Quantum

Two states where the counting says three

A particle of spin one has three states, and the photon has two. The missing one is not rare or weakly coupled — there is no such state of the electromagnetic field. What removes it is that a massless particle has no rest frame, so the rotations that would turn one projection into another are not available; and the state comes back the moment the particle acquires a mass, which is what a photon does in a plasma.

14 min read 5 figures Only some values fit What stays the same

Assumes: The angular momentum that is not a rotation · The turn that has to be made twice

The argument that spin is not a rotation establishes that spin is angular momentum that is not a rotation of anything, and that what survives of the mechanical analogy is the algebra. The algebra’s most useful consequence is a count: a particle of spin $j$ has $2j+1$ states, and an analyser splits a beam of them into that many.

The count works for everything that has been weighed. It fails for the photon.

The count that works for everything with a mass. How many beams a Stern-Gerlach analyser splits a particle into, for four spins, with the photon on the bottom row. For anything with a mass the answer is 2j+1 — go to the particle's rest frame, where its spin can point in any direction, and count the projections along whichever axis the magnet defines. A spin-one particle gives three: up, down, and a middle beam that is not deflected at all. The photon has spin one and gives two. The middle state does not exist, and it is not that it is hard to produce or weakly coupled — there is no such state of the electromagnetic field. A light wave has two polarisations and the third one, in which the field would oscillate along the direction of travel, is not a solution of Maxwell's equations at all. — Fig. 1 How many beams an analyser splits a particle into, for four spins with mass, and the photon on the bottom row. Spin one gives three — up, down, and a middle beam that is not deflected. The photon has spin one and gives two.

The missing state is not faint, not short-lived and not weakly coupled. It does not exist. There is no state of the free electromagnetic field in which the field oscillates along the direction the wave is travelling.

Where the count came from, and which step fails

The derivation of $2j+1$ has a step in it that is easy not to notice.

Go to the particle’s rest frame. There it is not moving, so nothing about its motion picks out a direction, and the rotations about all three axes are symmetries of its situation. The states then have to fall into a representation of the rotation group; the representations are labelled by $j$ ; and the one labelled $j$ has $2j+1$ members, which are the projections of the spin along whatever axis is chosen.

Every part of that is sound, and the first sentence is a step. A massless particle has no rest frame. It travels at $c$ in every frame, there is no boost that stops it, and the argument has nowhere to begin.

What survives is smaller. The transformations that leave the particle’s momentum alone are not the full rotation group; they are the rotations about the direction of motion, plus two odd transformations that act like translations in a plane. That group — the little group for a massless particle — has one-dimensional representations labelled by a single number, the helicity, which is the spin projection along the motion.

So a massless particle of spin $j$ has one state, of helicity $+j$ . A theory that also respects reflection symmetry contains its mirror image, of helicity $-j$ , and the two together make two. That is the photon’s count, arrived at without any of the algebra that makes a spin-half turn twice, and it is arrived at by a completely different route from the massive one.

The same argument covers the graviton, which has helicities $\pm 2$ and two states rather than five — which is why a gravitational wave has two polarisations and not five — and the neutrino, which would have one state if it were massless and turns out to have a mass and therefore two.

The classical version of the same statement

None of that requires quantum mechanics, and the classical version is shorter.

A plane electromagnetic wave in empty space has $\mathbf{E} \propto \mathbf{E}_0 e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ , and Gauss’s law in empty space says the divergence of $\mathbf{E}$ is zero. For that wave the divergence is $i\mathbf{k}\cdot\mathbf{E}$ , so

\mathbf{k}\cdot\mathbf{E}_0 = 0.

The field is perpendicular to the direction of travel, exactly, with no component along it. Two perpendicular directions remain, and they are the two polarisations.

That is the whole of it, and it is worth noticing which equation did the work: the one that counts what comes out of a closed surface, applied where there is nothing inside to come out. A longitudinal wave would require a charge density to support it, and in vacuum there is none.

Which immediately says where a longitudinal wave can exist: anywhere there is charge to move.

Three branches, where light has picked up a mass. The modes of an electromagnetic disturbance in a plasma of 1e+18 electrons a cubic centimetre at 10 electronvolts, with frequency in units of the plasma frequency. There are three, not two. Two of them are transverse and degenerate — the light wave, which cannot propagate below the plasma frequency and rises at the speed of light above it. The third is longitudinal: the electrons oscillating along the direction of travel, which is exactly the polarisation a light wave in vacuum does not have. All three meet at the plasma frequency at zero wavenumber, which is the signature of a massive vector particle rather than a massless one, and that is what light in a plasma is: a photon with an effective rest energy of 37.13 millielectronvolts. The longitudinal branch rises far more slowly than the transverse ones because its restoring force comes from the electrons' own thermal motion rather than from the field's tension. — Fig. 2 The modes of an electromagnetic disturbance in a plasma. There are three. Two are transverse and degenerate — the light wave, which cannot propagate below the plasma frequency. The third is longitudinal: electrons oscillating along the direction of travel, which is the polarisation vacuum does not have. All three meet at the plasma frequency at zero wavenumber, which is the signature of a massive vector particle.

Light in a plasma is the massive case, drawn

The figure above is worth dwelling on because it is not an analogy. It is the thing itself.

A photon in a plasma cannot propagate below the plasma frequency: the free charges respond and turn the wave back. Its dispersion relation is $\omega^2 = \omega_p^2 + c^2k^2$ , which is exactly the relation $E^2 = m^2c^4 + p^2c^2$ for a particle of rest energy $\hbar\omega_p$ . The photon in a plasma has a mass, it is a small one — a few millielectronvolts in a laboratory plasma — and it is a mass in every sense that can be measured.

And it has three states. The two transverse modes are the light wave’s two polarisations; the third is the plasma oscillation, the longitudinal mode in which the electrons surge back and forth along the wavevector. At long wavelength all three sit at the plasma frequency, which is precisely the degeneracy a massive spin-one particle has at rest.

So the passage from two states to three is not a discontinuity that happens at exactly zero mass. It is what happens when something gives the field a restoring force it did not have. In a plasma that something is the free charges; in the electroweak theory it is a field filling all of space, and the mechanism by which the massive vector bosons acquire their third state is the same mechanism by that name.

Three branches, where light has picked up a mass. The modes of an electromagnetic disturbance in a plasma of 1e+14 electrons a cubic centimetre at 2 electronvolts, with frequency in units of the plasma frequency. There are three, not two. Two of them are transverse and degenerate — the light wave, which cannot propagate below the plasma frequency and rises at the speed of light above it. The third is longitudinal: the electrons oscillating along the direction of travel, which is exactly the polarisation a light wave in vacuum does not have. All three meet at the plasma frequency at zero wavenumber, which is the signature of a massive vector particle rather than a massless one, and that is what light in a plasma is: a photon with an effective rest energy of 0.37 millielectronvolts. The longitudinal branch rises far more slowly than the transverse ones because its restoring force comes from the electrons' own thermal motion rather than from the field's tension. — Fig. 3 A much thinner and cooler plasma. The structure is identical and the scales are not: a thousandth of the density puts the plasma frequency down by a factor of thirty, and the longitudinal branch is flatter because the thermal speed that stiffens it is lower.

The other place the missing state shows up

Helicity is spin along the direction of motion, and for a massless particle it is the only spin label there is. That has a consequence for how a photon’s angular momentum behaves that is worth separating from the counting.

A massive particle’s spin can be tipped: boost to a frame where it moves the other way and a spin that pointed along the motion now points against it, so helicity is frame-dependent for anything with a mass. For a massless particle nothing can overtake it, so no boost reverses the direction of motion, and helicity is the same in every frame. It is an invariant label rather than a projection that depends on the observer.

What follows is that a photon’s two states are not two orientations of one thing. They are right- and left-handed circular polarisation, they are distinguished by a quantity every observer agrees about, and the only operation that exchanges them is a reflection. A linearly polarised wave is an equal superposition of the two, which is why a linear polarisation has no definite angular momentum and a circular one carries exactly $\pm\hbar$ per photon.

That is measurable and was measured in 1936, by hanging a small waveplate on a fine fibre and passing circularly polarised light through it: converting the light’s handedness delivers a torque, the fibre twists, and the twist is $2\hbar$ per photon converted. It is one of the few measurements in which a quantity that sounds purely formal is read off a mechanical instrument.

And it connects to the classification. The duality the four Maxwell equations nearly have is precisely a rotation between the two helicity states, and the conserved quantity it hands over is the difference in the number of right- and left-handed photons — a quantity with no analogue for a massive vector field, because there the two helicities are not separately invariant.

Why nothing observable turns on an exact zero

The plasma case suggests a question about the vacuum: how would anybody know the photon is massless rather than very slightly massive?

The third state fades and does not go away. The ratio of a massive vector particle's rest energy to its total energy, for a mass of 91.19 giga-electronvolts, on logarithmic axes. That ratio is how much of the particle's own rest frame is available: it is the reciprocal of the Lorentz factor, and everything that distinguishes a massive spin-one particle from a massless one scales with some power of it. At 92 giga-electronvolts it is 0.99 and at 5000 it is 0.0182, a Lorentz factor of 55. So the difference between two and three states is not a discontinuity at zero mass. A very light massive particle has three states, the third is hard to tell from the other two in any process at energies far above its mass, and the massless case is the limit rather than an exception to it. That is why a bound on the photon's mass is a bound and not a proof, and why every such bound is quoted with the length scale over which the experiment looked. — Fig. 4 The ratio of a massive vector particle’s rest energy to its total energy, for a mass of 91 giga-electronvolts. That ratio is how much of the particle’s own rest frame is available, and everything distinguishing a massive spin-one particle from a massless one scales with some power of it. At five tera-electronvolts it is 0.018.

The answer is that nothing does, sharply. The third state exists for any nonzero mass, and its effects fade as the rest energy over the total energy — so at energies far above the mass, a massive vector particle behaves in every measurable way like a massless one, with corrections that can be made as small as one likes by going to higher energy.

Turned round: a photon mass would show itself only at low energy and long distance. The signatures are that Coulomb’s law would acquire an exponential cutoff at the corresponding length, that a static magnetic field would not extend indefinitely, and that light of different frequencies would travel at slightly different speeds.

So the experimental bounds are all about long distances. The best come from the structure of planetary magnetic fields and from the behaviour of the solar wind, and they bound the photon’s mass below roughly $10^{-54}$ kilograms — which corresponds to a Compton wavelength larger than the Earth–Sun distance by a wide margin. That is an extraordinarily small number and it is a bound. Nothing in physics measures an exact zero.

The third state fades and does not go away. The ratio of a massive vector particle's rest energy to its total energy, for a mass of 80.4 giga-electronvolts, on logarithmic axes. That ratio is how much of the particle's own rest frame is available: it is the reciprocal of the Lorentz factor, and everything that distinguishes a massive spin-one particle from a massless one scales with some power of it. At 81 giga-electronvolts it is 0.99 and at 20000 it is 0.0040, a Lorentz factor of 249. So the difference between two and three states is not a discontinuity at zero mass. A very light massive particle has three states, the third is hard to tell from the other two in any process at energies far above its mass, and the massless case is the limit rather than an exception to it. That is why a bound on the photon's mass is a bound and not a proof, and why every such bound is quoted with the length scale over which the experiment looked. — Fig. 5 The same construction for the lighter of the two weak vector bosons over a wider span of energy. There is nothing special about the numbers: what the curve says is that the distinction between two states and three is a question of what energy the question is asked at.

What the missing state costs, and what it buys

The absence of the third state is not a deficiency; it is tied to the one property of electromagnetism that makes it calculable.

A theory of a massless spin-one field written with a four-component potential has four components and describes two states. The gap is closed by a redundancy: two different potentials that differ by the gradient of any function whatever give the same field, so the potential is not unique and the extra components are choices rather than facts. That redundancy is gauge invariance, and it is exactly what removes two of the four.

Two things come with it that would otherwise be separate assumptions.

Charge conservation stops being an extra law. A field with that redundancy can only be coupled to a current that is conserved, because otherwise the redundancy is not a redundancy and the choice of potential would affect the answer. So the conservation of charge is forced by the same structure that removes the longitudinal state.

And the force is long-ranged. A massive vector field’s force falls off exponentially over its Compton wavelength, and a massless one’s falls off as an inverse square, which is why electromagnetism reaches across the universe and the weak interaction reaches across a nucleus. The range and the count of states are the same fact seen twice.

So the trade is legible. Two states, a redundancy in the description, a conserved source and an unlimited range come together; three states, no redundancy, no requirement on the source and a finite range come together. There is no theory with two states and a finite range, and none with three states and an unlimited one.

Free particles, a non-relativistic plasma, and a structural analogy

The little-group argument is a statement about free particles. It classifies the states a particle can be in when nothing is acting on it, which is what asymptotic states in a scattering experiment are. Inside an interaction — between the moment a particle is made and the moment it is detected — the classification does not apply, and a photon exchanged between two charges is legitimately described with four components rather than two, of which two do not correspond to states at all.

The plasma case is not relativistic. The dispersion relations drawn come from a cold or warm plasma treated non-relativistically, they describe a plasma whose electrons are much slower than light, and the longitudinal branch’s stiffening term is the first correction in the thermal speed. In a relativistic plasma the branches are different and the degeneracy at zero wavenumber survives.

And the mechanism is not what makes the weak bosons heavy in the sense of being the same physics. What is shared is the structure: a field that would otherwise be massless acquires a rest energy and with it a third state, and the third state is supplied by a degree of freedom that was elsewhere before. In the plasma it was the electrons’ own motion; in the electroweak case it is a component of the scalar field. Saying the two are the same mechanism is a statement about the mathematics and not about the substance.

How the count is checked on things that do have mass

The 2j+1 rule is quoted so often that it is worth saying where it has actually been counted, since a rule confirmed only on the cases it was invented for is not much of a rule.

Spin one half was counted in 1922 on silver atoms: two beams, where a classical magnetic moment would have given a continuous smear and an orbital angular momentum would have given an odd number. Two is the count for a half-integer spin and it is the first evidence that spins can be half-integer at all — the experiment that cannot be done on a free electron at all.

Spin one was counted on nuclei by molecular-beam resonance in the 1930s and 1940s — deuterium has spin one, and its resonance splits into three components in a field, at the spacings the count requires.

Spin three halves and above appear in the hyperfine structure of heavy atoms, where the number of components into which a line splits gives the nuclear spin directly. Caesium-133 has spin seven halves and eight components, sodium-23 has three halves and four, and those counts are how the nuclear spins were first determined.

And the photon’s two were established by a different kind of measurement entirely, because there is no analyser that splits a beam of light into spin components. What was measured was the angular momentum carried — the waveplate torsion experiment above — together with the plain fact that no arrangement of polarisers has ever produced a third independent polarisation. A beam of light is described by two complex amplitudes and every optical measurement ever made is consistent with that and with nothing larger.

The asymmetry in that list is itself informative. The massive counts are read off the number of lines in a spectrum; the photon’s is read off the dimension of the space needed to describe a beam. The two are the same statement about the same rule, arrived at by methods with nothing in common.

What a helicity state is, which no count of branches draws

They cannot show the states. Every figure here counts branches, plots ratios and draws dispersion curves; what a helicity state actually is — a field configuration rotating about the direction of travel in one sense rather than the other — is not a line on a graph. The nearest thing to a picture of it is circularly polarised light, and it is a picture of the classical field rather than of a state.

Nor can they show why the count is one and not two for a massless particle before reflection symmetry is imposed. A theory of massless spin-one particles that does not respect reflection would contain only one helicity, and no such theory of electromagnetism is needed because electromagnetism does respect it. The weak interaction does not, and the consequence is that its massless limit would not be symmetric between the two helicities — which is a statement the figures have no way to draw.

And they cannot show that the third state is not “hidden”. There is a persistent way of speaking in which the longitudinal photon is present and unobservable, and it comes from the four-component description used in calculations. In that description two of the four components cancel against each other in every physical result, and what is left is two. They are not hidden states; they are bookkeeping that does not correspond to anything.

Still open: how small a photon mass can be tested for

The published bounds span several orders of magnitude and depend strongly on what is assumed. The tightest rely on the structure of large-scale magnetic fields — in the solar wind, in the galaxy, in clusters — because a photon mass would make a magnetic field fall off exponentially beyond its Compton wavelength, and the presence of ordered fields over a length puts a bound at that length.

The difficulty is that the same measurements can be read as bounds on quite different things. A field observed over a given distance constrains a photon mass, and it equally constrains several other modifications of electromagnetism at long range, and separating them requires assumptions about what else might be present. The bounds quoted by different authors for the same data differ by orders of magnitude for exactly that reason, and the honest summary is that the photon’s mass is smaller than anything that could matter for any known process by an enormous margin, with the precise number attached to a model.

The habit worth carrying away is about where a count comes from. A degeneracy is a consequence of a symmetry, so counting states means finding which symmetries the situation actually has. The count $2j+1$ is not a property of spin; it is a property of a particle that can be brought to rest. Remove the rest frame and the count changes, without anything about the spin changing at all.