Global gauge invariance—the Abelian case

We now consider internal symmetries that do not involve space-time. Each such symmetry will be expressed by the fact that there exists a field transformation which leaves C unaffected, and to each such ssunme-try there will correspond a conservation law and some quantity which is not measurable. 

One such symmetry, which is associated with charge conservation, is gauge invariance of the first kind or global gauge invariance. The field transformation is a phase transformation 

Clearly then is left invariant under (2.3.5). In other words does not depend on the phases of the f)j, which are therefore immeasurable. If 0 is infinitesimal, then (2.3.5) becomes 

If we define the current associated with the gauge transformation by 

The fact that an invariance of leads to a conserved current is known as Noether s theorem ind the currents are often called Noether currents (see, for example, Ramond, 1981). 

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