When a symmetry does not survive

Mathematical counterpart: Anomalies: when a classical symmetry does not survive quantisation.

A note about this page. As with the previous one, its counterpart derives nothing and marks its statements as unchecked. The statement below can be given plainly and is worth having. The mechanism cannot, and this page does not attempt it.

The statement

Unchecked. A classical theory has a symmetry. You build the quantum version of that theory. The symmetry is gone.

Not hidden, not approximate. Gone. And not because anyone was careless: for certain symmetries the failure is unavoidable, because the steps needed to make the quantum theory well defined are themselves incompatible with the symmetry. One or the other has to give, and it is the symmetry.

An event of this kind is called an anomaly. The best-known case makes a quantity that is conserved in the classical theory stop being conserved in the quantum one — and the effect was measured in the laboratory before it was properly understood.

The mechanism, which this page does not give

Why the steps are incompatible is a question about how infinities are handled in quantum field theory, and there is no honest short version.

This page could produce a paragraph. It would leave a reader believing they had been told something. That is worse than the gap.

The misreading this page exists to prevent

Here is the part that can be said plainly, and it is a point of logic rather than of physics.

An anomaly is not a failure of Noether's theorem.

Both of her theorems are conditional. Each says: if the scoring rule survives a certain symmetry, then certain things follow. Neither says that any particular scoring rule does survive any particular symmetry. That was never claimed and is not the theorem's business.

When a symmetry does not survive quantisation, the quantum theory simply fails the "if". A conditional whose condition is not met has not been contradicted. It has not been invoked.

The classical theorem remains exactly as true as it ever was, about the classical theory it was always about. What changed is that the quantum theory is a different theory, and it does not have the symmetry.

The correct sentence is: the symmetry did not survive. Not: the theorem broke.

Why some anomalies end a theory

One consequence is worth carrying, because it turns this from a curiosity into a working constraint.

If a symmetry of the one-choice-for-everywhere kind is anomalous, a classically conserved quantity stops being conserved. That is a prediction. It can be checked, and in the famous case it was.

If a symmetry of the choice-at-every-point kind is anomalous, the theory is broken. The slack in its description was never supposed to be physical, and the quantum theory needs it to stay that way. An anomaly means it does not, and the theory starts predicting nonsense.

So candidate theories are checked for this before anything else, and the requirement that such anomalies cancel is a severe restriction on what a theory may contain. Several features of the known particles are there because nothing else would cancel.

That is the second theorem's structure, a hundred years on, working as a design rule. It is as far downstream of 1918 as this volume follows.

Two ways a symmetry can stop being visible

Worth separating, because they are constantly confused and only one is on this page.

The symmetry is there and the situation is not. A pencil balanced on its point is symmetric about the vertical. Let it fall and it points one way. The laws did not change; the particular outcome simply had to choose. Physicists call this spontaneous breaking, and the symmetry is fully intact — it is now relating different possible outcomes to each other rather than being visible in any one of them.

The symmetry is not there at all. That is the anomaly above. Nothing chose anything. The quantum theory simply does not have the symmetry, and no rearrangement of the situation recovers it.

The two get run together because both are described as symmetry breaking, and both end with a conservation law you cannot use. They are entirely different events.

The test is what happens to the conserved quantity. Under spontaneous breaking it is still conserved — the pencil's fall conserves angular momentum perfectly well. Under an anomaly it is not conserved, and the failure is measurable.

One is a symmetric law with an unsymmetric outcome. The other is a law that lost its symmetry on the way from classical to quantum.

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