The turn that has to be made twice

Rotating something through a full circle puts it back. That is close to what “a full circle” means, and it is true of a chair, a planet, a molecule and a magnetic field. It is not true of the quantum state of an electron, a neutron, a proton or a silver atom, and the failure is not small or approximate: after one full turn the state is exactly minus what it was, and it takes a second full turn to return.

The claim needs care, because it is easy to state in a way that is false. Nothing about the neutron is different after one turn. It is not tired, not marked, not distinguishable from a neutron that never turned. What has changed is a sign in front of the state, and the whole of this essay is about the difference between those two statements — a difference that took forty-five years to close experimentally, and that closed with an interferometer rather than an argument.

Where the half comes from #

A rotation through angle $\theta$ about an axis $\hat n$ acts on a quantum state through the angular momentum operator that generates it: $U = \exp(-i\theta\, \hat n \cdot \mathbf{J}/\hbar)$. This is not a special rule for spin. It is the statement that angular momentum is the generator of rotations, which is the same statement as the conservation law a symmetry hands over read in the other direction: the operator that is conserved when the world is rotationally symmetric is the operator that performs rotations.

For a spin-½, $\mathbf{J} = \hbar\boldsymbol{\sigma}/2$, and that factor of a half is the whole subject. The exponent becomes $-i\theta\,\hat n\cdot\boldsymbol{\sigma}/2$, and because $(\hat n\cdot\boldsymbol{\sigma})^2 = 1$ the series sums in closed form to $\cos(\theta/2) - i\sin(\theta/2)\,\hat n\cdot\boldsymbol{\sigma}$. Every $\theta$ in the expression appears halved. At $\theta = 2\pi$ the cosine is $-1$ and the sine is $0$, so the operator is minus the identity: the state has been multiplied by $-1$ and nothing else has happened to it. At $\theta = 4\pi$ it is the identity.

The figure does not evaluate that formula. It integrates $d\psi/d\theta = -(i/2)\,\sigma_x\psi$ forward in small steps from the state pointing along z, checks that the norm survives, and reads the overlap and the three components of the spin’s direction off the trajectory it got. The closed form is what the trajectory is compared against at the two turning points, and never what is plotted. A figure that evaluates an expression and then compares it with itself has no way to be wrong.

The half is the spin, not the algebra. For a state of angular momentum $j$ the same construction gives a factor $\exp(-2\pi i j)$ after a full turn, which is $+1$ for integer $j$ and $-1$ for half-integer $j$. Photons, phonons and the orbital motion of an electron in an atom come back after 360°. Electrons, neutrons, protons and every other spin-½ object do not. The split is exactly the one that decides which particles crowd into a state and which refuse, and that is not a coincidence; it is the end of this essay.

The sign that nothing can see #

The reason the sign went unmeasured for so long is that it is not a measurable property of the spin carrying it.

Everything a measurement on a single spin can report comes from the density matrix $\rho = |\psi\rangle\langle\psi|$, built from the state and its conjugate. Multiply $|\psi\rangle$ by $-1$ and $\rho$ is unchanged, because the two minus signs meet. An analyser oriented along any direction $\hat m$ finds spin-up with probability $(1 + \hat m\cdot\langle\boldsymbol{\sigma}\rangle)/2$, so if the three components of $\langle\boldsymbol{\sigma}\rangle$ are unchanged, every experiment on that spin is unchanged.

The figure states the two facts side by side because they are usually stated apart, and apart they sound contradictory. The state has moved to the far side of the space it lives in. Nothing measurable has moved at all. Both are true, and the resolution is that the map from states to predictions is not injective: two states differing by a phase are one physical situation, and the whole space of physical situations for a spin-½ is the Bloch sphere rather than the sphere of unit vectors in $\mathbb{C}^2$.

This is not a nuisance to be worked around. It is a general feature, and it is why the potentials that are not unique is a subject at all: a description with more in it than the physics has is what makes the physics easy to write down, and the surplus has to be identified rather than removed.

A phase that multiplies everything is invisible; a phase that multiplies part of something is not. That sentence is the experiment.

One arm turned, and the other not #

A superposition is a sum of amplitudes, and a sum notices a sign on one of its terms.

Split a beam into two paths, rotate the spin in one of them by an angle $\alpha$, recombine, and count. What arrives is the sum of an amplitude that has been rotated and one that has not, and the interference term is the overlap between the spin state that went one way and the spin state that went the other. For a rotation through $\alpha$ that overlap is $\cos(\alpha/2)$ plus a term proportional to the component of the spin along the rotation axis — and for a beam with no net polarisation the second term averages to nothing, leaving $\cos(\alpha/2)$ exactly.

The dashed curve matters more than it looks. The two hypotheses under test — that a rotation acts through $\theta$ or through $\theta/2$ — do not differ by a small correction that a careful experiment might tease out. They differ by a factor of two in the period, which puts a maximum of one prediction on top of the minimum of the other. An experiment that can see fringes at all can decide between them.

The beam does not have to be polarised, and it is worth pausing on why. The interference term is the amplitude for the spin to come out of the field in the state it went in with, and for an unpolarised beam that amplitude is $\cos(\alpha/2)$ whatever the state happened to be. The 1975 experiments used unpolarised neutrons and did not have to prepare a spin direction at all — the sign is a property of the rotation, not of the state being rotated.

The experiment, in tesla #

The rotation is done with a magnetic field. A magnetic moment $\boldsymbol{\mu}$ in a field $\mathbf{B}$ has energy $-\boldsymbol{\mu}\cdot\mathbf{B}$, that energy generates precession about the field, and the precession is a rotation of the spin state at the Larmor rate $\gamma B$. So an angle becomes a field multiplied by a time, and the time is a path length divided by a speed. Those three numbers are all measurable, which turns the question into a number of tesla.

Those are laboratory numbers rather than heroic ones — a few tens of gauss over a couple of centimetres, and a neutron wavelength a monochromator supplies routinely. What made the experiment hard was not the field but the interferometer: a two-path device for neutrons has to hold a path difference stable to a fraction of a wavelength while the two paths are centimetres apart, which is why it had to wait for a perfect silicon crystal cut so that the two beams travel inside one piece of material and share its thermal expansion.

Two groups did it in 1975, independently, with the same kind of crystal and different fields. Both found the long period. The measured value came within about one per cent of 720°, and hundreds of degrees away from 360°, which is the comparison that matters: the experiment is not a precision test of a number but a decision between two possibilities that are nowhere near each other.

The same behaviour at other numbers #

A period measured once is a period; a period whose limits are known is an instrument. The precession angle is proportional to the time in the field and therefore to the wavelength, so a beam with a spread of wavelengths carries a spread of angles and the fringes fade as the angle grows.

The shape of the decay is the reason the measurement is possible and also the reason it does not generalise. One extra turn costs a beam almost nothing, because the accumulated spread of angles is still small compared with a radian. Twelve extra turns cost everything. So the 4π period is directly measurable and a count of turns is not: this apparatus cannot be used to ask whether the state comes back after forty turns, because by then there is no interference left to read. That limit is set by the monochromator rather than by the physics, and it is the ordinary situation — an effect visible in one window of a parameter and invisible everywhere else, as in how far a wave can remember, where the same competition between an accumulating phase and a spread of frequencies sets what can be seen at all.

The shape of the space underneath #

The factor of a half can be presented as an algebraic accident of the Pauli matrices. It is not one, and the geometry says why.

The rotations of three-dimensional space form a group, and it is not simply connected: there are loops in it that cannot be shrunk to a point. The clearest way to see this is to draw a rotation as an axis and an angle, with the angle running from zero to 180°, because turning by more than half a circle one way is turning by less than half a circle the other. That makes the group a solid ball of radius 180° whose opposite surface points are the same rotation — and a path that goes out to the surface and reappears at the opposite point is a closed loop that cannot be contracted, because the two ends cannot be brought together without leaving the ball.

The states live in the space that repairs this. Every loop in the rotations lifts to a path in the states, and a loop that cannot be contracted lifts to a path that ends where it did not start — at the antipode, which is the same physical situation with the opposite sign. Two such loops in a row lift to a closed path. That is the entire content of the minus sign, stated without any matrices: the rotations are covered twice, and the states know which of the two sheets they are on even though nothing measurable does.

This is the same kind of statement as the phase that is only a shape, where a phase is fixed by the geometry of a path rather than by what happened along it, and as the phase a magnet leaves on a path it never touched, where the topology of a region rather than the field in it decides an interference pattern. In all three the observable quantity is a phase difference between two branches, and in all three the thing that fixes it is a property of a space rather than a force.

What the belt does and does not settle #

The demonstration everyone remembers is the belt. Hold one end of a belt, rotate the other through 360°, and the twist cannot be removed by moving the free end about without turning it further. Rotate through another 360° and the double twist can be undone completely, by passing the belt around the held end.

It is a good demonstration of the topology and a bad explanation of the physics, and the distinction is worth being firm about. What the belt shows is that a full turn of an object connected to its surroundings is a loop in the rotation group that cannot be contracted, and two full turns is one that can. That is a fact about the rotation group, and the rotation group is the same whether the thing turning is a belt, a coffee cup or a neutron. It therefore cannot explain why a neutron picks up a sign and a photon does not, because both live in the same group.

What decides the sign is which representation of that group the system’s states carry, and that is a separate fact about the system. A spin-½ carries the two-dimensional representation, which exists only on the double cover; a spin-1 carries a representation that descends to the rotations themselves. The belt makes the double cover memorable. It predicts nothing.

Why this is the same fact as the exclusion principle #

The sign has one consequence that is not about rotations at all.

Take two identical particles and exchange them, by moving one halfway round the other and the other halfway round the first. Watched carefully, that exchange can be deformed into a rotation of the pair through 180° together with a rotation of each particle about its own axis — and following the deformation through, the amplitude for the exchange carries exactly the factor a full turn of one particle would give. Half-integer spin therefore comes with an antisymmetric two-particle state, and integer spin with a symmetric one.

That is the spin–statistics connection in outline, and the outline is all that is available without relativistic field theory, where it becomes a theorem. What matters here is that the sign measured in an interferometer and the rule that keeps no two in the same state are the same sign. The pressure that holds up a white dwarf, the volume of ordinary matter, the fact that the periodic table has rows: all of it descends from a factor of $-1$ that no measurement on a single particle can detect. Four states, and one of them is odd is where that antisymmetry becomes a spectrum, and the force with no force in it is where it becomes something that pushes.

Where the model stops #

The rotation here is a rotation of the spin alone. A real magnetic field acts on the whole neutron, and the identification of Larmor precession with the action of the rotation group on the spin state follows from the Hamiltonian rather than being independent of it. The experiment tests the periodicity of the interference pattern; calling the horizontal axis an angle imports a piece of theory that is well tested elsewhere.

Gravity and the interferometer are not separable in practice. A neutron interferometer is sensitive enough that turning it in the Earth’s field changes the fringe phase measurably, which is a famous experiment in its own right. Any measurement of the spin period has to hold that contribution fixed, and it is one of the reasons the apparatus is a single crystal.

The treatment is non-relativistic throughout. The half-angle survives in the relativistic theory, where the rotation group is a subgroup of a larger one and the spinor representation is where the Dirac equation lives, but the accounting of what a boost does to a spin is a separate subject with its own surprise in it, recorded in the turn that two pushes leave behind.

A phase is only ever relative. Every statement here about “the state changing sign” is shorthand for a comparison with something that did not change. The interferometer supplies that something. Without a second branch there is nothing to compare against, and the question of what sign the state has stops meaning anything at all.

What the pictures cannot show #

Nothing drawn here is a picture of a spin. The Bloch components are three numbers extracted from a state, and the arc in the last figure is a slice through a three-sphere chosen because the rotation stays inside it. A reader who takes the circle as a picture of where the neutron is will be misled in a way the figure cannot correct.

The interference curves are also idealised in one honest respect: they show a contrast that is constant with angle except for the wavelength spread, and a real interferometer loses contrast for a dozen other reasons — vibration, temperature gradients across the crystal, imperfect beam overlap. Those set the achievable contrast at any angle and would flatten the curves everywhere rather than change their period, which is why the period is the quantity the experiment reports.

Where the ladder goes next #

The spin ladder began with the angular momentum that is not a rotation, which is about the two values a measurement can return and the impossibility of building them out of anything spinning. It continued to four states, and one of them is odd, where two spins combine into three states that behave one way under exchange and one that behaves the other. This rung asks what a rotation does to the state rather than to the direction, and finds a sign that no instrument aimed at one spin can see and an interferometer can.

The rung after it is the one where the sign stops being about rotations: exchange two identical particles in a plane rather than in space, and the loop that takes one around the other is no longer contractible either, so the phase acquired need not be $\pm 1$ at all. That is a different topology and a different subject, and the habit that carries into it is the one this rung is built on — ask what space the loop lives in before asking what the phase can be.