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It is well known that not all symmetries are preserved when quantising a theory, as evinced by renormalising composite operators or by other means, which show that quantum corrections may alter a conservation law, such as with the chiral anomaly, or 'parity' anomaly of gauge fields coupled to fermions in odd dimensions.

However is the reverse possible: can a theory after quantisation gain a symmetry? Or if not, can it gain a 'partial symmetry'?

(For example invariance under $x\to x+a$ for any $a$ is translation symmetry, and invariance under $x\to x+2\pi$ would be said to be a partial symmetry. My question concerns whether a theory can gain a full symmetry, or a partial one at least through being quantised.)

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    $\begingroup$ Nice question. In principle, it is technically possible, but the variation of the action should compensate the variation of the measure -- something certainly non-trivial. I'm not sure how it could work while keeping the theory local. It will be interesting to see what others have to say. $\endgroup$ – AccidentalFourierTransform Oct 4 '17 at 15:55
  • $\begingroup$ There is a thing that has been studied in the past which is called "order-by-(quantum)disorder" that seems to be exactly what you are looking for. As far as I remember, it is discussed in the book "quantum field theory in condensed matter theory" by Tsvelik. $\endgroup$ – Fabian Oct 4 '17 at 16:07
  • $\begingroup$ @AccidentalFourierTransform maybe Chern-Simons theory is an ok example (I realize that is not 100% what OP is looking for, but still, tecnhically, it is not classicaly gauge-invariant, but is quantum-mechanically gauge invariant for integer levels $k$). $\endgroup$ – Prof. Legolasov Oct 4 '17 at 18:50
  • $\begingroup$ Well, the renormalization group certainly enhances or suppresses a symmetry in the UV or IR, and lots of model-building (Nielsen-Froggat) is predicated on it. Since the RG is predicated on quantization, this might serve as an example. For instance, supersymmetry is enhanced/achieved in the IR. $\endgroup$ – Cosmas Zachos Oct 4 '17 at 19:29
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    $\begingroup$ See my answer to this question for an example $\endgroup$ – jpm Mar 14 '18 at 6:33

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