Noether's theorems/Applications/Yang-Mills theory

Plain-language counterpart: The symmetry behind the strong and weak forces.

A gauge orbit and the constraint surface

This page describes. It does not work its case. Setting up the field content would take a track of its own, and this volume does not.

What is being generalised

Electromagnetism has one free shape and one identity. Yang and Mills asked in 1954 what happens with several, when the corresponding transformations do not commute with one another.

UNVERIFIED, all of the following. The answer is a family of theories with several gauge functions rather than one, whose field strengths acquire a term quadratic in the potentials, and whose currents are themselves charged. The strong and weak nuclear forces are theories of this kind. The count from the counting page applies unchanged: as many identities as free shapes, and as many field equations that follow from the rest.

Why this belongs in a volume on Noether

Two things can be said with confidence, and they are the reason the page exists.

The structure is the second theorem's and nothing else. Whatever the details, a theory with several free shapes has that many identities among its field equations, that many equations that are not independent, and that much description carrying no physical content. None of that depends on the field content, on the number of dimensions, or on which group is chosen. It follows from the second theorem given only the count.

The theorem preceded the physics by thirty-six years. The second theorem was proved in 1918 for a case — general relativity — that was then the only known example. It described the structure of theories nobody had thought of, and when they were thought of, the description already fitted.

That is worth stating plainly because it is unusual. Most mathematics used in physics is developed for the problem at hand. This was not.

What this page will not claim

It will not give the currents, the identities in explicit form, or the count for any particular force, because none of that has been derived here. The Standard Model page carries the same restriction and says so as well.

Provenance: no source was reached for this page and nothing on it is derived. The substantive physics is tagged UNVERIFIED above. The observation about the thirty-six-year gap rests on the dates, which are established elsewhere in this volume. See Sources.