# Are all gauge groups necessarily infinite dimensional?

If by a gauge group, I mean the Lie group corresponding to a local continuous symmetry of the Lagrangian of a system, is it true that the Lie group is necessarily infinite dimensional? If so, what is the proof?

By a local symmetry, I mean one that differs from one space-time point to another.

NOTE: This question arises from a study of Noether's Second Theorem which is a statement regarding a infinite dimensional group of transformations.

P.S. Maybe I am confusing between the group of gauge transformations and the Lie group associated with a gauge symmetry (local continuous symmetries). If so, please tell me the difference.

• Lie groups are by definition finite dimensional. Why do you think they are infinite-dimensional? or rather, what do you mean by dimension? Mar 4, 2017 at 18:06
• @AccidentalFourierTransform Well then, what infinite dimensional group does Noether talk about? And how is it related to the Lie group, if it is not that already? I hope you see my confusion. Mar 4, 2017 at 18:27
• @AccidentalFourierTransform: it's a Schwartz-Lie group, cf dx.doi.org/10.1063/1.526680 Mar 4, 2017 at 18:31
• I think what causes confusions is that for local gauge transformations, the representing matrices $\Lambda^a_{\ b}$ are functions, so they contain infinite degrees of freedoms. With that said, in a simpler perspective, the gauge group $G$ is a finite dimensional Lie group, and what we have is $U\subset M$ an open region of spacetime and a function $\Lambda:U\rightarrow\rho(G)$, where $\rho$ is a representation. In a more abstract perspective, a gauge transformation is a right action $P\times G\rightarrow P$, where $P$ is a principal fiber bundle whose structure group is $G$. Mar 4, 2017 at 18:33
• Nontheless, the group $G$ itself is strictly finite dimensional. Mar 4, 2017 at 18:33

1. In Yang-Mills theory the underlying gauge Lie group $G$ is finite-dimensional. E.g. the gauge Lie group $G$ in the standard model is $1+3+8=12$ dimensional, which is a finite number.
2. However, the corresponding group ${\cal G}= \Gamma(P\times_G G)$ of gauge transformations, i.e. the set of global sections in the associated bundle bundle $P\times_G G$ of the principal $G$-bundle $P$ over spacetime $M$, is necessarily infinite dimensional, if $\dim M >0$.
3. The latter group ${\cal G}$ (as opposed to $G$) is what is relevant for Noether's second theorem.
• re 1, $G$ is also known as the structure group of the principal bundle; re 2, the group of gauge transformations is not the set of maps $M\to G$ (or equivalently, global sections of the trivial bundle $M\times G$), but global sections of the associated bundle $P\times_G G$ ($P$ being the principal bundle in question) where the $G$-action is given by inner automorphisms Mar 4, 2017 at 18:59