  It is generally assumed that quantum field theory (QFT) is gauge invariant.
However it is well known that non-gauge invariant terms appear in various
calculations. This problem was examined in Refs. [3] and [4] and it was shown
that at the formal level QFT in the Schroedinger picture is not, in fact, gauge
invariant. It was determined that this problem was due to a mathematical
inconsistency related to the way the vacuum state is defined. In order to shed
further light on this problem we consider a simple field theory in 1-1D
space-time consisting of a quantized fermion field with zero mass in the
presence of a classical electromagnetic potential. It can easily be shown that
for this situation the equations of motion can be solved exactly in both the
Heisenberg and Schroedinger pictures. This allows us to easily identify the
source of the mathematical inconsistency that appears in QFT.
