What the Standard Model conserves

Mathematical counterpart: The conserved currents of the Standard Model.

A note about this page. Its counterpart describes and derives nothing, so this one does the same. The two structural statements below are secure. Everything specific about the Standard Model is reported as it is usually stated, not as anything checked here.

What holds no matter what the theory contains

Two results carry over from earlier in this volume and need nothing about particles at all.

The counting. A theory with symmetries you can dial differently at every point has exactly as many redundancies as it has such dials. That many identities among its equations, that many equations following from the rest, that much description that means nothing. From the counting page, and independent of what is being described.

The forcing. Anything coupled to such a symmetry must be conserved, or the equations have no solutions. Not permitted — required. This came out of a one-line argument about antisymmetry and carries across untouched.

Both apply to the Standard Model exactly as they apply to electromagnetism, and neither needed to be re-derived when the Standard Model was written down.

What is usually said, and is not checked here

Unchecked, all of the following. The Standard Model is built on three local symmetries of different sizes, whose conserved quantities are the electric charge, the weak charges and the colour charges. Alongside those it has global symmetries whose conserved quantities are the various particle-counting numbers — how many protons and neutrons, how many electrons, and so on.

None of that is derived in this volume, and the counterpart page declines to state it as established for the same reason.

The point this volume can actually make

It is not which currents the theory has. It is that nothing new was needed to know what shape they would take.

When physicists build a candidate theory of some force, they choose a symmetry first. The moment that choice is made, the count of conserved currents is fixed, the count of redundant equations is fixed, and the requirement that the coupled matter conserve those currents is fixed. All of it follows from a theorem proved before any of the relevant physics existed.

The theorem is not used to check the answer afterwards. It is used to constrain the question beforehand.

One qualification that is not optional

Everything above is classical.

A symmetry of the classical theory need not survive the move to quantum mechanics, and where it does not, a classically conserved quantity is simply not conserved. The particle-counting numbers are the standard example of quantities affected by this.

Worse, if a local symmetry fails to survive, the theory does not merely lose a conservation law — it stops making sense, because the redundancy in its description was never supposed to be physical and now leaks into the predictions.

Checking that this does not happen is one of the first things done to any candidate theory, and several features of the known particles are the way they are because nothing else would pass the check. See When a symmetry does not survive.

Exact, approximate, and merely apparent

The conserved quantities of the Standard Model are not all conserved in the same sense, and the differences matter.

Tied to a local symmetry. Electric charge and the colour charges. These are the most secure things in the theory. As the counting page showed, a local symmetry does not merely permit the conservation of what couples to it — it requires it, on pain of the equations having no solutions at all. A violation would not be a surprising measurement; it would be an inconsistent theory.

Tied to a global symmetry. The particle-counting numbers. These come from the first theorem in its ordinary form. They are conserved because the scoring rule happens to have a symmetry, and nothing forces the scoring rule to have it. A more complete theory could easily not.

Apparent. Some quantities look conserved because every process that would violate them is too slow or too weak to have been seen. That is a statement about experiments, not about symmetry, and it is the category most likely to be revised.

The three look identical in a table of conserved quantities and are entirely different claims. Knowing which is which is one of the more useful things the two theorems provide, and it is invisible unless you ask where each came from.

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