Casimir effect

In physics, the Casimir effect is a weak force exerted between separate objects, which is not due to charge, gravity, or exchange of particles, but instead is due to resonance in the intervening space between the objects, of all-pervasive energy fields. The force is only measurable when the distance between the objects is extremely small, since it falls off rapidly with distance.

Dutch physicist Hendrik B. G. Casimir first proposed the existence of the force, and an experiment to detect it in 1948 while participating in research at Philips Research Labs. His classic form of the experiment uses a pair of uncharged parallel metal plates in a vacuum, and has successfully demonstrated the force to within 15% of the predicted value according to Casimir's theory.

The van der Waals force between a pair of neutral atoms is a similar effect. 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 becoming of increasing importance in development of the ever-smaller, miniaturised components of emerging micro- and nano- technologies.

Overview

The Casimir effect can be understood by the idea that the presence of conducting metals and dielectrics alter the vacuum expectation value of the energy of the electromagnetic field. Since the value of this energy depends on the shapes and positions of the conductors and dielectrics, the Casimir effect manifests itself as a force between such objects.

Vacuum energy

The Casimir effect is an outcome of 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 naïve sense, 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. Canonically, 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, as this picture shows, even the vacuum has a vastly complex structure. 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, all 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. The removal of this infinity presents a challenge for theoretical particle physics, and as of 2005, there is no compelling explanation for how this infinity should be treated as essentially zero (as a non-zero value is essentially the cosmological constant; a large value causes trouble in cosmology).

The Casimir effect

Casmir forces on parallel plates.

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 sum 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 $E n$ . The vacuum expectation value 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 $E n$ depends on the shape, and so one should write $E n (s)$ 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 $δ s$ , at point p. That is, one has

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

Amazingly, this value is finite in many practical calculations.

Casimir's calculation

In the original calculation done by Casimir, he considered the space between a pair of conducting metal plates a 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 x-y 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, $k x$ and $k y$ 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

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

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

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

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 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 $F c / A$ 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

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 $F c / A$ is very small, and that furthermore, the force is inherently of quantum-mechanical origin.

Measurement

One of the first experimental tests was conducted by Marcus Spaarnay at Philips in Eindhoven, in 1958, in a delicate and difficult experiment, with results in general agreement with theory.

The Casimir effect was measured in 1997 by Steve K. Lamoreaux of Los Alamos National Laboratory and by Umar Mohideen of the University of California at Riverside and his colleague Anushree Roy. 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 of curvature.

Further research has shown that, with materials of certain permittivity and permeability, or with a certain configuration, the Casimir effect can be repulsive instead of attractive.

Regularization

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

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

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, however, it 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.

Analogies

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

A macroscopic effect analogous to the Casimir effect was observed by 18th century French sailors. Where two ships are rocking from side to side in conditions with a strong swell but light wind, and the ships come closer together than roughly 40 m, destructive interference eliminates the swell between the ships. The calm sea between the ships has a lower energy density than the swell to either side of the ships, creating a pressure that can push the ships closer together. If they get too close together, the ships' rigging can become entangled. As a countermeasure, a handbook from the early 1800s recommends that each ship should send out a boat rowed by 10 to 20 sailors to physically pull the ships apart.