The potentials that are not unique

The displacement current closes Maxwell’s equations and makes them a wave. That is where the first rung of this ladder stops, and it is also where the practice of the subject starts, because nobody solves those equations in the form they are written.

The reason is that two of the four are constraints rather than evolution equations, and they can be satisfied identically by writing the fields in terms of potentials:

$$\mathbf{B} = \nabla\times\mathbf{A}, \qquad \mathbf{E} = -\nabla\phi - \frac{\partial\mathbf{A}}{\partial t}$$

The divergence of a curl is zero, so the first of these makes $\nabla\cdot\mathbf{B}=0$ automatic; the curl of a gradient is zero, so the second makes Faraday’s law automatic. Four equations become two, and the two remaining ones are wave equations. Every calculation in the subject is done this way.

The price is that the potentials are not unique.

The freedom, and its exact size #

Add the gradient of any function $\chi$ to $\mathbf{A}$ and subtract its time derivative from $\phi$:

$$\mathbf{A} \to \mathbf{A} + \nabla\chi, \qquad \phi \to \phi - \frac{\partial\chi}{\partial t}$$

Neither field changes. $\mathbf{B}$ is unaffected because the curl of a gradient is zero. $\mathbf{E}$ is unaffected because the two changes — $-\nabla(-\partial\chi/\partial t)$ from the first term and $-\partial(\nabla\chi)/\partial t$ from the second — are equal and opposite.

That the freedom is exactly one arbitrary function, no more and no less, is worth checking rather than asserting.

Two potentials describing the same magnetic field differ by something with no curl; something with no curl is a gradient; and a gradient added to $\mathbf{A}$ can be absorbed by adjusting $\phi$. So the freedom is one function of position and time and there is nothing else.

Fixing that function is called choosing a gauge, and it is a choice rather than an approximation — nothing is lost or added, and every gauge gives the same answer for everything measurable. The choices in daily use are two.

The two gauges, and what each is for #

The Lorenz gauge imposes $\nabla\cdot\mathbf{A} + \frac{1}{c^2}\frac{\partial\phi}{\partial t} = 0$. Its virtue is that it decouples the two potentials completely: each satisfies its own wave equation with its own source, and the solutions are the retarded potentials — every point’s potential set by what the sources were doing a distance over $c$ ago. It is manifestly compatible with relativity, since the condition itself is a four-dimensional divergence, and it is what any relativistic calculation uses.

The Coulomb gauge imposes $\nabla\cdot\mathbf{A} = 0$. Its virtue is that the scalar potential then satisfies Poisson’s equation, $\nabla^2\phi = -\rho/\varepsilon_0$, whose solution is the instantaneous Coulomb potential of the charge distribution as it is now. It is what atomic and condensed-matter physics uses, because in a bound system the Coulomb interaction is the dominant term and having it appear directly is worth a great deal.

The second of those has a property that looks alarming.

The potential that is already everywhere #

Switch on a dipole. In the Coulomb gauge the scalar potential solves Poisson’s equation with the charge density as it is at this instant, so it takes its full value everywhere immediately — at a metre, at a light-year, at any distance at all.

Nothing has outrun light, and it is worth being precise about why rather than waving at it. The potential is not measurable. What is measurable is $\mathbf{E} = -\nabla\phi - \partial\mathbf{A}/\partial t$, and in the Coulomb gauge the vector potential contains a piece that is itself instantaneous and exactly cancels the instantaneous part of $-\nabla\phi$, everywhere outside the light cone, leaving zero. The two terms conspire, and the conspiracy is not a coincidence: it is enforced by the fact that they are two ways of writing the same gauge-invariant object.

So the statement “nothing travels faster than light” is a statement about the fields, and about anything else that can be measured. It is not a statement about the potentials, and a reader who has been taught it as a universal has to be shown this case or will be permanently confused by it.

The same care applies one level up. The field of a uniformly moving charge points at where the charge is now rather than where it was, and that too looks acausal and is not: the field is built from retarded information plus the charge’s velocity, and it happens to extrapolate correctly for a charge that has not changed what it was doing.

What each gauge is good at, in practice #

The choice is not aesthetic. Each gauge makes one part of a calculation trivial and another part awkward, and picking the wrong one turns a short problem into a long one.

The clearest case is a charge that stops. What propagates outward afterwards is a kink in the field lines, travelling at the speed of light and separating the region that has heard the news from the region that has not — and in the Lorenz gauge the retardation is written into the potentials themselves, so the kink is in the solution from the first line. In the Coulomb gauge the scalar potential updates instantaneously everywhere, and the retardation reappears only through a cancellation between two terms that individually travel too fast. Both give the same field. Only one of them makes visible why the field arrives when it does.

Radiation is a Lorenz-gauge subject. The retarded potentials come out directly, the fields follow by differentiation, and the whole apparatus of retarded time — including the kink a stopping charge leaves behind — is visible in the solution rather than hidden in cancellations. Anything relativistic is done here.

Bound states are a Coulomb-gauge subject. The instantaneous Coulomb interaction between electrons appears as an explicit term in the Hamiltonian, which is exactly what a calculation of atomic structure needs, and the vector potential is left describing the transverse radiation field alone. The price is that the theory is no longer manifestly relativistic, which for an atom is a price worth paying since the electrons are slow.

Waveguides and magnetostatics often want neither. A problem with translational symmetry along one axis is much easier in a gauge where the canonical momentum along that axis is conserved, which is what the Landau gauge above supplies, and the choice is made to match the symmetry of the problem rather than for any general reason.

The rule that emerges is a good general one for any redundant description: spend the freedom on whatever the problem’s symmetry suggests, and expect the answer to be the same either way. If two gauges give different physical answers, one of the calculations is wrong.

The quantity that is not gauge invariant, and matters anyway #

If gauge-dependent quantities were confined to the potentials themselves this would be bookkeeping. They are not.

The canonical momentum $\mathbf{p} = m\mathbf{v} + q\mathbf{A}$ is what Hamiltonian mechanics conserves, what appears in the Poisson brackets, and what quantum mechanics replaces with $-i\hbar\nabla$. It is gauge dependent.

That has three consequences worth stating.

A conservation law can be an artefact of a choice. In the Landau gauge the canonical momentum along one axis is constant, which is exactly why that gauge is chosen for problems with translational symmetry — and the constancy is a property of the description, not of the motion.

The wavefunction’s phase is gauge dependent. Changing gauge multiplies $\psi$ by $e^{iq\chi/\hbar}$, which changes nothing measurable because probabilities involve $|\psi|^2$. But it means the phase at a point has no meaning on its own.

And the line integral of $\mathbf{A}$ round a closed loop does not depend on the gauge, because the gradient added integrates to zero round any closed path. That integral is the enclosed magnetic flux, and it is therefore observable — which is the whole of the Aharonov–Bohm effect: an electron beam split round a region containing flux acquires a relative phase, even though the field is zero everywhere the electron goes.

That last case is the one that settles the status of the potentials. They are not measurable point by point, and a particular integral of them is. What is physical is neither the fields alone nor the potentials alone, but the gauge-invariant functionals — and the flux through a loop is one that the fields, in that geometry, cannot supply.

A test anybody can apply #

The practical residue of all this is a check to run on any calculation that has potentials in it.

If a quantity depends on the gauge, it cannot be measured. So a result that contains $\mathbf{A}$ or $\phi$ other than in a gauge-invariant combination — a curl, a difference of $\phi$ between two points at one time, a line integral round a closed loop, or the combination $-\nabla\phi - \partial\mathbf{A}/\partial t$ — is either wrong or incomplete.

This catches real mistakes. The energy density $\tfrac12\varepsilon_0 E^2$ passes; an expression involving $\phi\rho$ does not survive a gauge change unless the time dependence is handled. The momentum carried by a field passes when written with $\mathbf{E}\times\mathbf{B}$ and fails when written with $q\mathbf{A}$, which is why the angular momentum of a field is a real thing and the canonical version of it is a bookkeeping entry.

The check is easy to apply and is the fastest way to find an error in an electromagnetic calculation. It is also the reason gauge invariance is treated as a principle rather than a property: a theory in which it fails is one whose predictions depend on an arbitrary choice, and such a theory is not making predictions.

The counting, and where it leads #

Gauge freedom is often introduced as a nuisance and is better understood as a bookkeeping surplus that turns out to be structural.

The four components of $(\phi, \mathbf{A})$ describe a field with two polarisation states. Two components too many: one is removed by the gauge condition and one by the residual freedom the condition leaves. So the potentials over-describe the field by exactly the number of degrees of freedom the gauge function supplies, which is why they are convenient — an over-determined description is easier to write equations for than a constrained one.

That pattern is not confined to electromagnetism. Every fundamental interaction in the standard model is described by a gauge theory, with a redundancy of exactly this kind, and the structure of the redundancy determines the form of the interaction. Electromagnetism’s gauge function is a single real number at each point and the theory that results has one photon; the weak and strong interactions have larger groups and correspondingly more carriers, which do not commute with one another and therefore interact among themselves.

The historical order is worth noting. The gauge freedom was discovered as a nuisance in the 1860s, was named in the 1920s after a completely unrelated and wrong idea about rescaling lengths, and became the organising principle of fundamental physics in the 1970s. Very few pieces of notation have had a career like it.

A solenoid is the geometry that makes the surplus impossible to dismiss. Outside the winding the field is very nearly nothing, and the vector potential is not: it circulates round the solenoid at every radius, falling only as the inverse of the distance, because the flux threading the loop is fixed and the circulation of $\mathbf{A}$ round any loop equals it. So there is a region in which the field is zero and the potential is not, and the flux through a loop drawn in it is a real, measurable number.

Everything measurable, on the other hand, is built from gauge-invariant combinations. The energy flux in a field is $\mathbf{E} \times \mathbf{B}$ divided by $\mu_0$ — the potentials do not appear in it at all — and the same is true of the energy density, the momentum density and the force on a charge. That is the practical test for whether a calculation’s answer means anything: write it in terms of $\phi$ and $\mathbf{A}$, apply a gauge transformation, and see whether it moves. If it moves, it is not an answer about the world.

How the field is recovered from the arrows #

The number printed under each panel of the hero figure is not the field the generator was given; it is the field recovered from the arrows it drew, and the distinction is the whole reason the figure is worth having.

The curl is defined as a circulation per unit area: take a small closed loop, add up the component of $\mathbf{A}$ along it, divide by the area enclosed. Doing that on the drawn arrow field of each gauge — a square of side one-twentieth of the panel, at an arbitrary point away from the centre — returns 1.000000 tesla in every case.

That is a measurement anybody could repeat with a ruler on the printed page, at least in principle, and it is what makes the three panels an argument rather than an assertion. The symmetric gauge visibly circulates and the two Landau gauges visibly do not; if circulation were the physical content, the three would describe different fields. They do not, because the curl is a local circulation per area and a unidirectional field whose magnitude varies across the picture has one just as surely as a swirling one does.

That is also the answer to the most common objection to the vector potential — that a field pointing all one way “obviously” cannot describe something that goes round in a circle. It can, and the figure is the demonstration: what matters is the shear, not the swirl. Field lines are a choice about how to draw a vector field, and the potential’s lines are a choice about a quantity that is itself a choice.

The same three potentials, in a quantum problem #

The three panels of the hero figure are not decorative choices. All three are in daily use for the same physical situation — a charged particle in a uniform magnetic field — and comparing what each produces makes the essay’s point about what is and is not a property of the world.

Solve that problem in the Landau gauge, where the potential points along one axis and grows across the other, and the states come out as plane waves running along the first axis, each with a harmonic oscillator’s wavefunction across it, labelled by a momentum. Solve the same problem in the symmetric gauge, where the potential circulates, and the states come out as eigenstates of angular momentum about the origin, labelled by an integer and looking nothing like the first set.

Two completely different families of wavefunctions, with different quantum numbers, different shapes and different symmetries. And an identical spectrum: evenly spaced levels at $(n+\tfrac12)\hbar\omega_c$, each enormously degenerate.

What is gauge-independent is the count. Each Landau level holds exactly one state per flux quantum threading the sample — a density of $eB/h$ per unit area — and that number is the same however the states are labelled, because it is a property of the spectrum rather than of the basis. It is also the number the whole of the quantum Hall effect is built on: filling an integer number of levels is what makes the Hall conductance an exact multiple of $e^2/h$.

So a gauge choice decides which of infinitely many bases of a degenerate space one is handed, and does not decide anything anybody measures. The Landau gauge is used where the sample is a long strip, because it gives states that match the geometry; the symmetric gauge where the sample is a disc, for the same reason.

The combination a superconductor makes visible #

The essay’s rule — that what is physical is a gauge-invariant combination — has a case where the invariant object is not built from potentials alone but from a potential and something else, and where the something else is a phase that is itself meaningless.

A superconducting condensate is described by a wavefunction whose phase $\theta$ varies through the material. That phase is gauge-dependent: changing the gauge by $\chi$ shifts it by $2e\chi/\hbar$, so its value at a point means nothing. The vector potential is gauge-dependent too. What is invariant is the combination

$$\nabla\theta - \frac{2e}{\hbar}\mathbf{A},$$

and the supercurrent is proportional to it. Neither term alone is measurable and the difference is.

Now integrate that combination round a loop deep inside a thick superconducting ring, where the current is zero and so the integrand vanishes. The phase’s contribution must be a whole number of turns, because the wavefunction has to come back to itself; the potential’s contribution is the enclosed flux. Setting them equal gives

$$\Phi = n\,\frac{h}{2e},$$

which is the quantisation of magnetic flux — a measured effect, in webers, derived from the requirement that two individually meaningless quantities combine into one that is not.

That is the same structure as the Aharonov–Bohm phase and the same lesson stated in a material rather than in a beam. The gauge freedom is not removed by the superconductor; it is shared between the potential and the condensate’s phase, and the observable is what survives the sharing.

What the picture cannot show #

The three vector potentials are drawn over a bounded square and the field they describe is uniform everywhere. A real uniform field is produced by currents somewhere, and every one of these potentials would have to be joined onto something outside the picture. The gauge freedom is unaffected by that; the arrows are not, and their apparent centre — the origin, for the symmetric gauge — is a feature of the choice rather than of the physics.

The causality figure is one dimension of a three-dimensional field. It plots $\phi$ along the dipole’s axis, at one instant, and shows the Lorenz-gauge potential dropping abruptly to zero at the light cone. That abruptness is a consequence of switching the dipole on instantaneously, which nothing can do; a realistic source gives a smoothed edge.

The vector potential’s cancellation is asserted rather than drawn. The figure shows the two scalar potentials and states that in the Coulomb gauge $\partial\mathbf{A}/\partial t$ makes up the difference. That term is a solenoidal field with a source involving the whole current distribution and is not something a two-dimensional plot renders honestly.

And the gauges shown are two of infinitely many. The temporal gauge, the axial gauge and the radial gauge are all in use for particular problems, each with its own convenience and its own pathologies, and a general statement about “the gauge” usually means whichever one the speaker’s field uses by default.

The ladder from here #

Later rungs on this anchor: the Aharonov–Bohm effect worked out properly, with the interference pattern shifting by an amount fixed by a flux the electrons never touch; the potentials as a four-vector and the gauge condition as a four-divergence, which is where the whole structure becomes manifestly relativistic; gauge fixing in quantum field theory, where the redundancy has to be removed with care and the leftovers are called ghosts; the Wilson loop, which is the gauge-invariant object built from the potential round a closed path and is the natural variable of the non-Abelian theories; and the Berry phase, of which the Aharonov–Bohm phase is the earliest and simplest case.

The neighbouring ladders are the displacement current, which is where the four equations become closable, and where the energy of a field is, which is the standard example of a quantity built to be gauge invariant. The field that points where the charge is now is the other place in this collection where something looks acausal and is not.