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  1. Is compact $U(1)$ and non-compact $U(1)$ just two different representations for the same group or the same gauge theory?

  2. If not, what the difference of them? Or are there any properties that are special for compact $U(1)$?

These probably are naive questions, but I don't know a reference explain me these. Any hits or reference are warmly welcome.

Qmechanic
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    See What is the meaning of non-compactness in the context of U(1) in gauge theories?. Assuming I understand Luboš correctly the non-compact theory allows continuous, i.e. not quantised, charges. – John Rennie Oct 14 '14 at 10:25
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    If I also understand Luboš correctly, "non-compact $U(1)$" is a rather confusing other name of the universal cover of $U(1)$, i.e. the one parameter Lie group $(\mathbb{R}, +)$. Okay, so the two groups are locally isomorphic, but really I can't see a good motivation for this ugly way of describing an otherwise clear notion, unless it is to impress those of use mere mortals who don't know the non compact $U(1)$ secret handshake. Can anyone say what the reason for describing it this way is? I guess there is no succinct notation for $(\mathbb{R},,+)$: is this all there is to it? – Selene Routley Oct 14 '14 at 10:46
  • @WetSavannaAnimalakaRodVance Well, since $U(1)$ is the topological compactification of $\mathbb{R}$, topologically it makes sense to think of $(\mathbb{R}, +)$ as the "non-compact" version of $U(1)$. – Alex Nelson Oct 14 '14 at 13:19

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