Hamilton's principle/Explainers/Fields

Technical version: Classical field theory

This page explains how the method describes things that fill space. It uses no mathematics. The technical page it goes with is named above.

What a field is

Local conservation as a flow across a boundary.

Some things in physics are located at a point. A ball has a position.

Other things are spread out. The temperature in a room has a value at every point. So does the air pressure. So do the electric and magnetic fields, and the gravitational field.

Anything with a value at every point of space is called a field.

Fields are not exotic. A temperature map on a weather forecast is a picture of a field.

The same method, more of it

Locality: influence passing from neighbour to neighbour.

Hamilton's principle/Explainers/More than one variable showed the method handling several moving parts, and then infinitely many by treating a string as a row of beads and using more and more of them.

Fields are that idea in three dimensions. Instead of a value at each point along a line, a value at each point of space.

The recipe is unchanged. Write down a cost rule. Ask which arrangement makes the total stop changing. Out come the equations.

What changes is what you get: not a trajectory but a whole pattern, evolving in time.

One assumption doing most of the work

Modern physics adds one requirement, and it is worth stating plainly because almost everything follows from it.

The cost at a point may depend only on what is happening at that point and in its immediate vicinity.

Nothing in the rule may refer to what is happening far away.

This is the assumption of locality. It rules out any theory in which something here depends directly on something over there.

Two consequences follow at once.

Influence spreads. If nothing acts at a distance, disturbances must travel from neighbour to neighbour. That is why fields carry waves. Light is a disturbance of the electromagnetic field, passed from point to point.

Conserved quantities have to travel. For a ball, energy conservation means a number stays the same. For a field it means more: energy cannot vanish here and appear there. It has to flow across the space between.

That is why it makes sense to speak of energy travelling along a wire or arriving from the Sun. Conservation became local, and local conservation gives things routes.

Building a theory from what it must not care about

The most striking thing about this approach is how little you have to put in.

The usual procedure is:

1. Decide what the theory must be indifferent to. It should not matter where you are, which way you face, or how fast you are moving uniformly.
2. Write down the most general cost rule consistent with those indifferences.
3. Turn the handle.

Very often step two leaves almost no freedom. The requirements are so restrictive that only a couple of possibilities survive, and the equations follow.

The symmetry chose the theory. Nothing about the dynamics was assumed. This is the characteristic method of modern physics, and it is a direct consequence of treating the cost rule, rather than the equations, as the thing you write down first.

Two examples of the method working

Electromagnetism comes out of demanding that a certain quantity can be adjusted independently at every point without physical effect. Make that demand and the electromagnetic field is forced into existence to make it possible. That is at Hamilton's principle/Explainers/Electromagnetism from an action.

General relativity comes out of demanding that nothing depend on your choice of coordinates, anywhere. That is at Hamilton's principle/Explainers/Gravity from an action.

In both cases the requirement is about what the theory must ignore, and the content follows from it.

Conserved quantities again

Hamilton's principle/Explainers/Symmetry and conservation applies here without change, and produces something familiar.

Indifference to time gives conservation of energy. For a field, that means an energy density with an accompanying flow.

Indifference to position gives conservation of momentum, similarly as a density and a flow.

There is also a kind of symmetry that has nothing to do with space or time — an indifference to a quantity internal to the field itself. That gives conservation of electric charge.

Charge conservation is not an extra law. It is what a particular indifference produces, and there is no other reason for it.

Why energy is subtle in field theory

One honest complication.

The first expression for the energy of a field that the method produces is not always the right one to use. It can behave badly under changes that ought to make no difference.

It can be repaired, by adding something whose effect on the total is nothing at all. The repaired version is the one that couples to gravity.

This is not a defect being patched over. Noether's work classifies exactly this freedom, and identifies which parts of a conservation law carry information and which are bookkeeping. The subject of the repair and the subject of her second theorem are the same subject.

It becomes decisive for gravity, where the ambiguity cannot be repaired away at all. That is at Hamilton's principle/Explainers/Gravity from an action.

Where to go next

The first worked example is at Hamilton's principle/Explainers/Electromagnetism from an action.