The Heisenberg versus the Schroedinger picture and the problem of gauge invariance

TL;DR

This paper investigates the gauge invariance of quantum field theory in different formulations, revealing that the Heisenberg picture maintains gauge invariance while the Schrödinger picture does not, challenging common assumptions about their equivalence.

Contribution

It demonstrates that the Heisenberg picture preserves gauge invariance in QFT, whereas the Schrödinger picture does not, highlighting the importance of formulation choice.

Findings

Heisenberg picture is gauge invariant in QFT.

Schrödinger picture is not gauge invariant in QFT.

The two pictures are not necessarily equivalent in their gauge properties.

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 recently examined in [9] for a "simple" field theory and it was shown that for this case QFT in the Schroedinger picture is not, in fact, gauge invariant. In order to shed further light on this problem we will examine the Heisenberg and Schroedinger formulations of QFT. It is generally assumed that these two "pictures" are equivalent; however we will show that this is not necessarily the case. We shall consider a simple field theory consisting of a quantized fermion field in the presence of a classical electromagnetic field. We will show that, although the two pictures are formally equivalent, the Heisenberg picture is gauge invariant but that the Schroedinger picture is not. This suggests that the proper way to formulate QFT is to use the Heisenberg picture.