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Gauge transformations allowed by physical theories form groups. For example, a wave function in quantum mechanics can be multiplies by $e^{i\theta}$ and this won't change a thing. So the gauge group of this theory is $U(1)$.

Now, the Maxwell equations can be written in terms of a scalar $\phi$ and a vector potential $\vec{A}$ (I don't write them here since they are very well known). They admit gauge transformations in which $\vec{A}\to\vec{A}+\vec{\nabla} f$ and $\phi\to\phi+\partial_t f$ (up to some factors), where $f(\vec{r},t)$ can be any differentiable function.

Since I can take any function, I conclude that the gauge group of electromagnetism is infinite-dimensional. Is this correct?

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No, the gauge group of electromagnetism is still $\mathrm{U }(1)$. This is because there is a difference between the gauge group $G$ of a physical theory on a manifold $M$ and the group of gauge transformations $\mathcal{G}$ consisting of local functions $M\to G$ and the algebra of infinitesimal gauge transformations consisting of local functions $M\to\mathfrak{g}$ for $\mathfrak{g}$ the Lie algebra of $G$.

The group of gauge transformations is indeed typically infinite-dimensional, and in your example you have observed that the algebra of infinitesimal gauge transformation of a $\mathrm{U}(1)$-theory on flat $\mathbb{R}^4$ is just the (infinite-dimensional) algebra of smooth functions $\mathbb{R}^4\to\mathbb{R}$. Since $\mathfrak{u}(1)\cong\mathbb{R}$ (the tangent space of a circle is just the real line), this is indeed an infinitesimal gauge transformation in the above sense.

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  • $\begingroup$ So the gauge group is not the group of gauge transformations? What an unfortunate terminology. And what is the definition of gauge group then? $\endgroup$ – thedude Jan 28 '16 at 22:57
  • $\begingroup$ @thedude: It's even worse, some people call the group of gauge transformations the gauge group and the gauge group the global gauge group! (And I'm not claiming my terminology here is the better one - you just have to choose one and stick with it) In any case, if you've got the group of gauge transformations, the "gauge group" is simply the group you get when only admitting constant functions (if you've only got the algebra of infinitesimal transformations, then the constant functions there give you the Lie algebra of the gauge group). $\endgroup$ – ACuriousMind Jan 28 '16 at 23:08
  • $\begingroup$ Ah, I see. Its a local/global thing. Thank you $\endgroup$ – thedude Jan 28 '16 at 23:16

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