Illustrated Physics
physics, drawn
Concept

Gauge freedom — where it appears

The latitude to change the potentials without changing any field, which leaves every measurable quantity untouched. What survives it — a line integral round a closed loop, for instance — is what an experiment can be sensitive to.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Three vector potentials for one magnetic field. Three different vector potentials, drawn as arrow fields over the same square, every one of which describes the same uniform magnetic field of 1 tesla out of the page. The symmetric gauge circulates about the origin; the two Landau gauges are unidirectional and point in perpendicular directions, and neither has any circulation about anything a reader can see. Underneath each is the field recovered from its own arrows, by adding up the potential round a small square and dividing by the area enclosed — which is what the curl is. The three numbers 1.000000, 1.000000, 1.000000 agree to the last digit computed. The picture is the argument: a vector potential has no physical direction, no physical magnitude and no physical circulation of its own, because a change of gauge alters all three and alters nothing that can be measured. What survives is the curl, and the curl is the field.

The potentials that are not unique

Nobody solves Maxwell's equations for the fields. They are solved for potentials instead, and the potentials are not unique — three completely different vector potentials describe the same uniform magnetic field, and one of them changes everywhere the instant a charge moves, at any distance, without anything having outrun light.

electromagnetism · Maxwell equations The rotation that mixes electricity into magnetism. The two Lorentz invariants of a field — E² − c²B² across and 2E·cB up — as the field is rotated by the duality transformation that takes E into cB and cB into −E. Every configuration moves on a circle, so the combination of the two invariants is preserved while neither is. A light wave sits at the origin and stays there, which is why a wave cannot be turned into anything else by this rotation; a static charge starts on the positive axis and is carried round to a pure magnetic field a quarter turn later. The source-free equations are unchanged by the whole family, so a universe with no charges in it has no way to say which field is which.

The symmetry one missing charge would complete

Maxwell's equations with no sources are unchanged by rotating the electric field into the magnetic one. With sources they are not, and the only thing missing is magnetic charge — which, if one existed anywhere, would force every electric charge in the universe to be a multiple of a fixed unit.

electromagnetism · Maxwell equations

Named alongside it

The objects these essays reach for when they reach for this one.

Maxwell equationsVector potentialCanonical momentumCausalityCoulomb gaugeField transformationInvarianceLorenz gaugeMagnetic fluxMagnetic monopoleQuantisationScalar potential

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