1

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.

Nanashi No Gombe
  • 2,666

1 Answers1

1

  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.

Qmechanic
  • 220,844