case, the correct way to find the zero point energy of the field is to sum the energies of the standing waves of the cavity. To each and every possible standing wave corresponds an energy; say the energy of the nth standing wave is En. The vacuum expectation value of the energy of the electromagnetic field in the cavity is then

$\langle E \rangle = \frac{1}{2} \sum_n E_n$

with the sum running over all possible values of n enumerating the standing waves. The factor of 1/2 corresponds to the fact that the zero-point energies are being summed (it is the same 1/2 as appears in the equation $E=\hbar \omega/2$). Written in this way, this sum is clearly divergent; however, it can be used to create finite expressions.

In particular, one may ask how the zero point energy depends on the shape s of the cavity. Each energy level En depends on the shape, and so one should write En(s) for the energy level, and $\langle E(s) \rangle$ for the vacuum expectation value. At this point comes an important observation: the force at point p on the wall of the cavity is equal to the change in the vacuum energy if the shape s of the wall is perturbed a little bit, say by δs, at point p. That is, one has

$F(p) = - \left. \frac{\delta \langle E(s) \rangle} {\delta s} \right\vert_p\,$

This value is finite in many practical calculations.[10]

## Casimir's calculation

In the original calculation done by Casimir, he considered the space between a pair of conducting metal plates at distance a apart. In this case, the standing waves are particularly easy to calculate, since 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 parallel plates lie in the xy-plane, the standing waves are

$\psi_n(x,y,z;t) = e^{-i\omega_nt} e^{ik_xx+ik_yy} \sin \left( k_n z \right)$

where ψ stands for the electric component of the electromagnetic field, and, for brevity, the polarization and the magnetic components are ignored here. Here, kx and ky are the wave vectors in directions parallel to the plates, and

$k_n = \frac{n\pi}{a}$

is the wave-vector perpendicular to the plates. Here, n is an integer, resulting from the requirement that ψ vanish on the metal plates. The energy of this wave is

$\omega_n = c \sqrt{{k_x}^2 + {k_y}^2 + \frac{n^2\pi^2}{a^2}}$

where c is the speed of light. The vacuum energy is then the sum over all possible excitation modes

$\langle E \rangle = \frac{\hbar}{2} \cdot 2 \int \frac{dk_x dk_y}{(2\pi)^2} \sum_{n=1}^\infty A\omega_n$

where 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 regulator (discussed in greater detail below). The regulator will serve to make the expression finite, and in the end will be removed. The zeta-regulated version of the energy per unit-area of the plate is

$\frac{\langle E(s) \rangle}{A} = \hbar \int \frac{dk_x dk_y}{(2\pi)^2} \sum_{n=1}^\infty \omega_n \vert \omega_n\vert^{-s}$

In the end, the limit $s\to 0$ is to be taken. Here s is just a complex number, not to be confused with the shape discussed previously. This integral/sum is finite for s real and larger than 3. The sum has a pole at s = 3, but may be analytically continued to s = 0, where the expression is finite. Expanding this, one gets

$\frac{\langle E(s) \rangle}{A} = \frac{\hbar c^{1-s}}{4\pi^2} \sum_n \int_0^\infty 2\pi qdq \left \vert q^2 + \frac{\pi^2 n^2}{a^2} \right\vert^{(1-s)/2}$

where polar coordinates $q^2 = k_x^2+k_y^2$ were introduced to turn the double integral into a single integral. The q in front is the Jacobian, and the comes from the angular integration. The integral is easily performed, resulting in

$\frac{\langle E(s) \rangle}{A} = -\frac {\hbar c^{1-s} \pi^{2-s}}{2a^{3-s}} \frac{1}{3-s} \sum_n \vert n\vert ^{3-s}$

The sum may be understood to be the Riemann zeta function, and so one has

$\frac{\langle E \rangle}{A} = \lim_{s\to 0} \frac{\langle E(s) \rangle}{A} = -\frac {\hbar c \pi^{2}}{6a^{3}} \zeta (-3)$

But ζ( − 3) = 1 / 120 and so one obtains

$\frac{\langle E \rangle}{A} = \frac {-\hbar c \pi^{2}}{3 \cdot 240 a^{3}}$

The Casimir force per unit area Fc / A for idealized, perfectly conducting plates with vacuum between them is

${F_c \over A} = - \frac{d}{da} \frac{\langle E \rangle}{A} = -\frac {\hbar c \pi^2} {240 a^4}$

where

$\hbar$ (hbar, ħ) is the reduced Planck constant,
c is the speed of light,
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 $\hbar$ shows that the Casimir force per unit area Fc / A is very small, and that furthermore, the force is inherently of quantum-mechanical origin.

### More recent theory

Casimir's analysis of idealized metal plates was generalized to arbitrary dielectric and realistic metal plates by Lifshitz and his students.[11][12] 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' theory for two metal plates reduces to Casimir's idealized 1/a4 force law for large separations a much greater than the skin depth of the metal, and conversely reduces to the 1/a3 force law of the London dispersion force (with a coefficient called a Hamaker constant) for small a, with a more complicated dependence on a for intermediate separations determined by the dispersion of the materials.[13]

Lifshitz' 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.[14] For example, the force in the experimental sphere–plate geometry was computed with an approximation (due to Derjaguin) that the sphere radius R is much larger than the separation a, in which case the nearby surfaces are nearly parallel and the parallel-plate result can be adapted to obtain an approximate R/a3 force (neglecting both skin-depth and higher-order curvature effects).[14][15] However, in the 2000s 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.[14]

## Measurement

One of the first experimental tests was conducted by Marcus Sparnaay at Philips in Eindhoven, in 1958, in a delicate and difficult experiment with parallel plates, obtaining results not in contradiction with the Casimir theory,[16][17] but with large experimental errors. Some of the experimental details as well as some background information on how Casimir, Polder and Sparnaay arrived at this point[18] are highlighted in a 2007 interview with Marcus Sparnaay.

The Casimir effect was measured more accurately in 1997 by Steve K. Lamoreaux of Los Alamos National Laboratory,[19] and by Umar Mohideen and Anushree Roy of the University of California at Riverside.[20] In practice, rather than using two parallel plates, which would require phenomenally accurate alignment to ensure they were parallel, the experiments use one plate that is flat and another plate that is a part of a sphere with a large radius.

In 2001, a group (Giacomo Bressi, Gianni Carugno, Roberto Onofrio and Giuseppe Ruoso) at the University of Padua (Italy) finally succeeded in measuring the Casimir force between parallel plates using microresonators.[21]

## Regularisation

In order to be able to perform calculations in the general case, it is convenient to introduce a 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 or exponentially regulated sum is

$\langle E(t) \rangle = \frac{1}{2} \sum_n \hbar |\omega_n| \exp (-t|\omega_n|)$

where the limit $t\to 0^+$ is taken in the end. The divergence of the sum is typically manifested as

$\langle E(t) \rangle = \frac{C}{t^3} + \textrm{finite}\,$

for three-dimensional cavities. The infinite part of the sum is associated with the bulk constant 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 regulator

$\langle E(t) \rangle = \frac{1}{2} \sum_n \hbar |\omega_n| \exp (-t^2|\omega_n|^2)$

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

$\langle E(s) \rangle = \frac{1}{2} \sum_n \hbar |\omega_n| |\omega_n|^{-s}$

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

Not every cavity configuration necessarily leads to a finite part (the lack of a pole at 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 photons (such as X-rays), 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".)

## Generalities

The Casimir effect can also be computed using the mathematical mechanisms of functional integrals of quantum field theory, although such calculations are considerably more abstract, and thus difficult to comprehend. In addition, they can be carried out only for the simplest of geometries. However, the formalism of quantum field theory makes it clear that the vacuum expectation value summations are in a certain sense summations over so-called "virtual particles".

More interesting is the understanding that the sums over the energies of standing waves should be formally understood as sums over the eigenvalues of a Hamiltonian. This allows atomic and molecular effects, such as the van der Waals force, to be understood as a variation on the theme of the Casimir effect. Thus one considers the Hamiltonian of a system as a function of the arrangement of objects, such as atoms, in configuration space. The change in the zero-point energy as a function of changes of the configuration can be understood to result in forces acting between the objects.

In the chiral bag model of the nucleon, the Casimir energy plays an important role in showing the mass of the nucleon is independent of the bag radius. In addition, the spectral asymmetry is interpreted as a non-zero vacuum expectation value of the baryon number, cancelling the topological winding number of the pion field surrounding the nucleon.

## Wormholes

Exotic matter with negative energy density may be required to stabilize a wormhole.[22] Morris, Thorne and Yurtsever pointed out that the quantum mechanics of the Casimir effect can be used to produce a locally mass-negative region of space-time,[23] and suggested that negative effect could be used to stabilize a wormhole to allow faster than light travel.

In physics, a wormhole is a hypothetical topological feature of spacetime that would be, fundamentally, a "shortcut" through spacetime. For a simple visual explanation of a wormhole, consider spacetime visualized as a two-dimensional (2D) surface. If this surface is folded along a third dimension, it allows one to picture a wormhole "bridge". (Please note, though, that this is merely a visualization displayed to convey an essentially unvisualisable structure existing in 4 or more dimensions. The parts of the wormhole could be higher-dimensional analogues for the parts of the curved 2D surface; for example, instead of mouths which are circular holes in a 2D plane, a real wormhole's mouths could be spheres in 3D space.) A wormhole is, in theory, much like a tunnel with two ends each in separate points in spacetime, or it can be also known as two connecting black holes.

See also the Scharnhorst effect. This concept has been used extensively in science fiction.

## Analogies and the dynamic Casimir effect

A similar analysis can be used to explain Hawking radiation that causes the slow "evaporation" of black holes (although this is generally visualised as the escape of one particle from a virtual particle-antiparticle pair, the other particle having been captured by the black hole).

The dynamical Casimir effect is the production of particles and energy from an accelerated boundary, often referred to as a moving mirror[24] or motion-induced radiation.

Constructed within the framework of quantum field theory in curved spacetime, the dynamical Casimir effect has been used to better understand acceleration radiation; i.e. the Unruh effect.

Moving mirrors create entropy, particles, energy and gravitational-like effects. In analogy to the event horizon of a black hole, an accelerated mirror amplifies quantum field vacuum fluctuations.

An experimental verification of the dynamical Casimir effect was first achieved in May 2011 at Chalmers University of Technology, in Gothenburg , Sweden.[25][26]

## Repulsive forces

There are few instances wherein the Casimir effect can give rise to repulsive forces between uncharged objects. In a seminal paper, Evgeny Lifshitz showed (theoretically) that in certain circumstances (most commonly involving liquids), repulsive forces can arise.[27] This has sparked interest in applications of the Casimir effect toward the development of levitating devices. An experimental demonstration of the Casimir-based repulsion predicted by Lifshitz was recently carried out by Munday et al.[28] Other scientists have also suggested the use of gain media to achieve a similar levitation effect,[29] though this is controversial[citation needed] because these materials seem to violate fundamental causality constraints and the requirement of thermodynamic equilibrium.

## Applications

It has been suggested that the Casimir forces have application in nanotechnology,[30] in particular silicon integrated circuit technology based micro- and nanoelectromechanical systems, silicon array propulsion for space drives, and so-called Casimir oscillators.[31]

## References

1. ^ Cyriaque Genet, Francesco Intravaia, Astrid Lambrecht and Serge Reynaud (2004) "Electromagnetic vacuum fluctuations,Casimir and Van der Waals forces"
2. ^ The Force of Empty Space, Physical Review Focus, 3 December 1998
3. ^ A. Lambrecht, The Casimir effect: a force from nothing, Physics World, September 2002.
4. ^ American Institute of Physics News Note 1996
5. ^ Jaffe, R. (2005). "Casimir effect and the quantum vacuum". Physical Review D 72 (2): 021301. arXiv:hep-th/0503158. Bibcode 2005PhRvD..72b1301J. doi:10.1103/PhysRevD.72.021301.
6. ^ Photo of ball attracted to a plate by Casimir effect
7. ^ "The Casimir effect: a force from nothing". physicsworld.com. 2002-09-01. Retrieved 2009-07-17.
8. ^ Astrid Lambrecht,Serge Reynaud and Cyriaque Genet" Casimir In The Nanoworld"
9. ^ E. L. Losada" Functional Approach to the Fermionic Casimir Effect"
10. ^ For a brief summary, see the introduction in Passante, R.; Spagnolo, S. (2007). "Casimir-Polder interatomic potential between two atoms at finite temperature and in the presence of boundary conditions". Physical Review A 76 (4): 042112. arXiv:0708.2240. Bibcode 2007PhRvA..76d2112P. doi:10.1103/PhysRevA.76.042112.
11. ^ Dzyaloshinskii, I E; Lifshitz, E M; Pitaevskii, Lev P (1961). "GENERAL THEORY OF VAN DER WAALS' FORCES". Soviet Physics Uspekhi 4 (2): 153. Bibcode 1961SvPhU...4..153D. doi:10.1070/PU1961v004n02ABEH003330.
12. ^ Dzyaloshinskii, I E; Kats, E I (2004). "Casimir forces in modulated systems". Journal of Physics: Condensed Matter 16 (32): 5659. arXiv:cond-mat/0408348. Bibcode 2004JPCM...16.5659D. doi:10.1088/0953-8984/16/32/003.
13. ^ V. A. Parsegian, Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists (Cambridge Univ. Press, 2006).
14. ^ a b c Johnson, S.; Capasso, F.; Johnson, Steven G. (2011). "The Casimir effect in microstructured geometries". Nature Photonics 5 (4): 211–221. Bibcode 2011NaPho...5..211R. doi:10.1038/nphoton.2011.39.  Review article.
15. ^ B. V. Derjaguin, I. I. Abrikosova, and E. M. Lifshitz, Quarterly Reviews, Chemical Society, vol. 10, 295–329 (1956).
16. ^ Sparnaay, M. J. (1957). "Attractive Forces between Flat Plates". Nature 180 (4581): 334. Bibcode 1957Natur.180..334S. doi:10.1038/180334b0.
17. ^ Sparnaay, M (1958). "Measurements of attractive forces between flat plates". Physica 24 (6–10): 751. Bibcode 1958Phy....24..751S. doi:10.1016/S0031-8914(58)80090-7.
18. ^ Movie
19. ^ Lamoreaux, S. K. (1997). "Demonstration of the Casimir Force in the 0.6 to 6 μm Range". Physical Review Letters 78: 5. Bibcode 1997PhRvL..78....5L. doi:10.1103/PhysRevLett.78.5.
20. ^ Mohideen, U.; Roy, Anushree (1998). "Precision Measurement of the Casimir Force from 0.1 to 0.9 µm". Physical Review Letters 81 (21): 4549. arXiv:physics/9805038. Bibcode 1998PhRvL..81.4549M. doi:10.1103/PhysRevLett.81.4549.
21. ^ Bressi, G.; Carugno, G.; Onofrio, R.; Ruoso, G. (2002). "Measurement of the Casimir Force between Parallel Metallic Surfaces". Physical Review Letters 88 (4): 041804. arXiv:quant-ph/0203002. Bibcode 2002PhRvL..88d1804B. doi:10.1103/PhysRevLett.88.041804. PMID 11801108.
22. ^ M. Visser (1995) Lorentzian Wormholes: from Einstein to Hawking, AIP Press, Woodbury NY, ISBN 1-56396-394-9
23. ^ Morris, Michael; Thorne, Kip; Yurtsever, Ulvi (1988). "Wormholes, Time Machines, and the Weak Energy Condition". Physical Review Letters 61 (13): 1446–1449. Bibcode 1988PhRvL..61.1446M. doi:10.1103/PhysRevLett.61.1446. PMID 10038800.
24. ^ Fulling, S. A.; Davies, P. C. W. (1976). "Radiation from a Moving Mirror in Two Dimensional Space-Time: Conformal Anomaly". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 348 (1654): 393. Bibcode 1976RSPSA.348..393F. doi:10.1098/rspa.1976.0045.
25. ^
26. ^
27. ^ Dzyaloshinskii, I.E.; Lifshitz, E.M.; Pitaevskii, L.P. (1961). "The general theory of van der Waals forces†". Advances in Physics 10 (38): 165. Bibcode 1961AdPhy..10..165D. doi:10.1080/00018736100101281.
28. ^ Munday, J.N.; Capasso, F.; Parsegian, V.A. (2009). "Measured long-range repulsive Casimir-Lifshitz forces". Nature 457 (7226): 170–3. Bibcode 2009Natur.457..170M. doi:10.1038/nature07610. PMID 19129843.
29. ^ Highfield, Roger (2007-08-06). "Physicists have 'solved' mystery of levitation". The Daily Telegraph (London). Retrieved 2010-04-28.
30. ^ Capasso, F.; Munday, J.N.; Iannuzzi, D.; Chan, H.B. (2007). "Casimir forces and quantum electrodynamical torques: physics and nanomechanics". IEEE Journal of Selected Topics in Quantum Electronics 13 (2): 400. doi:10.1109/JSTQE.2007.893082.
31. ^ Serry, F.M.; Walliser, D.; MacLay, G.J. (1995). "The anharmonic Casimir oscillator (ACO)-the Casimir effect in a model microelectromechanical system". Journal of Microelectromechanical Systems 4 (4): 193. doi:10.1109/84.475546.

### Temperature dependence

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