Another look at the problem of gauge invariance in QFT

Abstract

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.