Three vector potentials for one magnetic field

At its defaults it draws 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.

gauge-freedom is one function in lib/figures/fields.js — charge, current, flux and the lines drawn between them. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

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.

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 field left over when one vector potential for a uniform magnetic field is subtracted from another — here the symmetric gauge less the Landau gauge — drawn as arrows over the contours of the function it is the gradient of, χ = ½Bxy. The arrows cross every contour at right angles and point uphill, which is what makes them a gradient rather than merely a field. The circulation of that field round loops of four different sizes, measured on the arrows themselves, is 0.0e+0, 0.0e+0, 0.0e+0, 0.0e+0 — zero to the precision the arithmetic has, at every size. That is the whole theorem: two potentials describing the same magnetic field can differ only by something with no curl, something with no curl is a gradient, and a gradient added to A is absorbed by subtracting its time derivative from φ. The freedom is exactly one arbitrary function of position and time, no more and no less, which is why fixing it is called choosing a gauge rather than making an approximation.

The scalar potential along the axis of a point dipole switched on at t = 0, drawn at the instant light has travelled 1.4 units, in two gauges. In the Lorenz gauge the potential is the same expression evaluated at the retarded time, so it is zero beyond the light cone: nothing has arrived there yet, and the picture says so. In the Coulomb gauge the scalar potential solves Poisson's equation with the charge density as it is *now*, so it takes its full value everywhere the instant the dipole exists, at any distance. Both are correct, both are in daily use, and they give identical electric and magnetic fields — because in the Coulomb gauge the vector potential carries a term that cancels the instantaneous part of −∇φ exactly, everywhere outside the cone, leaving nothing. What the pair shows is that a potential is not a thing that can be watched propagating. Only the fields are, and the fields are zero outside the cone in both.

A charge on a circular orbit in a uniform magnetic field, integrated once for 1.25 turns, with three quantities read off the same trajectory. The x component of the velocity is a cosine and belongs to the motion, so it is the same whatever potential was used to describe the field — the integrated speed holds constant to 2.1e-11, as it must, since a magnetic force does no work. The other two curves are the x component of the canonical momentum, mv + qA, in the symmetric gauge and in the Landau gauge. They are different functions of time: the first swings over 1.000 and the second over 0.000, and in the Landau gauge it is constant, which is the whole reason that gauge is chosen for problems with translational symmetry. The canonical momentum is what Hamiltonian mechanics conserves and what quantum mechanics turns into −iħ∇; it is also gauge dependent, which means a conserved quantity can be an artefact of a choice. What is never an artefact is the difference of the canonical momentum round a closed loop, which is the flux enclosed.

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.

physicscheck asserts something about gauge-freedom that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.