Conformally symmetric critical systems are often, though not always, studied by modeling them by conformal field theories, which do involve an action obeying a conformal symmetry. A good reference about conformal field theories is any book or paper by John Cardy who is exceptionally clear. Here is a good one where he shows an action of a CFT at the beginning of section 3.0.1: https://arxiv.org/abs/0807.3472
For some reason the Wikipedia page on CFT’s says they are quantum which can be true but is certainly not necessarily the case, and a huge amount of the beautiful mathematics of CFTs was worked out for purely classical systems (classical statistical mechanics). The whole subject is a triumph of the 20th century and deserves to be better known.
> I’m pretty sure Noether’s theorem does apply to CFTs
I'm pretty sure, too. :)
Then again, the paper only demonstrates "rotational invariance at large scales". So if anything, any associated conserved charge would only be conserved (and defined!) approximately, i.e. in the large-scale limit.
For some reason the Wikipedia page on CFT’s says they are quantum which can be true but is certainly not necessarily the case, and a huge amount of the beautiful mathematics of CFTs was worked out for purely classical systems (classical statistical mechanics). The whole subject is a triumph of the 20th century and deserves to be better known.
I’m pretty sure Noether’s theorem does apply to CFTs and is used quite heavily to analyze them. See for example these notes: https://www.google.com/amp/s/tracingcurves.wordpress.com/201...