Local gauge invariance—the Abelian case

In the case of global gauge invariance the phase 9 is not measurable and can be chosen arbitrarily, but once chosen it must be the same for all times everywhere in space. Could it happen that one can fix the phase locally and diflferently at diflferent places It turns out that in electrodynamics one can do this. It possesses a local gauge symmetry (or gauge 

Those terms of C that contain only the fields and their Hermitian conjugates are invariant as before, but terms involving gradients must be handled with more care since, from (2.3.11), we have 

The second term is new and did not occur for the global case. The main 

Electrodynamics is locally gauge invariant because all derivatives occur in special combinations D, called covariant derivatives which do have the property that 

It should be stressed that gauge invariance plays a vital role in proving that the theory is renormalizable. Technically, as mentioned in Section 1.1, it gives rise to the Ward-Takahashi identities amongst matrix elements (Ward, 1950 Takahashi, 1957). 

---