Fields fill all space
Faraday invented the field to avoid action at a distance. A century later it turned out to be the thing that actually exists — and the emptiest possible space, with nothing in it at all, is still not quiet.
- Say what it means to quantise a field, and why every mode becomes a harmonic oscillator
- Explain why the vacuum has zero average field and a non-zero spread
- State what the Casimir effect measures, and what it does and does not prove
We began with Faraday in 1831, and an idea he could not write down: that the space between two magnets is not empty but in a condition, and that the condition is what does the pushing. For a century that was treated as a convenience. The field was a way of bookkeeping forces without believing in action at a distance — a map of where a particle would be pushed if you put one there. The particles were the things that existed. The field described what happened between them.
The move that gets you there is simpler than its reputation. Take a field: one number at every point of space. Ask what its small oscillations look like, and break them into modes — patterns with a definite wavelength, the way a guitar string has a fundamental and its harmonics. Now the key fact. Each mode, taken on its own, is a harmonic oscillator: something that wobbles about zero with a restoring force proportional to how far it has been pulled. Not something like one. Mathematically one. And we already know what happens when you quantise a harmonic oscillator.
We met it in the uncertainty lesson, as the reason liquid helium will not freeze. A quantum harmonic oscillator cannot sit still. Its lowest state has energy ½ħω, not zero, and it has a spread in position even there — because sitting exactly at the bottom, exactly motionless, would fix position and momentum simultaneously, and that is not available. So it jiggles in its ground state, permanently, and cooling does not stop it. That is a fact about one oscillator. A field is infinitely many of them.
Read ⟨φ⟩ and √⟨φ²⟩ together, then press "Take ħ to zero" and watch the field go flat. That flat line is the classical vacuum, and it is the picture most people are carrying.
Hendrik Casimir worked out the consequence in 1948, at the Philips laboratory in Eindhoven, after a conversation with Niels Bohr who suggested the zero-point energy might be the way to think about it. Take two conducting plates and face them at each other, very close. Between them, only modes with the right wavelengths fit — nodes at both surfaces, exactly as for a string fixed at both ends. Outside, every wavelength is allowed. So there are fewer modes inside than outside. Less zero-point energy inside than outside. And the plates are pushed together.
Measuring the force between two plates with nothing between them
- The question
- Is there really an attractive force between two uncharged conductors in vacuum, of the size Casimir predicted from counting modes?
- The apparatus
- Sparnaay used a spring balance and flat plates. Lamoreaux used a torsion pendulum with a gold-coated spherical lens facing a flat plate, at separations of 0.6–6 µm. Mohideen and Roy used an atomic force microscope with a polystyrene sphere, reaching 100–900 nm. Both later experiments used a sphere against a plate rather than two plates, because keeping two flat surfaces parallel to within a fraction of their separation is not achievable.
P = π²ħc/240d⁴, with no free parameters — though real surfaces need corrections for finite conductivity, roughness and temperature, which is much of the experimental difficulty.
Sparnaay: consistent with the prediction, but with uncertainties too large to constitute a confirmation. Lamoreaux 1997: agreement to about 5%. Mohideen and Roy 1998: about 1%.
How sure could they be? Modern experiments reach the 1% level, where the corrections for real metals matter more than the ideal formula.
The force is real, it is the predicted size, and it depends on nothing but geometry and fundamental constants. Whatever the vacuum is, it is not inert.
The Casimir effect proves that empty space is full of zero-point energy.
It is weaker evidence than it is usually made out to be, and this is worth being straight about. Robert Jaffe showed in 2005 that the Casimir force can be computed entirely as an interaction between the charges in the two plates — an ordinary QED calculation in which zero-point energy is never mentioned — and that the force vanishes if you switch off the coupling between the field and matter, which a true vacuum-energy effect should not do. So the effect is real, measured, and correctly predicted. What it does not do is settle the interpretation. Both accounts give the same number, which is a normal situation in physics and an uncomfortable one for anybody who wants the experiment to prove a picture.
The better evidence that a quantum field fluctuates is older, and it is spectroscopic. In 1947 Willis Lamb and Robert Retherford measured two levels of hydrogen — 2S½ and 2P½ — which the Dirac equation says have exactly the same energy. They are separated by about 1058 MHz. Accounting for that number requires treating the electron as interacting with a field that has fluctuations, and the effort to calculate it is what produced modern quantum electrodynamics within about two years. It is the measurement that forced the theory of this act into existence.
And now the problem, which is worth meeting early even though it belongs at the end. If every mode carries ½ħω and there are infinitely many modes, the vacuum has infinite energy density. Cut the sum off at the Planck length — the scale below which nobody claims to know what space is doing — and the answer comes out around 4.6 × 10¹¹³ joules per cubic metre. Empty space does have an energy density, and it has been measured, because it is what makes the expansion of the universe accelerate. It is about 5.4 × 10⁻¹⁰ joules per cubic metre.
- 1831Faraday: space between objects is in a condition. The field as a picture.
- 1927Dirac quantises the electromagnetic field; the photon becomes a mode excitation.
- 1947Lamb and Retherford measure a splitting the Dirac equation forbids.
- 1948Casimir predicts a force between plates from the modes that do not fit.
- 1958Sparnaay: consistent with Casimir, but cannot claim a confirmation.
- 1997–98Lamoreaux, then Mohideen and Roy, measure it to 5% and then 1%.
- 1998The expansion of the universe is found to be accelerating: the vacuum has energy after all, and far too little of it.
- 2005Jaffe: the Casimir force can be derived without invoking vacuum energy.