Quantum fluctuations and symmetries?

Question

While reading this piece about symmetry breaking, in section 3 I came across the term "anomalous symmetry breaking", which happens when a symmetry is broken by quantum fluctuations:

Let us turn to another kind of symmetry breaking, which has so far not received much philosophical discussion, namely anomalies. Anomalies label instances, where the symmetry of the classical theory turn out not to remain the symmetry of the corresponding quantum theory. While “anomaly” may sound very serious, maybe even something that can give rise to scientific revolutions, the name should rather be understood as the consequence of the bafflement physicists found themselves in when they realized that quantum fluctuations can break classical symmetries. A more suitable name may be “quantum mechanical symmetry breaking”.

I have a questions about this: To what extent can quantum fluctuations break symmetries? Can all symmetries (like the Poincaré, Lorentz, diffeomorphism, CPT and translation invariances) be broken by quantum fluctuations in certain contexts?

Answer 1

Quantum fluctuations, normally summarized by the triangle chiral fermion loop diagram in perturbation theory in your QFT text, violate symmetries by current divergence pieces of $O(\hbar)$, $$ \partial_\mu J^\mu= O(\hbar). $$ So transformations effected by the charge Q arising out of the current $J^\mu$ are not real symmetries if this right hand side fails to vanish. They are only symmetries for the classical theory, (when $\hbar \to 0$). Such violations are provably one-loop exact, by a theorem of Adler and Bardeen.

Your text details when they are "bad" (invalidate important gauge symmetries), or "good/safe" (invalidating global or non-existent symmetries), such as in neutral pion decay, or the renormalization group violating the scale anomaly through the RG β-function ("trace anomaly").

All specific symmetries you are asking about are not anomalous, i.e. preserved by quantum fluctuations, so they are good cornerstone quantum symmetries of our full quantum world. (CPT is not a Lie symmetry, so there is no point in discussing them in the same breath as the rest.)

Answer 2

In the modern physics, there are several anomaly.

1. Gauge anomaly : this terminology is used to describe the situation that classical theory has some gauge symmetry but it becomes anomalous in quantum theory. If gauge symmetry has an anomaly, a theory does not hold an unitality or renormalizability, so Gauge anomaly must be cancelled. (See anomaly cancellation.)

2. Global anomaly by a dynamical gauge field: this type of anomaly is known as ABJ anomaly. It describes the situation that classical theory has some global symmetry $G$ but it becomes anomalous in quantum theory by some dynamical gauge field. Here, we note that this dynamical gauge field is not necessarily the gauge field for the classical symmetry $G$ that we are now considering. This is main deference compared to the ‘t Hooft anomaly that we consider at the third pert. Also, this anomaly does not make a serious problem like a breaking of unitality. In this sense, it simply tells that classical symmetry is not necessarily a symmetry of quantum theory.

3. Global anomaly by a background gauge field: this type of anomaly is called as the ‘t Hooft anomaly. It describes the situation that classical theory has some global symmetry $G$ but it becomes anomalous in quantum theory by a background gauge field about $G$. This anomaly is RG invariant. Thus it is used to check a consistency of the theory. (See Anonaly matching)

There are other anomalies that should be separately mentioned, such as Witten anomaly or conformal anomaly(trace anomaly), but broadly speaking, these three anomalies are the ones you should know at least to understand modern field theories. I guess your text basically only refer to the first two types of anomaly.

We here note that we can consider the ‘t Hooft anomaly not only for continuous symmetry but discrete symmetry. (e.g. this article)

In this context, mixed ‘t Hooft anomaly about C, P and T symmetry is discussed. (e.g. this article) It is a little bit different context from your original text, but we should know about this type of anomaly so I mentioned.

About Poincaré symmetry, this forum already has one post for it, so please look that post.

Anyway, my main point is that if we take anomaly in a broad sense, discrete symmetry can also have anomaly. Of course, CPT symmetry, etc., can also be anomalous in principle.