Local and Global Symmetries

Question

Could somebody point me in the direction of a mathematically rigorous definition local symmetries and global symmetries for a given (classical) field theory?

Heuristically I know that global symmetries "act the same at every point in spacetime", whereas local symmetries "depend on the point in spacetime at which they act".

But this seems somehow unsatisfying. After all, Lorentz symmetry for a scalar field $\psi(x)\rightarrow \psi(\Lambda^{-1}x)$ is conventionally called a global symmetry, but also clearly $\Lambda^{-1}x$ depends on $x$. So naively applying the above aphorisms doesn't work!

I've pieced together the following definition from various sources, including this. I think it's wrong though, and I'm confusing different principles that aren't yet clear in my head. Do people agree?

A global symmetry is a symmetry arising from the action of a finite dimensional Lie group (e.g. Lorentz group, $U(1)$)

A local symmetry is a symmetry arising from the action of an infinite dimensional Lie group.

If that's right, how do you view the local symmetry of electromagnetism $A^{\mu}\rightarrow A^{\mu}+\partial^{\mu}\lambda$ as the action of a Lie group?

Answer 1

Your proposed definitions are not quite correct. I'll sketch correct definitions, but I won't actually give them because I don't know how you choose to define classical field theory.

A group of local symmetries is a group of symmetry transformations where you get to change the system differently at different places in space/time.

A symmetry is global (in the context of field theory) if it acts in the same way at every point.

Local symmetries are necessarily infinite-dimensional, unless the spacetime manifold consists of finitely many points (which happens in lattice gauge theory). Global symmetries are usually finite-dimensional. Field theories which have infinitely many global symmetries are either very interesting, or not very interesting, depending on who you hang out with.

Gauge symmetries are usually local symmetries. They don't have to be. You can gauge a global $\mathbb{Z}/\mathbb{2Z}$, if you're in the mood to. But the most useful gauge symmetries are the ones which allow us to describe the physics of electromagnetism and the nuclear forces in terms of variables with local interactions. Our description of gravity in terms of a metric tensor also involves gauge symmetries. This is perhaps more puzzling than useful.

Let $\Sigma$ be the spacetime, probably $\mathbb{R}^{3,1}$. The local symmetry of the $1$-form description of electromagnetism is an action of the group $\mathcal{G} = \{ \lambda: \Sigma \to U(1) \}$ on the field space $\mathcal{F} \simeq \Omega^1(\Sigma)$, in which $\lambda$ sends the 1-form $A$ to the $1$-form $\lambda \dot{} A$ given at each $x$ in $\Sigma$ by $$ (\lambda \dot{} A)_\mu(x) = A_\mu(x) + \lambda^{-1}\partial_\mu \lambda(x). $$ The group of gauge transformations is the subgroup $\mathcal{G}_0$ of functions which become the identity at infinity. We apparently can't measure anything about electromagnetic phenomena which depends on $\mathcal{F}$ and $\mathcal{G}$, except through the quotient $\mathcal{F}/\mathcal{G}_0$.