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§ Phase xiv · Fields as Particles · Chapter 02

Vacuum fluctuations

The vacuum, in quantum field theory, is not the absence of things. It is the lowest-energy state of every field in the universe, and the energy of that state is not zero. The uncertainty principle does not let any mode sit perfectly still. Pull two mirrors close together and you can measure the consequence. Measure a hydrogen atom precisely enough and you can measure it again. This chapter is about the quiet weather inside an empty box.

Aristotle wrote in the fourth century BC that nature abhors a vacuum. He turned out to be a better physicist than he gets credit for. Two and a half millennia later, his slogan is closer to literal truth than he could possibly have imagined. The vacuum of quantum field theory is not the calm, silent void that Newton and Maxwell inherited from him. It is a humming, restless medium in which every field is forced by the rules of quantum mechanics to wiggle even when there is nothing in it to wiggle about. Empty space, in QFT, is the lowest-rent apartment in town, but it is not vacant. There is always somebody home.

To see why, remember where the previous chapter left us. A quantum field, like the electromagnetic field, can be written as a giant collection of independent harmonic oscillators, one for each plane-wave mode. Excite a mode and you have added a photon to the universe; excite it twice and you have two photons of the same momentum. Leave every mode in its ground state and you have no photons. That is the vacuum.

But a harmonic oscillator, even in its ground state, is not at rest. We saw this all the way back in Phase iii, when we solved the quantum harmonic oscillator and found that its ground-state energy is not zero but ½ℏω. The uncertainty principle forbids the perfect knowledge of both position and momentum that classical rest would require. Even at absolute zero, the oscillator jitters. Each mode of the electromagnetic field is an oscillator. Each one carries that same irreducible ½ℏω of zero-point energy. Sum over every mode of every field and you get a vacuum energy that is, taken naively, infinite.

Most of the time we throw the infinity away. Energy in non-gravitational physics is measured by differences, and the constant background drops out of every calculation. The vacuum just sits there and does no work. The reason this chapter exists is that under the right conditions, the differences in vacuum energy show up. Change the geometry, change the boundary conditions, change the background of the universe, and the bookkeeping shifts. Light bills come due. And the universe, it turns out, has paid them, in places we can point at.

The first concrete way the vacuum gives itself away is geometric. Imagine taking two perfectly conducting, electrically neutral metal plates and bringing them parallel to each other in an evacuated chamber. Classically, the situation is a non-event: no charges, no currents, no fields, nothing happens. In QFT, you have just done something subtle. A perfect conductor forces the parallel component of the electric field to vanish at its surface. Between the plates, the electromagnetic field is no longer free to take whatever mode shapes it likes; it has to fit standing waves that vanish at both plates, exactly the way a vibrating guitar string has to vanish at both bridges. Only specific wavelengths fit. Outside the plates, in the open air of the chamber, every wavelength still fits.

The zero-point energy of the modes between the plates is therefore different from the zero-point energy of the modes outside. Both numbers are formally infinite, but their difference is finite. Hendrik Casimir, working at Philips Research in Eindhoven on the colloidal stability of paint, was the man who first did the subtraction in 1948. He was thinking about why suspended particles in a fluid attract each other at short range. Niels Bohr, on a visit, suggested he look at it as a problem of vacuum energy. Casimir went home, did the sum, and got an extraordinary result.

The crucial physical fact in that passage is the second sentence. The quantum vacuum is the lowest-energy state, and contains no particles, but it is not empty. The fields themselves cannot stop fluctuating. Photons in particular do not need to be present for the electromagnetic field to have measurable properties at every point of space. You cannot ask a field operator to be quietly zero everywhere; the same commutation relations that quantize photons also forbid the field’s value and its time derivative from being simultaneously known with arbitrary precision. There is always some residual wiggle. Casimir’s number was the first proof that the wiggle has consequences.

The figure below shows the geometry. Between two plates separated by distance L, only standing waves with nodes on both surfaces survive. The longest mode has wavelength 2L, then L, then 2L/3, and so on, an evenly-spaced ladder of allowed wave numbers. Outside, in the unconfined region, every wavelength contributes to the vacuum. The difference in zero-point energy per unit area, after a careful regularization that throws away the infinite common background, is −ℏc π² / (720 L³). Take the derivative with respect to L and you get the force per unit area between the plates.

Casimir’s prediction sat in the literature largely unverified for almost half a century. The geometry is demanding. To see a force of a few piconewtons per square millimeter you need optically flat metal surfaces brought to within a micron of each other without touching, in a vacuum, with vibration isolation that would shame an interferometer. Marcus Sparnaay attempted it in 1958 with about 100 percent error bars and called it consistent with the theory. The clean confirmation had to wait until 1997, when Steve Lamoreaux at the University of Washington used a torsion pendulum and a spherical lens against a plate (a geometry that finesses the parallelism problem) and pinned the effect to about 5 percent. Umar Mohideen in 1998 squeezed the error to 1 percent with an atomic-force-microscope tip. Casimir’s number, calculated on a notepad in 1948, is now a textbook precision measurement.

A second consequence of the busy vacuum, predicted in the same heroic year as Casimir’s calculation, was observed slightly before it. In April 1947 at Columbia University, Willis Lamb and his graduate student Robert Retherford used a beam of metastable hydrogen atoms and a microwave cavity to look for an effect Dirac’s relativistic theory said should not exist. According to Dirac, the 2s and 2p energy levels of hydrogen with the same total angular momentum (2S₁/₂ and 2P₁/₂) are exactly degenerate. Lamb found that they were not. The 2s level lay about 1000 megahertz above the 2p, a shift small in absolute terms (roughly four parts in ten million of the binding energy) but large enough to wreck the relativistic prediction. He had measured the influence of the vacuum on the electron’s orbit.

Hans Bethe, a German emigré at Cornell, was on the train home from the Shelter Island conference in June 1947 where Lamb had announced the result. By the time he stepped off at Schenectady the next morning, he had a nonrelativistic estimate of the shift on the back of an envelope: about 1040 megahertz, within five percent of Lamb’s measurement. His trick was a cutoff at the electron’s rest energy mc² that would, in retrospect, be the first appearance of renormalization in particle physics. Within a year, Julian Schwinger, Sin-Itiro Tomonaga, and Richard Feynman had built up the systematic machinery that explained the rest of the shift, plus the anomalous magnetic moment of the electron, plus a handful of other small effects with the same physical origin: the electron does not orbit in vacuum, it orbits inside a swarm of virtual photons that it has emitted and is about to reabsorb. Those virtual photons jostle it, smear out its position, and shift its binding energy. The vacuum nudges the atom.

E(L)/A = ℏc · Σₙ ∫ d²k∥/(2π)² · ½ √(k∥² + (nπ/L)²)

ΔE(L)/A = − ℏc π² / (720 L³)

P(L) = − ∂(ΔE/A)/∂L = − ℏc π² / (240 L⁴)

By the mid-1950s, the picture that had emerged was vivid. The vacuum is filled with what physicists came to call virtual particles. Roughly speaking, a region of empty space at any instant contains a thin soup of electron-positron pairs, photons, and (now) every other quantum field, popping in and out of existence on time scales that the uncertainty principle permits. Energy ΔE can be borrowed for a time Δt ~ ℏ/ΔE without bookkeeping consequence. A bare electric charge polarizes those pairs the way an ion polarizes a dielectric: virtual positrons crowd a little closer, virtual electrons retreat a little farther, and the field you measure at long distance is reduced. When you say the electron has charge e, you are quoting a value renormalized by this screening. The bare charge, before vacuum polarization, is much larger and depends sensitively on how close you stand. Get to within an attometer and you can measure the difference; the running of the electromagnetic coupling has been confirmed at colliders to better than a percent.

The convenient cartoon for visualizing all this is the vacuum bubble diagram, a small closed loop drawn in spacetime in which a virtual particle and its antiparticle are created from nothing, propagate briefly, and annihilate again, leaving no external lines. Every region of space, all the time, is decorated with countless such loops. Feynman diagrams with no external lines are the QFT equivalent of a doodle in the margin: they do not affect any transition probability between asymptotic states (they cancel against the normalization of the vacuum) but they are physically real in the sense that they represent the energy density that filled the space.

So far the story is QED, electrons and photons. The story does not stop there. The Standard Model contains one more field whose vacuum is profoundly weirder. The Higgs field, unlike the electromagnetic field, does not settle at value zero in its ground state. Its potential energy looks like the inside of a wine bottle: a hump in the middle and a circular trough at finite field value. The lowest-energy configuration of the universe is therefore not “no Higgs field,” but “the Higgs field equal to about 246 GeV, everywhere, all the time.” Every electron, every quark, every W and Z boson moves through this nonzero background, and its interaction with the background is what makes its rest mass. Switch off the Higgs vacuum expectation value and every charged fermion in the Standard Model becomes massless. The 2012 discovery at the LHC of an excitation of this field (the Higgs boson at 125 GeV) was, in the language of this chapter, the discovery of the lowest-lying ripple on top of the Higgs vacuum. The vacuum itself we walk through every day.

We are left with the cleanest mystery in modern physics. If every mode of every field carries ½ℏω of zero-point energy, the natural value of the total vacuum energy density is the Planck-scale value (M_Pl)⁴, around 10¹¹³ joules per cubic meter. The observed value, inferred from the accelerating expansion of the universe and identified as the cosmological constant, is roughly 10⁻⁹ joules per cubic meter. The natural prediction overshoots the observation by 122 orders of magnitude, the worst quantitative discrepancy between theory and experiment in the history of physics. This is the cosmological constant problem, and no convincing solution exists. The vacuum, having spent a chapter being elegant, ends it by being the source of the deepest open puzzle in fundamental physics.

For our immediate purposes, the takeaway is humbler. We started with Planck quantizing oscillator energies in a hot oven. We end with the realization that the same quantization, applied to every mode of every field in the universe, fills “empty” space with measurable structure. The Casimir effect drags two plates together. The Lamb shift bumps the levels of hydrogen by one part in ten million and can be measured to one part in a thousand. Vacuum polarization screens the electron’s charge and runs the coupling constant. The Higgs vacuum gives every massive particle its mass. The inflaton vacuum may have launched the universe. None of these were available in any classical theory. All of them follow from one quiet sentence: the field is a sum of oscillators, and an oscillator can never sit perfectly still.

The next chapter takes the formal leap that brings all this together. We will move from the single-particle Schrödinger equation that has carried us through fourteen phases to a genuine quantum field at every point of space. That step is what makes everything in this chapter calculable, and what binds the rest of the book together.