# What does it mean to gauge a group?

I'm starting to learn about gauge theories and Goldstone bosons. What does it mean for a group to be gauged?

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Begin with theory of fields $\phi$ on a manifold $M$ with target space $V$ (which we take to be a vector space in this discussion) $$\phi:M\to V$$ that exhibits a global symmetry under some group $G$. This means that there is a representation $\rho$ of the group $G$ acting on the target space $V$ under whose action the theory is invariant, in other words the transformation $$\phi(x) \to \rho(g)\phi(x)$$ is an invariance of the theory. By gauging the group, we mean that we allow for transformations of the fields under this representation for which the group element $g$ depends on the point $x$ on the manifold where we are applying the transformation; $$\phi(x)\to\rho(g(x))\phi(x)$$ Such a transformation is called local. We now look to see how we can modify the theory (the standard way is to include an auxiliary field called a gauge field along with a gauge covariant derivative) such that the new theory exhibits invariance under these local transformations. The result is a gauge theory.

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It means that instead of global transformations

$\Psi(x,t) \rightarrow U \Psi(x,t)$

for some group element $U$, you allow local transformations

$\Psi(x,t) \rightarrow U(x,t) \Psi(x,t)$

where $U(x,t)$ is a space-time dependent group transformation. Any textbook on quantum field theory will go into detail about the implications of this. :) A free pre-publication version of Srednicki's (pretty good) book is available here if you don't already have access to one.

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+1: My apologies for the duplicate answer; I was too far in not to post when I saw your answer had posted. –  joshphysics Feb 18 '13 at 0:57
@joshphysics No worries –  Michael Brown Feb 18 '13 at 1:53