Second quantization and gauge invariance

TL;DR

This paper investigates how second quantization affects gauge invariance in Dirac field theories, revealing that gauge invariance is broken when transitioning from single particle to field theory in 1+1 dimensions.

Contribution

It demonstrates that second quantization can break gauge invariance in Dirac field theories, a previously unexamined aspect of gauge symmetry in quantum field formulations.

Findings

Second quantization breaks gauge invariance in Dirac fields.

Gauge transformation alters the current density operator after second quantization.

The effect is demonstrated explicitly in 1+1 dimensional zero mass Dirac fields.

Abstract

It is well known that the single particle Dirac equation is gauge invariant. This means that observable quantities, such as the current density, are not affected by a gauge transformation. However what happens when the method of second quantization is applied to convert a single particle theory into a field theory? In this case it will be shown that the theory is no longer gauge invariant. This will be shown by considering the second quantization of a zero mass Dirac field in 1+1 dimensions and examining the change of the current density operator due to a gauge transformation.