What does it mean to gauge a group?

Question

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

Answer 1

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.

Answer 2

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.