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My first question is about gauge invariance. I can see that the lagrangian is usually invariant under a global gauge transformation, and I can see that if introduce the covariant derivative we have a local gauge invariant lagrangian. I don't understand why we would want that however. I found the explanation that a local gauge invariance means that the theory is renormalizable, however that's just words to me. What does this mean, and is that why we want a lagrangian to be local gauge invariant? Where does this local gauge invariance origin from?

My second question is about spontaneous symmetry breaking. I'm a little lost about this concept. It appears that if the potential part of the lagrangian doesn't have a minimum at the origin you have to choose one of the minima and introduce a new field with respect to this minimum. I can see that this breaks the original symmetry in the lagrangian. I don't understand why you have to do this however? Why is it you have to translate your field so it has a minimum at the origin? I'm quite confused.

I can see that if you impose these conditions on the lagrangian

[tex]L = D_\mu\phi(D^\mu\phi)^*-\frac{1}{4} F^{\mu\nu} F_{\mu\nu}-V(\phi)[/tex]

with [tex]D_\mu[/tex] as the covariant derivative and [tex]V(\phi)=-\mu^2\phi^*\phi+\lambda(\phi^*\phi)^2[/tex] yields 2 fields with masses if you pick a particular gauge. Where does the potential come from however? A good guess?

And lastly, if know some good links I'd appreciate that very much. Don't hold back even if you think it's too advanced, I can always discard it if I find it too difficult.

Thank you for your time