The conserved currents of the Standard Model

Plain-language counterpart: What the Standard Model conserves.

This page describes. Nothing on it is derived, and the specific content is tagged below.

What can be said from the theorems alone

Two structural statements follow from the count and require no knowledge of the theory's content.

A theory with local symmetries has as many identities among its field equations as it has free shapes, and that many equations that follow from the others. And a current coupled to a local symmetry must be conserved, or the field equations have no solutions — the argument of the electromagnetic page, which used only antisymmetry and does not depend on the group.

Those hold for the Standard Model as for any theory of its kind.

What requires the theory, and is not derived

UNVERIFIED, all of the following. The Standard Model's local symmetries are usually given as three factors, with one, three and eight free shapes respectively, and the conserved currents associated with them are the electric, weak and colour currents. Beyond those there are global symmetries whose conserved quantities — baryon number, the three lepton numbers — are conserved in the classical theory and are subject to the qualifications on the anomaly page.

None of that is derived here, and the counts above are reported as they are usually stated rather than as anything this volume has checked.

Why the page is thin, and why it exists anyway

Working these cases needs the field content, the representation theory, and the covariant derivative — a track this volume does not have.

What the page can do is mark that the structure carries all the way to current physics without modification, and that the two structural statements above hold for every theory in the family, including ones not yet written down.

The interesting fact about the Standard Model, from this volume's point of view, is not which currents it has. It is that nothing new was needed to say what shape they must take.

The place where the classical statement fails

One qualification is not optional. A symmetry of the classical theory need not survive quantisation, and where it does not, a classically conserved quantity is not conserved. See Anomalies, which also records why an anomaly in a local symmetry is fatal rather than merely interesting.

Provenance: no source was reached and nothing is derived. The two structural statements follow from results proved elsewhere in this volume; everything specific to the Standard Model is tagged UNVERIFIED. See Sources.