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This is a philosophical question which has intrigued us for many years – is it possible to prove that something can come out of nothing? The answer, according to the latest scientific research, is “yes”.

This particular page of our website is intended to pique the interest of some of our readers and we trust you find the following exposition of the Casimir effect interesting and stimulating.

CAN something come of nothing?

Philosophers debated that question for millennia before physics came up with the answer—and that answer is yes. For quantum theory has shown that a vacuum (ie, nothing) only appears to be empty space. Actually, it is full of virtual particles of matter and their anti-matter equivalents, which, in obedience to Werner Heisenberg's uncertainty principle, flit in and out of existence so fast that they cannot usually be seen.

 Alamy Virtual percussion

However, in 1948, a Dutch physicist called Hendrik Casimir realised that in certain circumstances these particles would create an effect detectable in the macroscopic world that people inhabit. He imagined two metal plates so close together that the distance between them was comparable with the wavelengths of the virtual particles. (Another consequence of quantum theory is that all particles are simultaneously waves.) In these circumstances, he realised, the plates would be pushed together. That is because only particles with a wavelength smaller than the gap between the plates could appear in that gap, whereas particles of any wavelength could appear on the other sides of the plates. There would thus be more particles pushing in than pushing out, and the plates would clash together like a pair of tiny cymbals.

A neat idea. And in 1996, it was shown experimentally to be true. But so what? The answer is that now things in the computing industry have become so small that the Casimir force is starting to affect them. Computer engineers talk of “stiction”—the phenomenon of microscopic components sticking together—and spend a lot of time trying to figure out ways to avoid it. The Casimir effect is not the only cause of stiction, but it is an important one.

Engineers are now beginning to understand how Casimir forces depend on a device's design—an essential starting point if Casimir-related stiction is to be eliminated. A forthcoming paper in Physical Review Letters by Ho Bun Chan of the University of Florida, Gainesville, shows how the force changes when one of the flat metal plates imagined by Casimir is replaced with a corrugated silicon one. Dr Chan found that the effect's size depends on the spacing of the trenches. This suggests that if you designed microelectronic components carefully, you could reduce stiction.

Stiction is not, however, always a problem, and work is also going on to turn the Casimir effect from a liability to an asset, by using it in what are known as microelectromechanical systems (MEMS). In 2001 Dr Chan and his collaborator Federico Capasso, who were then working at Bell Labs, published a paper showing one way to exploit the effect in MEMS. They made a tiny, gold-plated seesaw and slowly lowered a gold-plated ball towards it. At a distance of about 300 nanometres (a nanometre is a billionth of a metre), the seesaw began to tilt towards the ball. The effect was particularly strong at distances less than 150 nanometres. What Dr Chan and Dr Capasso had shown was that the Casimir force could be used to make a simple lever that might act as a tiny but accurate stress sensor in nanoscale machines.

What really excites researchers, though, is the idea that the Casimir effect may sometimes be repulsive. That would knock the problem of stiction firmly on the head and, according to Dr Capasso, would open up the possibility of making things like frictionless ball-bearings.

According to calculations made several decades ago by Evgeny Lifshitz, a Russian physicist, Casimir repulsion should be possible—but you have to replace the vacuum with a fluid. The maths suggest that in order for this to happen, both liquid and plates must be made of carefully chosen substances. This is because the repulsion is caused by electromagnetic charges induced in the plates and the liquid by the virtual particles, and the forces produced by these charges depend on the materials you use.
As with Casimir's original suggestion, the gap between the theory and its practical demonstration has been a long one. But Dr Capasso, who is now at Harvard, and his student Jeremy Munday, have started investigating Lifshitz's ideas.

They recently performed an experiment in which they looked at what happens when the plates are made of gold and the fluid is ethanol. Although they did not succeed in reversing the attractive Casimir effect completely, they did reduce it by 80%—a reduction in line with Lifshitz's predictions. The right combination of materials, if it can be found, should therefore produce the desired reversal. That would be very good news for MEMS indeed.

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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 second quantized 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

Main article: 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 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. 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, 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. 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.

The Casimir effect

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) \ranglefor 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.

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, 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 0is 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^2were introduced to turn the double integral into a single integral. The q in front is the Jacobian, and the 2π 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 \hbarshows 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

A very complete analysis of the Casimir effect at short distances is based upon a detailed analysis of the van der Waals' force by Lifshitz. 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. In addition to these factors, complications arise due to surface roughness of the boundary and to geometry effects such as degree of parallelism of bounding plates.

For boundaries at large separations, retardation effects give rise to a long-range interaction. For the case of two parallel plates composed of ideal metals in vacuum, the results reduce to Casimir’s.

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, but with large experimental errors.

The Casimir effect was measured more accurately in 1997 by Steve K. Lamoreaux of Los Alamos National Laboratory and by Umar Mohideen and Anushree Roy of the University of California at Riverside. 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 at the University of Padua finally succeeded in measuring the Casimir force between parallel plates using microresonators.

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} + \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.

Casimir effect and wormholes

Exotic matter with negative energy density is required to stabilize a wormhole. 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, and suggested that negative effect could be used to stabilize a wormhole to allow faster than light travel. This was used in the novel Warp Speed by Travis S. Taylor.

 
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