View source for Noether's theorems/Explainers/The symmetry behind the strong and weak forces

{{DISPLAYTITLE:The symmetry behind the strong and weak forces}}
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''Mathematical counterpart: [[Noether's theorems/Applications/Yang-Mills theory|Yang-Mills theory]].''

[[File:NT-F-08.png|thumb|right|360px|alt=A shaded surface labelled the constraint surface, with three horizontal rings drawn on it labelled gauge orbits. Three marked points on one ring are annotated as the same physical state described differently. A caption says solving the theory means choosing one point per ring, which is gauge fixing.|A gauge orbit and the constraint surface]]

'''A note about this page.''' Its counterpart describes rather than works its
case, so this page does the same. What follows is the shape of the idea and what
Noether's second theorem says about it regardless of the details.

== One dial, then several ==

[[Noether's theorems/Explainers/Drifting centres and electric charge|Electric
charge]] came from a symmetry that rotates a pair of numbers into each other. One
dial, one conserved quantity.

Electromagnetism is what you get when that dial may be set differently at every
point — the local version, from
[[Noether's theorems/Explainers/Symmetry that can differ from place to place|the
local symmetry page]]. One free shape, one identity among the field equations.

In 1954 Yang and Mills asked the obvious next question. What if the thing being
rotated has three components rather than two, so that there are several
independent ways to rotate it — and, crucially, what if those rotations do not
commute?

== Why not commuting matters ==

Recall the globe from
[[Noether's theorems/Explainers/Groups of continuous motions|the Lie page]]. Turn
it about the vertical axis, then towards you; start again and do it the other way
round. Different results.

Rotating a pair of numbers has no such feature — two rotations of a dial in either
order give the same answer. Rotating three or more does.

'''Unchecked, the following.''' When the rotations do not commute, the resulting
theory changes character. The force-carrying field becomes charged under the very
symmetry it belongs to — it interacts with itself, in a way the electromagnetic
field does not. That self-interaction is why the strong force behaves so
differently from electromagnetism at short and long range.

The strong and weak nuclear forces are theories of this kind.

== What the second theorem says without any of that ==

Two things, and neither needs the details above.

'''The count carries over unchanged.''' As many free shapes as the symmetry has,
that many identities among the field equations, that many equations that follow
from the others, and that much of the description carrying no physical content.
That follows from
[[Noether's theorems/Explainers/Counting the equations a gauge theory really has|the
counting result]] and from nothing else. It does not depend on how many
components are being rotated, whether they commute, or what the fields mean.

'''A current coupled to such a symmetry has to be conserved.''' Not may be —
has to. If it were not, the field equations would have no solutions at all. That
argument was worked for electromagnetism and used only antisymmetry, so it carries
across untouched.

== The thing worth taking away ==

The second theorem was proved in 1918, for a subject that then had exactly one
example: general relativity.

Thirty-six years later, people invented a family of theories nobody in 1918 had
imagined, describing forces nobody in 1918 knew existed, and the theorem already
described their structure. Not approximately. The counting is exact, and no part
of it had to be adjusted.

'''That is not how mathematics usually relates to physics.''' Most of the
mathematics physicists use was developed for the problem in hand, or adapted
afterwards. This was proved for one case, generalised because Noether generalised
by habit, and turned out to have been about the future.

== What this page will not tell you ==

Which currents any particular force has, or how many free shapes each one carries.
None of that has been derived in this volume, and the counterpart page declines to
state it as fact for the same reason.

== The theorem as a design tool ==

There is a habit of thought in modern physics that the second theorem makes
possible, and it is worth naming because it inverts the usual order.

The old way: write down equations describing what you think happens, then examine
them and see what symmetries they turn out to have.

The way that replaced it: choose the symmetry first. Decide what may be dialled
freely at every point. Then ask what theories are consistent with that choice, and
the answer is usually a very short list — sometimes a single theory, up to a few
numbers that experiment must supply.

'''The symmetry is the input and the theory is the output.'''

This works because the second theorem makes the consequences of a symmetry choice
exact rather than approximate. Pick the symmetry and you have already fixed how
many redundancies the theory has, how many of its equations are independent, and
what must be conserved by anything it couples to. There is very little freedom
left over.

The strong and weak forces were both found this way. So, arguably, was general
relativity, which is what you get from demanding that the laws not care how you
label the points of spacetime.

None of that was Noether's purpose. She was answering a question about energy in
one theory. But the reason the method works is her theorem, and the method is now
close to being how fundamental physics is done.

''No mathematics is used on this page. The mathematical treatment is on the
counterpart page named above.''