  It is well known that the single particle Dirac equation is gauge invariant.
This means that observable quantities, such as the current density, are not
affected by a gauge transformation. However what happens when the method of
second quantization is applied to convert a single particle theory into a field
theory? In this case it will be shown that the theory is no longer gauge
invariant. This will be shown by considering the second quantization of a zero
mass Dirac field in 1+1 dimensions and examining the change of the current
density operator due to a gauge transformation.
