Field

A Value at Every Point

A field is a quantity that has a value at every point of a region. A weather map of temperature is one: every place has a temperature, the countryside between the cities as well as the cities. A map of the wind is another, with a speed and a direction at every point. The field round a magnet is of the second kind. At every point near it there is a direction, the way a compass needle there would turn, and a strength, how hard it would turn.

Iron filings make it visible. Sprinkle them on a sheet of glass over a magnet and tap the glass. Each filing turns to lie along the field where it is, and the filings link end to end into strands that run from one end of the magnet round to the other.

The strands trace something real that fills the space round the magnet: the magnetic field. The filings show the field only where it is strong enough to turn them against the grip of the glass. A compass needle, balanced on a jewel, turns where they cannot. The field is there whether or not anything is there to feel it: between the filings, above and below the sheet, through the glass.

The News Takes Time

A map of values could still be bookkeeping, a way to work out pushes that really leap from one body to another. The first test is to change something and see when the change arrives. Take two electric charges and jerk one of them sideways. If each pushed the other directly, the second would feel the change at the same instant. It does not. The change spreads outward through the field at the speed of light, 299,792,458 meters a second. Until it arrives, the second charge goes on being pushed by the old field, as if nothing had happened.

Close to the moved charge, the field already runs out from its new place. Farther off, the news has not yet arrived, and the field still runs out from the old one. Between them a ripple races outward. The ripple is not a side effect. It is light, in the broad sense of a wave in the electromagnetic field. From a charge jerked by hand it would be a radio wave; only a charge shaken hundreds of trillions of times a second makes visible light.

The equations for electric and magnetic fields allow such waves, and give them the measured speed of light. Made with sparks, the waves reflect, bend and line up as light does. Radio, light and X-rays are the same kind of ripple in the same field, differing only in how often they repeat.

The Field Holds What It Carries

A push that leapt straight across, only late, could still explain the delay. What it cannot explain is this. While the news is on its way, something has left one charge and not yet reached the other. Energy has, and so has momentum, the quantity that collisions never lose. Neither charge holds them. The field does.

This is where the old picture of pushes between bodies breaks. Two charges moving past each other at right angles push and pull each other with magnetic forces that are not equal and opposite. Newton’s third law fails between them, and the momentum of the two charges alone is not kept. Count the momentum held in the field round them, and the books balance exactly, as they do in every collision.

Light, which is field and nothing else, carries momentum too, and pushes on what it hits. The push of a lamp’s beam has been measured on tiny mirrors hung on fine fibers in glass cases. In space near Earth, sunlight on a mirror one meter square, facing the Sun, pushes with about nine millionths of a newton. A space probe on its way to Venus unfolded a sail 14 meters on a side, and sunlight pushed it along with about a thousandth of a newton.

The toy light mill, black and silver vanes in a glass bulb, is not this push. It spins the wrong way for it, driven by the thin gas left in the bulb. The real push is far too weak to turn it.

So a field is not a calculating device. It holds energy and momentum, carries them from place to place at a set speed, and pushes what it reaches. It is as real as the bodies it moves.

Whole Lumps

Look closely enough at light and it turns out to come in lumps. Shine a laser through a pinhole and the light spreads into rings on a screen. Now dim it a billion times, and watch the screen with a camera sensitive enough to see the least light there is. The rings do not simply fade. The light lands as separate dots, each in one place, each carrying the same energy. Dimmer light brings fewer dots, never fainter ones.

Added up, the dots rebuild the rings. They land often where the wave is strong and never where it cancels: the light travels as a wave, and each lump of it is caught whole, in one spot.

Dots alone could come from the camera, whose atoms take up energy in steps. The proof that the light itself comes in lumps is a half-silvered mirror, which lets half the light through and reflects the rest, with a detector on each side. Send it light from a source that gives out one lump at a time, such as a single atom. Each lump sets off one detector or the other, never both, which no spread-out wave could do. Each lump is a photon.

The glow of a hot, dark body, the same whatever it is made of, can be explained only if light’s energy at each frequency comes in steps of a fixed size. The size of a step is the light’s frequency times a fixed number, Planck’s constant: 6.62607015 × 10⁻³⁴ joule-seconds, exact by definition. Blue light comes in bigger lumps than red. That is why, from a metal such as sodium, blue light knocks out electrons and ordinary red light does not, however bright: one red lump is too small, and only an intense pulsed laser can land two on one electron at once.

This is what makes a field a quantum field. At any one frequency, it can take or give energy only in whole steps of that size. Measure how much it holds above its quietest state and you always find a whole number of steps. One step is one photon, two are two, never one and a half. Before the measurement the number need not be settled: a laser’s beam is a blend of different numbers, and each measurement finds one.

Every Particle Is a Lump

Matter works much the same way. An electron is a lump of the electron field, as a photon is a lump of the electromagnetic field. Fired one at a time, electrons too land as single dots that build up the stripes only a wave can make. One difference shapes everything made of matter: no two lumps of the electron field can share the same state. So electrons never pile up into a beam like a laser’s, and the electrons of an atom cannot all crowd into its lowest state, so they stack outward in layers, which is why matter takes up room.

That answers a question that would otherwise have no answer: why every electron in universe is exactly like every other, with the same mass and the same charge. They are not copies made one by one. Each is the same smallest step of one field, which fills all of space.

A field can also make lumps and unmake them. A gamma ray with enough energy, passing close to an atom’s nucleus, can turn into an electron and a positron, its antimatter twin. These are two lumps of the electron field, one of each charge, so the charge still adds up to zero. When an electron meets a positron, both lumps are unmade, and their energy leaves as light. A particle is not a little permanent thing. It is a way a field can be stirred.

The Standard Model, the theory that holds every particle seen so far, is a list of these fields and of how they act on one another. Its usual chart has seventeen kinds of particle, each the lump of a field of its own. Counted more finely, the fields number nearer sixty. Gravity is the one force left out.

Never Quite Still

Take every lump away and the fields are still there, each in its lowest state. That state is the vacuum, empty space. Even then a field cannot be perfectly flat. The uncertainty principle forbids it a sharp value and a sharp rate of change at once. Measure it again and again, averaged over a small patch, and the answers scatter about its resting value: zero for most fields, and for the Higgs field a set value that it holds even in empty space. The smaller the patch, the wider the scatter. The state itself does not change in time. What scatters is the measurements.

The spread has effects that are measured. Two energy levels of hydrogen, which the simpler theory makes equal, in fact differ by about a millionth of their energy. Almost all of the difference comes from the electron’s tie to the electromagnetic field, whose spread jiggles it, in a rough picture that works here, so that it feels the nucleus a little less.

The electron’s magnetism is a tenth of a percent stronger than the simplest theory gives, for a reason of the same kind: the electron keeps acting on itself through its own field. The jiggle picture that worked for hydrogen, tried on this one, gets even its sign wrong. Today the magnetism is measured to 0.13 parts in a trillion, and prediction and measurement agree to about one part in a trillion. At that level the test is held back by how well the strength of the electric force is known without leaning on this same theory. The two best such measurements, which time how cesium and rubidium atoms recoil as they absorb light, disagree with each other.

The muon, a heavier cousin of the electron, is more sensitive to fields too heavy to make directly, so its magnetism was watched for more than twenty years as a possible crack. The measurement sat above the prediction. The final measurement of that extra, precise to 127 parts per billion, now agrees with a new prediction built on supercomputer calculations of the strong force. One puzzle is left inside the prediction. Decades of older collider data give a smaller strong-force part than the supercomputers and the newest collider data do.

Popular accounts describe empty space as full of virtual particles popping in and out of existence. That is a picture of a calculation, not a report of events. In the theory’s sums, the effect of one field on another is written as terms that look like particles passed back and forth. None can be caught on its own, and nothing pops anywhere. What is real, and measured, is the effects.

The picture has a known edge. Gravity has a field too, the shape of space and time. It can be treated as a quantum field only at energies far below the point where its effects on single particles grow strong, and no theory that works all the way down has been confirmed: string theory and loop quantum gravity are the leading proposals, and no experiment has confirmed either. And the lowest energies of all the fields, added up, should give empty space an energy of its own. Every estimate gives far too much, by a factor of about 10⁵⁵ in the most careful one, and the theory leaves the true amount to be set by hand.