13 - Anomalies

Summary

A characteristic feature of relativistic quantum field theories is that symmetries of the classical theory are not always present after quantization. We do not mean here the spontaneous breaking that is characterized by a non-invariant vacuum and by the presence of the Goldstone bosons. Rather we mean a situation where there is no conserved current for the symmetry despite the absence of any terms in the action that appear to break the symmetry. Such breaking of a symmetry is called anomalous.

If the classical action is invariant, then a naive application of Noether's theorem gives us a conserved current. That is, there is no anomalous symmetry breaking. What prevents the argument from being correct is the presence of UV divergences. The current is a composite operator, i.e., a product of elementary fields at the same point, and to define it, some kind of regularization and renormalization is needed. The renormalization may invalidate the equations used to prove Noether's theorem.

For simplicity, we will consider only global symmetries, as opposed to local, or gauge, symmetries. The simplest cases of global symmetries were considered in Chapter 9. These could be treated by using an ultra-violet regulator that preserved the symmetry. The proof of Noether's theorem can then be made in the cut-off theory.