Hamilton's principle/Explainers/Electromagnetism from an action

Technical version: The electromagnetic Lagrangian

This page explains how electromagnetism can be derived rather than discovered. It uses no mathematics. The technical page it goes with is named above.

A theory that was found twice

Counting physical polarisations from gauge redundancy.

Electromagnetism was worked out over a century, from experiments. Coulomb measured forces between charges. Ørsted found that currents deflect compasses. Faraday found that moving magnets make currents. Maxwell assembled it.

That is how it happened, and it took a hundred years.

There is a second route. Start from a single requirement about what a theory must ignore, and electromagnetism comes out. Nothing about electricity or magnetism has to be put in.

This page follows the second route. It is not how the subject was discovered, and it is how the subject is now taught to people who will build the next theory.

The requirement

Maxwell's equations split into dynamics and identities.

Start with a description of matter that involves a quantity with no direct physical meaning — something like the setting of a dial, where only differences in the setting matter, not the setting itself.

If only differences matter, you should be able to turn all the dials by the same amount everywhere and change nothing. That is an ordinary symmetry, and by Symmetry and conservation it gives a conserved quantity. That quantity is electric charge.

Now make the harder demand. You should be able to turn each dial by a different amount, independently at every point in space, and still change nothing.

Why that demand cannot be met alone

It fails immediately, and the reason is easy to see.

Physics cares about how things change from place to place. If you turn the dial by different amounts at neighbouring points, you have manufactured a change between them. The theory notices.

So the demand is impossible as it stands.

Unless something else changes to compensate.

The field appears

Suppose there is another quantity, spread through space, whose job is to cancel exactly that manufactured difference. When you turn the dials by varying amounts, this quantity shifts in a matching way, and the two effects cancel.

Then the demand can be met.

That compensating quantity is the electromagnetic potential. The electric and magnetic fields are built from it.

The logic is worth stating slowly, because it is easy to skate past. You did not put in electricity. You did not put in magnetism. You demanded that a certain adjustment have no physical effect, discovered that the demand was impossible, and found that it becomes possible if a new field exists.

The field is forced into existence by the requirement.

Its own behaviour

The new field needs its own rules. Those are fixed the same way: write the simplest cost rule that respects the same requirement.

There is very little freedom. Essentially one possibility survives, and turning the handle produces Maxwell's equations.

Or rather, half of them. And the split is interesting.

Two halves of Maxwell, two kinds of statement

Maxwell's equations come in two groups, and the standard presentation treats them as four equations of equal standing. They are not.

One group is dynamics. It says how the field responds to charges and currents. It comes out of the cost rule, and it could have been otherwise.

The other group is bookkeeping. It says there are no magnetic monopoles, and it gives the law of induction. These follow automatically from the fact that the field was built out of the compensating quantity. They are true by construction, for any conceivable field, before any physics is put in.

That asymmetry looks arbitrary when the four equations are simply listed. Seen this way it is not arbitrary at all. Two of them are laws and two of them are consequences of how the description is built.

Charge conservation is not optional

One more result comes free, and it is a strong one.

If you couple the field to a flow of charge, the whole construction only works if that charge is conserved. A theory in which charge could appear or vanish fails the requirement outright. It is not a theory at all.

So charge conservation is not an experimental fact that the theory happens to respect. It is a precondition for the theory existing.

The price of the description

The compensating quantity is not unique. Several different versions of it give exactly the same electric and magnetic fields, and hence the same physics.

That is the redundancy described at When symmetry is local. The description carries spare parts.

Two consequences, both predicted by Noether's second theorem.

Some of the equations are not independent — they follow from the others as a matter of arithmetic.

And the tidy reformulation of the Hamiltonian does not apply cleanly, because it needs a description without spare parts.

The redundancy also explains a number. Counting the description gives four quantities. Removing the redundancy and the dependent equation leaves two — and light has exactly two independent polarisations, which is why sunglasses work.

That count came out of an argument about what a theory must ignore.

Why this route matters

Electromagnetism did not need deriving. It was already known.

The point is that the method generalises. Make the same kind of demand about a different quantity and you get a different force. The strong and weak nuclear forces were constructed this way in the twentieth century, and neither could have been found by the patient experimental route that worked for electricity.

Electromagnetism is the worked example that showed the method is trustworthy.

Where to go next

The same construction, applied to the choice of coordinates rather than a dial, gives gravity. It is at Gravity from an action.