Gauge invariance and the Dirac free field energy

Dirac field theory is assumed to be gauge invariant. However it is well known that a calculation of the polarization tensor yields a non-gauge invariant result. The reason for this has been shown to be due to the fact that for Dirac theory to be gauge invariant there must be no lower bound to the free field energy. Here an additional proof will be presented to show that this is, indeed, the case.