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It is folklore that quantum gravity cannot have any exact global symmetry (see Global symmetries in quantum gravity). This follows for example from thought experiments involving black holes (no-hair). Yet electrically charged "hair" is allowed. Gauge symmetries seem to be excepted (due to long range forces). But gauge symmetries imply global symmetries. (Invariance under $\phi \to e^{i \theta(x)} \phi$ implies invariance under $\phi \to e^{i \theta} \phi$.) Therefore it seems like global symmetries are allowed, namely the global parts of the local symmetries. That would mean a $U(1)$ global symmetry corresponding to electric charge is allowed for example. What is wrong with this reasoning?

Qmechanic
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Marten
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  • The example given in the question assumes a classical spacetime background, and in that context, the argument is fine. Is the question asking what goes wrong with it when gravity is quantized? That's a hard question. The point of the no-global-symmetries folklore is that (1) something must go wrong with the reasoning and that (2) we don't yet know what goes wrong. The paper Symmetries in Quantum Field Theory and Quantum Gravity spends many pages just to define what global symmetry should mean in quantum gravity before "proving" that AdS/CFT can't have any. – Chiral Anomaly Jun 03 '20 at 03:24
  • @Chiral, thanks, I will read it. But as a quick reply: Semi classical black hole arguments are often at the basis of the no-go- theorems for global symmetries in quantum gravity. Indeed the very claim is that it holds for any realization of quantum gravity not just in the specific cases we understand a bit better (String Theory/ AdS-CFT). So even if you say there might be other arguments, would you say that those semi-classical thought experiments about black holes say nothing to forbid the global parts of local symmetries (while they do say something already about other global symmetries). – Marten Jun 03 '20 at 09:07
  • @Chiral, a quick glance seems to suggest that also in this case Harlow means that if there is a global U(1) there has to also be a local U(1) (in a AdS bulk corresponding to a CFT). So indeed this would mean that what is meant is that global symmetries ARE ALLOWED as long as they are the local part of a global symmetry. Is this reading wrong? – Marten Jun 03 '20 at 10:51
  • If by semi-classical you mean that the spacetime background is fixed, then a no-global-symmetry argument can't be purely semi-classical. For example, one argument involves black hole evaporation: we can use a semi-classical model to derive Hawking radiation, but black holes don't evaporate in such a model. The assumption that the black hole would evaporate in a quantum gravity theory is an essential input to the argument, otherwise the no-go result would not follow. – Chiral Anomaly Jun 03 '20 at 12:55
  • Not sure which part of Harlow and Ooguri you're referring to (it's a long paper), but the if-then statement you wrote doesn't say that global symmetries must be allowed in the bulk, which is what the no-go folklore is about. In fact the abstract says "We first show that any global symmetry, discrete or continuous, in a bulk quantum gravity theory with a CFT dual would lead to an inconsistency in that CFT, and thus that there are no bulk global symmetries in AdS/CFT." The CFT can have global symmetries, but the no-go folklore doesn't claim to exclude that. – Chiral Anomaly Jun 03 '20 at 12:56
  • @Chiral, I meant that while they do state that global symmetry (in the bulk) is not allowed it does not seem that they actually mean to exclude what we would normally call a global symmetry in QFT. Because normally we would conclude that if a theory has a local symmetry it also has a global symmetry. And thus since local symmetries in the bulk are allowed that global symmetry must also be. (Of course they complicate things by I think finding new definitions for global and local symmetries but I assume that in the low energy QFT these definitions should match the conventional ones.) – Marten Jun 03 '20 at 12:59
  • @Chiral, and I made no statement about the CFT theory in the previous comments. I was talking about the bulk. – Marten Jun 03 '20 at 13:00
  • Sorry, I misread your second comment. I see now that you were just talking about the bulk theory. But then I don't see how you reach the conclusion that a global symmetry must be allowed. The implication global $U(1)\to$ local $U(1)$ doesn't imply that a global $U(1)$ must exist. (The reverse implication would.) Did I misunderstand your comment again? – Chiral Anomaly Jun 03 '20 at 13:03
  • In any case, the point of my reply to that comment was that Harlow and Ooguri explicitly claim to show that the quantum gravity theory cannot have global symmetries. Not sure what you mean by "what we would normally call a global symmetry in QFT", and in fact they show that the definition we would normally use in QFT isn't meaningful in quantum gravity (at least not in the AdS side of AdS/CFT). That's why they were compelled to propose a new definition. – Chiral Anomaly Jun 03 '20 at 13:07
  • @Chiral. Well the whole point of my question basically is that at least for what we understand on the level of a (low energy effective) QFT any local symmetry always implies a global symmetry (you can just fix the parameters describing the transformation to a fixed value for all x). It seems there might be a nuance that destroys this implication on the non-perturbative level and the point of the question would be to bring out this nuance. ... – Marten Jun 03 '20 at 13:10
  • ... Note that while the idea of global and local symmetry we have in QFT might not translate easily to Quantum Gravity those concepts seem to be perfectly consistent in the low energy QFT and so any new definition should overlap in that regime. So I believe the question still stands but maybe you are happier with the rephrasing: Why does a local symmetry not imply a global symmetry (at the very least on the level of the effective QFT). – Marten Jun 03 '20 at 13:10
  • At the level of effective QFT, a local symmetry can imply a global one, but that doesn't necessarily say much about quantum gravity because an IR effective QFT can have symmetries that its UV origin doesn't have. That was the motive for my original request-for-clarification. If the question is asking how global symmetries can emerge in the effective QFT starting from quantum gravity, then I think the answer is unknown -- that's why the folklore is interesting. (Maybe Harlow and Ooguri implicitly answered it in the special case of AdS/CFT, but that question wasn't their focus.) – Chiral Anomaly Jun 03 '20 at 13:48

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