The Casimir Force is a consequence of zero-point energy

## Casimir Force

### What Is The Casimir Force?

If you bring two flat, parallel metal plates close to one another, you will find that they are pulled together or pushed apart -- that is, the exert equal and opposite forces on each other. If the plates are not at the same electrostatic potential, charges on the surfaces (charges in a conductor always go to the surface) will either attract or repel one another depending on whether the plates are oppositely or similarly charged. This electric force is usually a strong one, and varies relatively slowly with the distance between the plates.

If the temperature is above absolute zero, the electrons in the two plates will be more or less mobile, and move about randomly. The resulting fluctuations in their density will give the same effect as a positive or negeative charge, which will attract or repel electrons in the other plate. This weak, temperature-dependent force is similar to the van der Waals forces between two uncharged atoms. It decreases rapidly as the plates are separated.

Should you measure the force very accurately, and subtract from it the sum of the two forces just mentioned, you will find that there is an unexplained force of attraction, intermediate in magnitude between the ones you expect. This is the Casimir Force, the necessity of whose existence was deduced by H. B. G. Casimir in 1948 [Koninkl. Ned. Akad. Wetenschap. Proc. 51, 793 (1948)]. It was first observed, rather roughly, in 1958 by M. J. Spaarnay [Physica 24, 751 (1958)]. The force received much theoretical study, but a good, conclusive observation was only made much later by S. K. Lamoreaux [Phys. Rev. Lett. 78, 5 (1997)], clearly described in an excellent paper so unlike most modern ones. It is well worth reading!

### What Is The Origin Of The Casimir Force?

The Casimir Force is an electric force, but its origin is different from that of ordinary electric forces. It is a purely quantum-mechanical effect arising from the zero-point energy of the harmonic oscillators that are the normal modes of the electromagnetic field. The electromagnetic field must satisfy certain boundary conditions at the surfaces of our conducting plates, and these boundary conditions rule out some of the modes (oscillators) that would otherwise exist in unbounded space. Since there are fewer oscillators between the plates, there is less zero-point energy in this region. If the plates are brought closer together, this volume of smaller energy density is decreased, while the volume of normal zero-point energy density is increased. Since this results in an overall gain of energy to the universe, a force pressing the plates together is the result.

To see this better, suppose the region between the plates was at a lower pressure than outside, so that it possessed less pV energy. If we write E = pAx, where p is the difference in pressure, A the area of a plate, and x the separation, then the force, by the usual rule, is F = -dE/dx = -pA, as we expect. This is just 'nature abhors a vacuum,' as they used to say.

The force can also be explained as the difference in attraction of the fluctuating zero-point fields between the plates and outside them for the electric charges induced in the plates by the zero-point fields. These zero- point fields are just those necessary to account for the zero-point energy.

### Zero-Point Energy

The harmonic oscillator is the first system a student of quantum mechanics analyzes. Its energy is found to vary by increments of hf, where h is Planck's constant, and f is the frequency of the oscillator. In addition, it has a zero- point energy of hf/2, and corresponding fluctuations in its position and momentum, even when it is in its lowest, unexcited state. This energy can never be taken from the oscillator; it is necessary for its existence.

The medium that carries electromagnetic waves, called the vacuum, can be considered as a collection of harmonic oscillators or normal modes, distributed evenly in space with frequencies from zero to infinity. An electromagnetic wave is an excitation of these oscillators. A photon is a unit excitation of an oscillator, and has energy hf. An actual wave is usually a superposition of many photons. If you have followed the argument, you now realize that every volume of space has an infinite zero-point energy!

Energy itself is an odd and difficult concept. It cannot be regarded as a substance, though most people think of it as one. Energy is not a property of a point, but of a complete system, as beauty is of a painting. Of course, it can be so considered at times, but this is not fundamental. Only differences of energy are significant, not absolute values, as in the case of the simpler concept of entropy. Energy is always differentiated to get forces, and what we call the energy depends on how we intend to do the differentiation.

Zero-point energy adds mystery to an enigma. The fact that it is infinite for the electromagnetic field is uncomfortable, since we are unaccustomed to infinite values as reference levels, but in this case this is valid. When an elctromagnetic wave exists, the energy is infinite plus wave. Subtracting, we find the wave energy as infinite plus wave minus infinite = wave. This is the price to be paid for the convenience of continuous variables.

### Measuring The Casimir Force

The Casimir Force between plates a distance x micrometers apart is 0.016/a4 dynes per square centimeter. This is much smaller than unavoidable electrostatic forces, but since these vary inversely as the first power of x, the Casimir force can be separated by the very different distance dependence. In Lamoreaus's experiment, these were found to be the only significant forces.

It is impractical to use plane plates, because they must be brought closer than a micrometer apart, making the maintenance of parallelism and accuracy of spacing and motion very difficult. Lamoreaux solved this problem by using one plane plate, and one spherical plate. Although the force is weaker, all problems of alignment vanish, and it is easy to alter the spacing.

The forces that had to be measured were in the region of 10- 4 dyne, so that the method had to be sensitive practically to microdynes! A torsion pendulum was the solution. The fiber was 76 micron Tungsten wire, 66 cm long, suspending a triangular aluminum plate with the Casimir plates on one side, and a pair of circular conducting plates on the other. The latter plates were used to exert electrostatic forces on the pendulum to keep its orientation constant by a feedback control. As the separation of the Casimir plates was changed, changing the force, the feedback voltage was an accurate measurement of the force. Full details are given in the paper.

At closest approach, the Casimir Force was about 20% of the observed force, but the result agreed to 5% with the expected value. There were no adjustable parameters in the theory, so this is a remarkably good result. The van der Waals force was low because the plates were plated with the yellow metal gold, that has a low plasma frequency. This, and the temperature correction derived by Schwinger, were estimated as at most 3%, so were not evident within the accuracy of the experiment.