Part 8 of 13. Rotating the phase of a wave function is a symmetry that moves nothing in space, and its conserved quantity is charge. Making the rotation local leads to gauge theory.
Every symmetry so far moved something. The Schrödinger wave function has one that moves nothing: multiply it everywhere by one unit complex number,
and nothing measurable changes. Noether's theorem turns this internal symmetry into conservation of total probability, or of electric charge when the field is charged.
A global rotation turns the arrow at every point by the same angle.
Figure 1. The packet , rotated by α. The strip paints the phase as hue, the grey curve is , the pink fill is , and the dial shows the arrow . The readout recomputes the norm, momentum and kinetic energy from the rotated values on a 601-point grid, by finite differences.
The Schrödinger equation comes from a Lagrangian density in which and count as independent fields:
Every term holds one and one , so the factors cancel. Feeding , into the field form of Noether's theorem gives a current
which here is, up to a constant,
These are the probability density and current that textbooks introduce by hand. For a field the law is local: the charge in any region changes only by what the current carries across its edges.
Figure 2. A packet with (units ) meets a barrier of height 5 and splits. Crank–Nicolson on 600 grid points; its lattice current obeys a discrete continuity equation exactly. Drag the box edges or focus one and use the arrow keys. Below, the pink curve is the charge inside the box and the blue dashes are its starting value plus everything the current carried in across its two edges.
Add to , a term that depends on the phase itself. Time translation is still a symmetry, so energy holds, but phase rotation is not. Noether's argument with then predicts the drift of :
Figure 3. A packet released off-centre in the trap , integrated by the implicit midpoint rule, which conserves the energy of a linear system exactly. Left: and the energy , each over its starting value. Right: one dot every fourth step, the measured against the Noether prediction. At λ = 0 the dots collapse to the origin and turns flat.
Let depend on position and the derivative no longer passes through the phase factor:
The extra term changes the kinetic energy. It cancels if you add a field that shifts as and use the covariant derivative . That field is the electromagnetic potential; with a relativistic electron field in place of , the theory is QED.
On a grid the derivative compares neighbours, and the gauge field lives on the links between them as a phase applied to before the subtraction. Wilson built lattice gauge theory this way in 1974.
Figure 4. Twelve sites, each an arrow for , with the link phase drawn as a tick on each link. Each bar is , the square root of that link's kinetic term, against an outline at its original height; the readout sums the squares. Click a site (or focus it and press Enter; arrow keys move between sites) to rotate its phase by 45°. Sites 4 and 8 start rotated. With the field on, each link turns by .
A phase gradient is momentum, so with no field is a real kick. With the matching field it only relabels the state.
Figure 5. Three runs of the Figure 2 solver with no barrier. Pink: the original packet (). Red: the packet times , run with no field. Green dashes: the same product, run with link phases . Below, on a log scale, each run's largest density difference from the original.
Across particle physics, internal symmetries give charges and making them local requires gauge fields.
| Symmetry group | Noether charge | Gauge boson | Theory |
|---|---|---|---|
| Weak isospin, hypercharge | W±, Z, photon | Electroweak theory | |
| Color | Eight gluons | QCD |
Electroweak symmetry breaking leaves one combination unbroken, . It is this article's phase symmetry: its charge is electric charge and its gauge boson the photon.
Noether's 1918 paper, written before quantum mechanics, also has a second theorem about local symmetries like this one. It gives identities instead of charges, and it is the next article.