View source for Noether's theorems/Applications/The conserved currents of the Standard Model

{{DISPLAYTITLE:The conserved currents of the Standard Model}}

''Plain-language counterpart: [[Noether's theorems/Explainers/What the Standard Model conserves|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
[[Noether's theorems/Second theorem/Fewer independent field equations than fields|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
[[Noether's theorems/Applications/Electromagnetism and charge conservation|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
[[Noether's theorems/Mathematics/Anomalies|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
[[Noether's theorems/Mathematics/Anomalies|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
[[Noether's theorems/Sources|Sources]].''