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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
    Commented Oct 4, 2017 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
    Commented Oct 4, 2017 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
    Commented Oct 4, 2017 at 18:50
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    $\begingroup$ One example could be Liouville CFT. In the Lagrangian description there is a single coupling constant $b$. Upon quantising the theory one finds a symmetry $b\to 1/b$ which was not manifestly present in the Lagrangian description. However, it's important to bare in mind that there are generally many ways to specify the same QFT, consequently symmetries that may be manifest in one description may not be manifest in another. $\endgroup$
    – Thomas Bourton
    Commented Oct 9, 2017 at 20:34
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    $\begingroup$ See my answer to this question for an example $\endgroup$
    – jpm
    Commented Mar 14, 2018 at 6:33

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Maybe not the answer you are looking for but, remember that (Wilsonian) QFTs are defined at a certain scale $\mu$.For example we can take Yang-Mills theory with various matter fields added with a certain set of coupling constants/masses $a_i$. Classically this theory can be made to have conformal symmetry by choosing the couplings in such a way that all the coupling constants are dimensionless. For concreteness let us take $SU(N)$ Yang-Mills theory with 6 scalars in the adjoint representation with a general quartic potential and 4 Dirac fermions with general Yukawa couplings. It is well known that conformal symmetry is broken by quantum effects generically. But it is also known that at a point in parameter space, $\partial_\mu a_i=0$, this theory is superconformal at the quantum level. So it can indeed happen that quantum/loop corrections conspire among each other to enhance symmetries. Another example is ABJM which appears to only have $SU(2)\times SU(2)$ flavor symmetry but actually has $SU(4)$ or even $SO(8)$ symmetry depending on the ranks of the gauge group.

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