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I've a question about the definition of 'large' gauge transformations. There are two competing definitions:

  1. small gauge transformations are equal to the identity at spatial infinity while large gauge transformations are not

  2. small gauge transformations are continuously connected to the identity while large gauge transformations are not

It seems to me that these definitions are not equivalent. For example consider a SU(2) gauge transformation on 3-space: $$\Omega_n(\bf{x})=\exp(i\omega(|\bf x|) ~\sigma_i \hat x^i/2)=\cos(\omega/2)+i\sin(\omega/2)\sigma_i \hat x^i$$ where $\omega(\cdot)$ is any monotonic function such that $\omega(0)=0$ and $\omega(\infty)=4\pi n$ (this is taken from (2.37) of http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html).

This gauge transformation is not continuously connected to the identity (in fact it has winding number $n$); but this gauge transformation is the identity at spatial infinity. So is $\Omega_n(x)$ a small or large gauge transformation?

dennis
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  • Any more information about your second statement? Because in the first definition, for a gauge transformation to go smoothly to the identity at infinity it needs to be continuously connected to the identity as well. It can't suddenly make a discrete jump when considering spatial infinity. So there might be something missing where you found the second statement. – Guliano Jan 14 '23 at 15:57
  • @JulianDeV. I found the second statement on p23 of https://www.classe.cornell.edu/~pt267/files/documents/A_instanton.pdf – dennis Jan 14 '23 at 16:03
  • Homotopic to the identity indeed implies continuously connected to the identity. However, large vs small only has to do with its properties at spatial infinity. Taking your example, the fact that it is equal to the identity at the origin makes it indeed continuously connected to the identity, but it is not the identity at spatial infinity. It can happen that both small and large gauge transformations are connected to the identity, but the question is whether that happens at spatial infinity or somewhere else. In that light, I think the second statement from those notes is incomplete. – Guliano Jan 15 '23 at 02:41
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    There are at least two different things that are sloppily called "large gauge transformations", see this answer of mine – ACuriousMind Jan 15 '23 at 11:31
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    As far as I'm aware the two concepts of Large Gauge Transformation you present are different and it's just unfortunate that they get the same name. The first refers to asymptotic symmetries, which are local transformations that turn out to be physical because of their behavior at infinity, instead of mere redundancies. For more details on why these exist see my answer here https://physics.stackexchange.com/questions/719053/can-residual-gauge-symmetries-have-compact-support/725592#725592. – Gold Jan 15 '23 at 14:09
  • @Gold I agree! What do you think the conserved charge for the asymptotic symmetry given by gauge transformations not equal to the identity on spatial infinity is? – dennis Jan 15 '23 at 14:29
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    Well there are various such transformations in various theories and what the charge means certainly depends on the specific case. In electrodynamics, the conservation of the large gauge charge in the scattering problem means "charge conservation at every angle" and likewise in gravity the conservation of supertranslation charge means "energy conservation at every angle". In the quantum theory conservation of the charges implies in soft theorems, an extremely important result for holography in asymptotically flat spacetimes. – Gold Jan 15 '23 at 14:34
  • I suggest you see this review on the subject https://arxiv.org/abs/1703.05448 and also this one https://arxiv.org/abs/1801.07064, although the second does not talk much about the quantum theory and soft theorems, while it is really the focus of the first. If you want to know the implications on holography then I suggest https://arxiv.org/abs/2107.02075. – Gold Jan 15 '23 at 14:38

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