1. The problem statement, all variables and given/known data I need to gauge the symmetry: [tex] \phi \rightarrow \phi + a(x) [/tex] for the Lagrangian: [tex] L=\partial_\mu\phi\partial^\mu\phi [/tex] 2. Relevant equations 3. The attempt at a solution We did this in class for the Dirac equation with a phase transformation and I understood the method, but can't seem to figure out a way to apply the same approach to this field. All I have done so far is plug the transformation into the Lagrangian, but what I'm not sure about is what form the gauge field [tex]A_\mu[/tex] needs to take, or even where in the Lagrangian to put it! I have a load of QFT books too but the only mention of gauge invariance in them is very brief, not at all comprehensive, and is way further into the book than I am currently. Any guidance on where to start or where to read further would be really appreciated. Thanks as always!