# Local and Global Symmetries

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?

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Your example (v3) is called global symmetry because $\Lambda$ does not depend on $x$. –  Qmechanic Jan 2 '13 at 22:32

## 1 Answer

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 $\mathcal{F}/\mathcal{G}_0$.

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Can you point to an example of a gauged global $\mathbb{Z}/\mathbb{2Z}$ please? Sounds interesting. –  Michael Brown Jan 3 '13 at 2:38
Great answer. I might add that many physicist use a slightly different definition of "gauge group". Instead of considering the space of maps (which is infinite-dimensional) $M\rightarrow G$ as the gauge group, people usually call $G$ the gauge group (which is usually finite-dimensional). –  Heidar Jan 3 '13 at 2:40
@MichaelBrown: A silly example: Suppose you want to describe the classical mechanics of a particle on $\mathbb{R}P^2$. Maps to $\mathbb{R}P^2$ are probably most easily thought of as maps to $S^2$, modulo the obvious global action of $Z/2$. –  user1504 Jan 3 '13 at 2:42
@Heidar: Yes. I avoided using this term. But I agree: It's a good convention to say gauge group for target space, and group of local transformations for maps to this space, and group of gauge transformations for the subgroup of the local transformations which are actually gauge transformations (the ones which vanish at infinity). –  user1504 Jan 3 '13 at 2:46
@EdwardHughes: Note that I made a typo in that last equation. Forgot a factor of $\lambda^{-1}$, which makes the second term equal to $\partial_\mu \operatorname{ln}(\lambda(x))$. $\lambda$ shouldn't take values in $\mathbb{C}^\times$, because then we'd be adding complex-valued forms to real forms, after already deciding to represent the gauge field by real-valued 1-forms. –  user1504 Jan 3 '13 at 14:01