Hi, excuse the funny title :). In his book on quantum field theory Zee says (pag 245, fouth line) that QED gauge simmetry follows from the conservation of the current j=ψ γ^μ ψ (with the bar on the first spinor). I'm confused because that current is the noether current resulting from the invariance of QED lagrangian under a global phase trasformation, in which the spinor trasforms under U(1) and the vector potential A do not transform, and this global symmetry wouldn't be spoilt if the photon had a mass! So what I get from this observation is that whatever mass m the photon has, j is a conserved quantity. On the other hand, gauge invariance (that is a local transformation, under which A transforms too) holds only if m=0. So is Zee right or wrong?