Could we list examples of symmetries that preserve only the combination of the measure $\mathcal{D}\phi$ together with $e^{-S}$ but not each on their own? (That is, symmetries which have no classical analogue).
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$\begingroup$ OK - I just saw this: physics.stackexchange.com/questions/388153/… which gives the wonderful example of the Green-Schwarz anomaly cancellation. $\endgroup$– SvenForkbeardCommented Oct 30, 2019 at 9:25
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$\begingroup$ Nevertheless, can we compile a bigger list than done in that question? $\endgroup$– SvenForkbeardCommented Oct 30, 2019 at 9:26
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$\begingroup$ I suppose that discrete symmetries could provide simple examples. $\endgroup$– SvenForkbeardCommented Oct 30, 2019 at 9:47
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2 Answers
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Liouville theory possesses conformal invariance on the quantum level, but its classical action isn't conformally invariant (because it contains dimensionful constants such as $\lambda'$).
answered Oct 30, 2019 at 7:34
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$\begingroup$ Thanks, this is an interesting example. It's not clear to me how one should interpret this from a path integral point of view (although this is not too problematic for me). Can we give other examples then, in addition to this, where there is a very clear path integral expression where the transformation of the measure cancels precisely with the change in the action? $\endgroup$ Commented Oct 30, 2019 at 9:22
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$\begingroup$ @SvenForkbeard the interpretation from the path integral point of view is quite interesting, and very similar to the toy model with the axion. You start with the Polyakov path integral and observe that the measure isn't invariant under conformal transformations leading to the conformal anomaly. This can be re-interpreted as a non-invariance of the auxiliary term in the action (the Liouville action). This paper seems to do a good job at explaining this: iopscience.iop.org/article/10.1209/0295-5075/7/8/009 $\endgroup$ Commented Oct 30, 2019 at 9:33
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I would like to submit a partial answer to my question.
In general, any examples where it is only the combination of the measure and action which are invariant would fall under the umbrella of `anomaly cancellation'. That is, any anomaly cancellation mechanism would give rise to such a phenomenon as I describe, and conversely, any such case can be thought of as arising from applying anomaly cancellation to an otherwise anomalous theory.
The Green-Schwarz `anomaly cancellation' mechanism (in 4d, 10d or whatever you like) is a perfect example.
The Liouville theory is another excellent example, where as the comment above describes, it can be thought of as a Polyakov action (in eg 0 dimensions) where there is a conformal anomaly. Adding the dilaton terms to make the full Lioville action would then be another example of anomaly cancellation.
Notes:
1) My definition of anomaly cancellation is `adding dynamical fields to an anomalous action in order to make it anomaly-free'.
2) All I am doing is giving the well-known name to the phenomenon I was thinking about, but at the time I did realise that these were the same concepts. So googling `anomaly cancellation' will bring up countless other examples presumably.
Hope everything I say is correct, otherwise please correct!