I have recently heard that non-compact supported gauge transformations of non-abelian gauge theories have a non-trivial effect on the Hilbert space of states. This is a topic I can not find anywhere online, I am not sure if I am missing keywords but I am very interested to find out more! I believe this is related to the strong CP problem? (Please correct me if I am wrong)
2 Answers
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Goggle "large gauge transformation". It's not the non-compactness but the non-trivial homotopy that matters.
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$\begingroup$ I’ll just add that since 2013, the term large gauge transformation has also been used for gauge transformations of non-compact support which have nothing to do with the ones mentioned in the answer. Make sure you know which type gauge transformation you are reading about when you google it. $\endgroup$– PraharNov 18, 2021 at 13:39
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$\begingroup$ I don't think this is what the OP is referring to. Large gauge transformations are still gauge transformations and the Hilbert space is invariant under them. The OP is referring to gauge transformations that are not equal to the identity at the spatial boundaries (see the discussion between 2.3 and 2.4 of Remarks on the canonical quantization of the Chern-Simons-Witten theory) $\endgroup$ Nov 18, 2021 at 13:40
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$\begingroup$ So the OP means "gauge transformations" that actually change the bundle? I.e change the Chern number? I'd not call such things "gauge transformations" $\endgroup$ Nov 18, 2021 at 13:44
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$\begingroup$ @mikestone yes, those shouldn't be called gauge transformations, I agree. And to be fair, in the literature they are mostly not. $\endgroup$ Nov 18, 2021 at 13:45
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$\begingroup$ @Prahar May I ask, what role do non-compact gauge transformations play in QFT? I have heard about large gauge transformations but I was confused as papers seem to use large gauge transformations and non-compact gauge transformations despite having different definitions $\endgroup$ Nov 18, 2021 at 14:39
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This is certainly not a good enough answer but in here https://arxiv.org/abs/1906.08616 it is shown with examples that gauge transformations that don't vanish at the spatial boundaries have a non-trivial effect on the symplectic form. Accordingly, they alter the commutation relations and have a non-trivial effect on the Hilbert space.