The symmetry one missing charge would complete

Written with no charges and no currents in them, Maxwell’s four equations have a symmetry that is easy to miss because there is nothing to see. Swap the electric field for $c$ times the magnetic one, and $c$ times the magnetic one for minus the electric, and every equation turns into another of the equations. Rotate part of the way and the four turn into four combinations of themselves, which are the same four.

The circle in that figure is not drawn as a circle. At each angle the fields are rotated and the two invariants recomputed from them, and what comes out lies on a circle to machine precision. So the invariants are a pair rather than two separate quantities: what the Lorentz group leaves alone individually, the duality rotation mixes, and only the combination survives both.

That has a use quite apart from monopoles. The two invariants decide whether a frame exists in which the field is purely electric or purely magnetic, and the duality rotation says something stronger — that for a field with $\mathbf{E}\cdot\mathbf{B} = 0$ and $E \ne cB$, a duality rotation can also make it purely one or the other. The two operations are different and their fixed points are the same: a light wave, for which both invariants vanish, is the field that no boost and no duality rotation can turn into anything else.

Where the asymmetry actually is #

Put the sources back and the symmetry breaks, and it breaks in exactly one place. Two of the four equations acquire a term — the divergence of $\mathbf{E}$ gets the charge density and the curl of $\mathbf{B}$ gets the current — and the other two do not. Written out with hypothetical magnetic sources included, the four are symmetric again:

$$\nabla\cdot\mathbf{E} = \rho_e/\varepsilon_0, \qquad \nabla\cdot\mathbf{B} = \mu_0\rho_m,$$

$$\nabla\times\mathbf{B} = \mu_0\mathbf{J}_e + \tfrac{1}{c^2}\partial_t\mathbf{E}, \qquad -\nabla\times\mathbf{E} = \mu_0 c^2 \mathbf{J}_m + \partial_t\mathbf{B}.$$

Nothing in that set is new physics. It is the same four laws with two source densities put where the symmetry says they belong, and if $\rho_m$ and $\mathbf{J}_m$ are zero everywhere it reduces to what is measured. The minus sign in front of the curl of $\mathbf{E}$ — the sign that makes Lenz’s law an opposition rather than an assistance — is exactly the sign the duality rotation requires, which is a small piece of evidence that the symmetry is real rather than imposed.

There is a further consequence that is worth stating because it removes an apparent problem. If every particle in the universe had the same ratio of magnetic to electric charge, a duality rotation could set all the magnetic charges to zero at once, and the world would be indistinguishable from one with none. So the observable statement is not “there are no magnetic charges” but “every particle has the same ratio”, and the ratio can then be rotated away by convention. Monopoles matter physically only if the ratio differs between particles.

The potential a monopole cannot have #

A magnetic charge’s field is the simplest field there is: radial, falling as the inverse square, the same in every direction. The trouble is not with the field.

The argument that no regular potential exists is three lines and is worth having. The flux out of a sphere surrounding the charge is not zero — that is what a magnetic charge means. The flux through the northern cap is the circulation of $\mathbf{A}$ round the equator, taken with the cap; the flux through the southern cap is the circulation round the same equator, taken the other way. If one $\mathbf{A}$ worked everywhere, those two circulations would be equal and opposite, and the total flux would be zero. It is not, so no such $\mathbf{A}$ exists.

What does exist is a pair of potentials, each regular on one hemisphere, differing on the overlap by a gradient — which is to say by a gauge transformation. That is the modern description and it is the cleanest one: a monopole is not a field with a singular potential but a field whose potential cannot be given by a single formula, and the two formulas needed are related by exactly the freedom the potential already has.

Seeing the freedom laid out is what makes the monopole’s difficulty precise. A vector potential is never unique — anything whose curl vanishes can be added to it — and the three shown here differ by large amounts while producing an identical field. The usual moral is that the potential is a calculational convenience with more in it than the physics needs.

The monopole turns that on its head. The freedom is exactly what makes a description possible at all, because the two hemispherical potentials are related by it; and the global structure of that freedom — whether a single choice can be made everywhere — is not a convenience but a physical fact, since it is what distinguishes a field with a magnetic charge in it from one without. A quantity that appeared to be pure bookkeeping turns out to carry the one piece of information the field alone cannot express.

Dirac’s original description keeps one potential and puts the singularity on a line — the string — running from the charge to infinity. The string is not a physical object: its position can be moved anywhere by a gauge transformation, so nothing about it can be measured. That last clause is a requirement rather than an observation, and meeting it is what produces the famous result.

The condition that quantises charge #

The string carries all of the monopole’s flux — it has to, since the flux has to arrive from somewhere. A charge taken round a line of flux picks up a phase equal to the enclosed flux times the charge over $\hbar$, and it does so even though the field where it travels is zero: that is the effect a solenoid has on an electron passing outside it, and it is the reason the string is not automatically invisible.

For the string to be undetectable, that phase must be a whole number of turns. So

$$q g = n h,$$

with $g$ the monopole’s strength in webers. The smallest allowed monopole therefore carries $h/e$ of flux — and this is the same quantum, up to the famous factor of two, that a superconducting ring traps, for a reason that is not a coincidence: both are conditions that a wavefunction be single-valued round a loop.

Read the other way, the condition is the striking one. Suppose one magnetic charge $g$ exists, anywhere in the universe, ever. Then every electric charge $q$ must satisfy $qg = nh$, so every electric charge must be a multiple of $h/g$. Charge quantisation — the fact that every charge ever measured is an integer multiple of one third of the electron’s, and that the proton’s and the electron’s agree to twenty-one decimal places despite being made of entirely different things — is not explained by anything else. It is put in by hand.

That is the argument’s whole force, and it is worth being precise about what it does not do. It does not predict that monopoles exist; it says what would follow if one did. It does not say how many there are; one suffices. And it does not say what the unit is: the condition relates the two charges, so a monopole’s strength is fixed by the electron’s charge rather than the reverse.

The potential’s other promotion #

The monopole is one of two places where the vector potential stops behaving like bookkeeping, and it is worth putting the other beside it because they say opposite things.

Here the potential’s excess is on display: in one common gauge, a component of it changes everywhere the instant a source is switched, faster than light, and nothing is wrong because nothing measurable does. That is the standard argument for regarding the potential as a device — it contains information that is not in the world, and different choices of it describe the same world.

The monopole says the reverse. Whether the potential can be defined by one formula over a whole sphere is not a choice; it is decided by whether a magnetic charge is inside. So the potential carries information the field does not, and that information is topological — a fact about coverings and overlaps rather than about values at points. The two observations together give the modern position: the potential’s local values are conventional and its global structure is not, and the quantities that survive both statements are the ones a charge can measure by going round a loop.

Why they would be hard to miss and are missing #

The condition also fixes how strongly a monopole would interact, and the number is alarming. The electron’s coupling is the fine structure constant, about $1/137$. The minimum monopole’s coupling comes out as $1/4\alpha$, about thirty-four — roughly forty-seven hundred times the electron’s, and larger than one, which means no perturbation expansion in it converges.

Two consequences follow. A monopole passing through matter would ionise it thousands of times more heavily than a fast proton, so a single one crossing a detector would be unmistakable. And the strong coupling means the theory of monopoles cannot be done by the methods that work for everything else — a monopole is intrinsically a strong-coupling problem, which is part of why the subject has stayed theoretical.

Searches have accordingly been sensitive and unsuccessful. The most famous positive result is a single event recorded on a superconducting loop at Stanford in February 1982 — a step in the trapped flux of exactly the size a monopole would produce, in an instrument built for the purpose — which has never been repeated in far larger later experiments and is generally not believed. The instrument was the right one: a monopole passing through a superconducting ring changes the trapped flux permanently, and the change is a quantum whatever the speed, which is the only monopole signature that does not depend on how fast it is going.

Grand unified theories predict monopoles, at masses so large that none could be made in any accelerator and so few that none may be within reach; the resulting shortage was one of the original arguments for cosmological inflation. That is a long way from the equations at the top of this essay, and the chain from one to the other is a single missing source term.

What a monopole would do to the rest of the subject #

It is worth listing what would and would not change, because the answer is unbalanced in an instructive way.

What would not change: every measurement ever made. The equations with magnetic sources reduce to the familiar four wherever the magnetic sources are absent, and they are absent everywhere anyone has looked. No result in electromagnetism would be revised.

What would change is a set of statements that are currently theorems and would become approximations. That the magnetic field has no divergence, which is why field lines have no ends and can be drawn as closed loops; that a vector potential exists globally, which every calculation in magnetostatics assumes; that magnetic flux through a closed surface is zero, which is what makes flux through an open surface depend only on its boundary and therefore makes the flux rule a rule at all. Each of those is used constantly and each is a consequence of one missing term.

And one thing would change that is not about electromagnetism at all: charge would be quantised for a reason. The present situation is that the electron and the proton have charges equal to better than one part in $10^{21}$, that quarks come in thirds, and that no principle requires any of it. A single monopole would supply the principle, and would do so retroactively — the constraint applies to charges everywhere and at all times, not only to those that meet one.

Where the model stops #

The duality figure treats fields at a point and the rotation as a global one. A duality rotation applied to a field configuration is a symmetry of the equations, but a rotation that varies from place to place is not, and nothing here says what the rotation does to the sources beyond mixing them. The symmetric equations above are the statement about sources, and they are a definition rather than a result.

The monopole is a point, and the string argument needs it to be. In a theory where a monopole is an extended object — a knot in a field, which is what unified theories produce — the singular string is replaced by structure of a definite size, and the quantisation condition survives while the singularity does not. That is worth knowing before treating the string as a physical difficulty: it is an artefact of idealising the source.

And the quantisation argument is quantum-mechanical throughout. It is a statement about a phase, so it exists only because charges have wavefunctions. In a purely classical electrodynamics a monopole of any strength is perfectly consistent and quantises nothing. The condition connecting the two charges contains Planck’s constant, and the constant is not decoration.

Dyons, and the lattice the condition really describes #

The condition as stated relates one electric charge to one magnetic one, and the general case is slightly richer and worth a paragraph because it is where the argument becomes a statement about a set of particles rather than a pair.

Suppose two objects each carry both kinds of charge — $(q_1, g_1)$ and $(q_2, g_2)$, which are called dyons. Applying the same phase argument to one moving round the other gives $q_1 g_2 - q_2 g_1 = nh$: an antisymmetric combination rather than a product. So the allowed charges form a lattice in the plane of electric and magnetic charge, with the area of the unit cell fixed by Planck’s constant and its shape not fixed at all.

That has a consequence the simple version hides. A world with a single kind of monopole quantises electric charge in units of $h/g$; a world with dyons of two different ratios quantises it more finely, and the observed unit is the lattice’s, not any single particle’s. So the argument’s prediction is that charges lie on a lattice, and the electron’s charge is a lattice vector rather than a fundamental one — which is the honest statement of what the condition delivers.

What the pictures cannot show #

The duality figure draws the invariants and not the fields, so it cannot show what a partially-rotated field looks like — a configuration that is neither electric nor magnetic but a mixture, which has no everyday name because nothing in the world is one. The figure’s claim is only about what is preserved.

The potential figure plots one component against latitude and hides the fact that the two gauges differ, on their overlap, by the gradient of an angle — the azimuthal angle itself, multiplied by the flux. That function is not single-valued: it increases by a whole turn going round. The quantisation condition is precisely the requirement that a many-valued gauge function still produces a single-valued wavefunction, and that statement is a topological one that no plot of a component against an angle can carry.

Where the ladder goes next #

The Maxwell ladder began with the term that made light, went through the potentials that are not unique and the two equations that are not laws of motion. This rung asks what symmetry the four nearly have. The rungs after it: the equations written as one statement about a two-form, in which the two source-free equations become an identity rather than a law; the conserved quantities the symmetry hands over, which for duality is a helicity rather than an energy; and the boundary-value problems where the missing source is imposed by hand, since a solenoid’s end is a monopole as far as anything outside it can tell.

The habit worth carrying away is to look at where a symmetry fails rather than at where it holds. An equation set that is symmetric except in one term is telling something about that term, and the two possibilities — that the asymmetry is fundamental, or that the missing piece exists and has not been found — are both worth taking seriously enough to compute the consequences of.