Locality and local gauge invariance

Question

I was reading this question on the Physics Stack Exchange, and I'm still not quite sure how I can understand the relationship between locality and local gauge invariance using this example. Consider the Lagrangian density for the electromagnetic field $A_\mu$, coupled to a complex scalar field $\phi$: \begin{equation*} \mathcal{L}_{{\rm gauge}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D_\mu\phi)^*D^\mu\phi, \end{equation*} where $D_\mu = \partial_\mu-iqA_\mu$, and $F_{\mu\nu} = \partial_\mu A_\nu-\partial_\nu A_\mu$. Is it right to say the local gauge invariance $\phi\rightarrow e^{i\theta(x)}\phi$ preserves the Lagrangian density $\mathcal{L}_{{\rm gauge}}$, which makes the corresponding action local, therefore the field theory is local?

I think I don't quite understand how we define 'locality' in the context of field theory. Also for this example, are we assuming local gauge invariance, instead of global symmetries?

Answer 1

"Local" gauge invariance has nothing, per se, to do with locality of the Lagrangian. Locality of the Lagrangian is about having all the products of fields and derivatives in the Lagrange density be evaluated at the same point. If you allow for combinations of fields at different spacetime points, like $\phi(x)\phi(y)$, you are almost certain to violate causality, with faster-than-light signaling from $x$ to $y$ and vice-versa.

On the other hand, what is "local" about a gauge transformation of the second kind* is that the parameter $\theta$ in the gauge transformation $\psi\rightarrow\phi'=e^{i\theta(x)}\phi$ can be chosen to be an arbitrary function of spacetime (subject to suitable differentiability conditions). The transformation is "local" in the sense that it is allowed to be completely different in different local neighborhoods. Concretely, you could choose $\theta(x)$ to be zero outside some small spacetime region, in which case the transformation would clearly be "local" in the sense that only in that local neighborhood is the field actually transformed in a nontrivial way.

*"Gauge transformation of the second kind" is probably a better name, since it does not introduce this confusion about with another meaning of "local."