Another look at the problem of gauge invariance in QFT

TL;DR

This paper investigates the gauge invariance issue in quantum field theory, revealing that the problem stems from a mathematical inconsistency related to the vacuum state definition, demonstrated through an exactly solvable 1+1D fermion model.

Contribution

It provides a detailed analysis of gauge invariance problems in QFT, identifying the vacuum state definition as the source of inconsistency using an exactly solvable model.

Findings

Gauge invariance issues are linked to vacuum state definition.

Exact solutions in a 1+1D fermion model clarify the inconsistency.

Mathematical inconsistency affects formal QFT calculations.

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.