The one medium that was supposed to add exactly

Waves add because the wave equation is linear, and what that does to the energy ends by naming the one thing that would destroy the whole accounting: a medium whose response depends on what is already in it.

Every material medium is such a medium at some amplitude. A sound wave loud enough steepens until it breaks; a crystal in a strong beam generates frequencies nobody supplied; a spring stretched far enough stops being a spring. Superposition is a property of an approximation, and the approximation is that the amplitude is small.

Empty space was the exception, and it was a clean one. Maxwell’s equations in vacuum are linear exactly — there is no amplitude at which they acquire a nonlinear term, no material to respond, and two beams crossing anywhere pass through each other with no interaction whatsoever.

That is false, and the scale on which it is false is the subject of this essay.

Why the vacuum is a medium #

The mechanism needs only one idea and it is the idea that light arrives in lumps made to do work.

A photon of enough energy can make an electron–positron pair. Below that energy it cannot make a real pair, and it can still make a virtual one — a pair that exists for a time short enough that the energy shortfall is hidden by the uncertainty relation, and then annihilates back into the photon. That is not a picturesque description: the process contributes measurably to the electron’s magnetic moment and to the hydrogen atom’s energy levels, and both are among the most precisely verified predictions in physics. The pair is real in every sense except that it does not last, which is the same licence an evanescent field takes to exist where nothing propagates.

Now put two photons in the same place. The first makes a pair; the pair, while it exists, can absorb the second; and the result is two photons going somewhere else. Light has scattered light, through an intermediate state of matter that was never there.

And put a strong field across empty space. The virtual pairs are polarisable — they are charges, briefly — so the field polarises them, and a polarised medium has a refractive index. The vacuum therefore has one, and because the pairs are polarised along the field rather than across it, it has two: the vacuum is birefringent.

The scale of both effects is the field at which the vacuum stops being a perturbation, and it is fixed by one quantity. A pair costs $2m_ec^2$ to make, and the field does work over a distance of order the Compton wavelength $\hbar/m_ec$, so the field at which a pair pays for itself is

$$E_S = \frac{m_e^2c^3}{e\hbar} = 1.3\times10^{18}\ \text{V/m}$$

or $4.4\times10^9$ tesla. The correction is of order $(E/E_S)^2$, which is what the hero figure draws, and the whole difficulty of the subject is in that number.

What the linear rule was doing for the energy accounting #

Before leaving the linear case it is worth saying exactly which conclusions of the interference accounting depend on it, because the answer is all of them.

That average is zero because the cross term is a cosine, and the cross term is a cosine because the fields add. Introduce a nonlinearity and the two waves generate a third at a frequency neither of them had, energy leaves the two original beams for good, and no amount of integrating over fringes recovers it.

So the vacuum’s nonlinearity, small as it is, is a correction to every statement about linear superposition rather than an unrelated curiosity — and a correction of $10^{-8}$ at the strongest field ever made is the reason those statements can be treated as exact.

What a laboratory has managed #

The birefringence is the effect within reach, and the experiment is beautiful and has not yet worked.

Send polarised light along a cavity through a strong transverse magnet. If the vacuum has two indices, the two polarisation components travel at different speeds, and light that went in linearly polarised comes out slightly elliptical. The ellipticity is the signal, it can be built up by bouncing the light back and forth through the magnet a million times, and it is a rotation of order $10^{-10}$ radians.

The PVLAS experiment, and BMV before it, have set limits a factor of a few above the predicted value — which is the most delicate kind of null result: near enough that everybody expects the next iteration to see it, far enough that nobody has. The scaling is what makes it hard. The birefringence goes as the square of the field, so doubling a magnet that is already at what superconductors will do buys a factor of four, and the path length can be extended only as far as a cavity’s mirrors will keep the light.

Where it has been seen is somewhere nobody built. A magnetar’s magnetic field is of order $10^{11}$ tesla, twenty-three times the critical value, so the vacuum around one is not weakly birefringent but strongly so — and X-rays leaving its surface are polarised by it on the way out. The X-ray polarimetry mission IXPE has measured the polarisation of magnetar emission, and the degree found is consistent with the vacuum birefringence being large; it is not yet a clean measurement of the effect alone, because the emission’s own polarisation is not independently known.

The sixth power #

The other effect — light scattering light — is much further out of reach, and the reason is an exponent.

Ten to the minus sixty-eight square metres is the smallest cross-section in physics. Two of the strongest laser beams available, crossed at their focus, would scatter one photon out of each beam in a time enormously longer than the age of the universe. There is no prospect of measuring photon–photon scattering with light.

The sixth power is why, and it is worth seeing where it comes from. The process has four interaction vertices, so its amplitude carries four powers of the charge and the cross-section carries $\alpha^4$ — a factor of $3\times10^{-9}$. And the amplitude must vanish as the photon energy goes to zero, because a photon of no energy cannot do anything; dimensional analysis then forces the remaining energy dependence, and it comes out as the sixth power of the ratio to the electron’s rest energy.

So the only way to raise the answer is to raise the photon energy, and the only source of photons at tens of gigaelectronvolts in any quantity is a relativistic nucleus’s own field. A lead nucleus at the LHC carries a field of $10^{25}$ volts per metre, which in its own frame is an intense pulse of hard photons, and two nuclei passing near each other without colliding therefore collide their photons.

That is the measurement. ATLAS and CMS observed light-by-light scattering in ultraperipheral lead–lead collisions and reported it in 2017, at rates consistent with quantum electrodynamics. The vacuum’s nonlinearity is confirmed, at the one energy where the sixth power lets it be.

Why the exponent had to be six #

The sixth power is worth deriving rather than quoting, because once it is derived the whole shape of the experimental programme follows from it.

Start with what cannot happen. A single photon cannot scatter off nothing, so the amplitude for light–light scattering has to vanish when either photon’s energy does. It also has to vanish if either photon’s field strength does, which is the same statement. And the cross-section is a squared amplitude, so whatever power of the energy the amplitude carries gets doubled.

Now count what the process needs. Four photons meet the pair — two in, two out — so there are four vertices and four powers of the charge, giving $\alpha^4$ in the cross-section. The only length available is the electron’s Compton wavelength, so the cross-section’s dimensions are supplied by its square. What is left is a dimensionless function of the ratio of the photon energy to the electron’s rest energy, and the vanishing requirement forces it to be a power.

Working the power out needs the calculation and the answer is six. But the structure — a fourth power of the coupling, a Compton area, and a steep power of the energy ratio — is available from the counting alone, and it says immediately that the only useful knob is the photon energy.

That is what decided where the measurement was made. Neither the intensity nor the beam quality nor the interaction length appears in the cross-section to any useful power; the energy appears to the sixth. So the experiment had to move from an optical bench to a collider, and no improvement in lasers would have changed that.

What the same nonlinearity does that is easier to see #

Two other consequences of the same vacuum polarisation are measured routinely, and mentioning them keeps the scale in proportion.

The running of the coupling. The vacuum’s polarisability screens a charge, so the effective fine-structure constant depends on the distance at which it is measured: 1/137 at long range and about 1/128 at the mass of the Z boson. That is the same virtual pairs, measured at colliders to high precision, and it is not controversial in the slightest.

Delbrück scattering. A gamma ray passing a heavy nucleus can scatter off the nucleus’s own field through a virtual pair, which is light scattering light with one of the photons supplied by a static field. It was predicted in 1933, was confirmed in the 1970s, and is a standard correction in gamma-ray work, sitting in the same tables as the absorption edges a photon meets in matter.

So the vacuum’s nonlinearity is not in doubt at all, and what has not been done is the specific measurement that shows it as a failure of superposition between two freely propagating beams. That is a narrower claim than “the effect is unobserved”, and it is the accurate one.

The scale, met from the other side #

The critical field has an equivalent statement in terms of length and intensity that is worth having because it is what an experimenter thinks in.

A field of $1.3\times10^{18}$ volts per metre corresponds to an intensity of $4.6\times10^{29}$ watts per square centimetre. The strongest focus achieved is about $10^{22}$, so the shortfall is seven decades of intensity — and because the intensity goes as the square of the field, closing four decades of field costs eight of intensity, which is worse than it first sounds.

The other route to the critical field is the one relativity supplies. A field that is $10^{-4}$ of critical in the laboratory is critical in the frame of an electron moving with a Lorentz factor of ten thousand, because an electric field transforms. So a laser pulse meeting a high-energy electron beam head-on is, from the electron’s point of view, a field far stronger than anything static — and that is how the frontier is actually being approached, with the boost doing the work no magnet or laser can.

A low-energy expansion, with one species in it #

The Euler–Heisenberg description is a low-energy expansion. It treats the vacuum as an effective nonlinear medium with a correction in powers of $(E/E_S)^2$, which is valid for fields well below critical and for photon energies well below the pair threshold. At or above either, the expansion fails and real pairs are produced — which is a different phenomenon with a different threshold, and is what a field at $E_S$ actually does.

Only electrons are counted. Heavier charged particles contribute too, with a critical field scaling as the square of their mass, so a muon’s contribution is forty thousand times weaker and the hadrons’ are weaker still. That scaling is the same statement as a heavier particle having a shorter Compton wavelength, which is the distance the field has to do its work over. At fields near critical for electrons those are genuinely negligible; the expansion’s coefficient is an electron result.

The birefringence figure assumes a uniform field over the whole path. A real magnet has ends, the field falls off, and the effective length is shorter than the physical one by a factor that has to be computed for the particular magnet. Every published limit carries that correction and it is not small.

And the magnetar evidence is indirect. What is measured is the polarisation of X-rays that have left a surface whose own emission properties are modelled rather than known. The measurement is consistent with a strongly birefringent vacuum and does not isolate it, and saying otherwise would be overstating a real and difficult observation.

A tensor effect drawn as a single number #

The hero figure draws a correction against a field as though a single number characterised the nonlinearity. It does not: the effect is a tensor, it depends on the angle between the fields and on their relative phase, and a beam crossing another at a small angle experiences something different from one crossing at a right angle. A scalar plotted against a scalar is the order of magnitude and nothing about the structure.

The birefringence figure draws a rotation and hides what is being fought. The signal is $10^{-10}$ radians and the apparatus has to distinguish it from the birefringence of the mirrors, of the vacuum windows, of the residual gas, and of the mirror coatings under the stress of their mounts — every one of which is larger, and each of which has to be separated by modulating the magnetic field and looking only at the component that follows it. None of that difficulty is in a plot of the prediction.

And the scattering figure draws a cross-section for two photons of the same energy, which is not the case that was measured. The nuclei’s photons have a spectrum, the collisions are characterised by an impact parameter rather than by a beam intensity, and the measured quantity is a rate in a detector after a calculation of the photon flux. The curve is the physics and the experiment is a different quantity computed from it.

Still open: whether the vacuum can be made to break down #

The figures stop at the critical field because the description does, and what happens at or above it is the frontier.

At the Schwinger field the vacuum does not merely become nonlinear; it becomes unstable. Pairs are produced in earnest, out of the field, at a rate with an exponential in it — $\exp(-\pi E_S/E)$ — so the production is utterly negligible a factor of ten below critical and catastrophic at it. That is the Schwinger effect, it has never been observed, and observing it would be the first direct demonstration that the vacuum has a breakdown field in the same sense a dielectric does.

Reaching it with a laser requires an intensity of order $10^{29}$ watts per square centimetre, which is seven decades above the best focus achieved. Several proposals avoid the brute-force route: colliding a laser pulse with an ultrarelativistic electron beam, so that the field in the electron’s own frame is boosted by the Lorentz factor, brings the requirement down by the boost and is the basis of experiments now being built. Whether the effect will be seen in a laboratory this decade is a genuine question rather than a rhetorical one.

The habit worth carrying away is the one this whole essay is. When a rule follows from an equation being linear, ask what the equation left out. Superposition is not a property of waves; it is a property of a particular equation, and the equation is always an approximation to something. For light in vacuum that approximation is extraordinarily good — a part in a hundred million at the strongest field anybody has made — and it is an approximation, and the scale on which it fails is a number computable from the mass of the electron.