# Can a theory gain symmetries through quantum corrections?

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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.)

This post imported from StackExchange Physics at 2019-01-11 12:59 (UTC), posted by SE-user user2062542
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.

This post imported from StackExchange Physics at 2019-01-11 12:59 (UTC), posted by SE-user AccidentalFourierTransform
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.

This post imported from StackExchange Physics at 2019-01-11 12:59 (UTC), posted by SE-user Fabian
@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 $k$).

This post imported from StackExchange Physics at 2019-01-11 12:59 (UTC), posted by SE-user Solenodon Paradoxus
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.

This post imported from StackExchange Physics at 2019-01-11 12:59 (UTC), posted by SE-user Cosmas Zachos
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.

This post imported from StackExchange Physics at 2019-01-11 12:59 (UTC), posted by SE-user Tom Bourton
en.m.wikipedia.org/wiki/Accidental_symmetry ... also many forms of enhanced or emergent symmetry

This post imported from StackExchange Physics at 2019-01-11 12:59 (UTC), posted by SE-user Mitchell Porter
See my answer to this question for an example

This post imported from StackExchange Physics at 2019-01-11 12:59 (UTC), posted by SE-user jpm

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