A note on Gauss operators and quantizations of Yang-Mills theories

TL;DR

This paper examines the role of Gauss operators in the quantization of Yang-Mills theories, highlighting the limitations of Lorentz covariance and classifying possible Gauss operators with residual gauge symmetries.

Contribution

It introduces a classification of Gauss operators in Yang-Mills theories, emphasizing their physical significance over gauge fixings and analyzing constraints imposed by Lorentz covariance.

Findings

Lorentz covariance cannot be maintained with certain gauge properties in non-abelian theories

Gauss operators can be classified based on residual gauge symmetries

Relaxing Lorentz covariance allows for a broader set of Gauss operators

Abstract

The quantization of Yang-Mills field theories requires the introduction of a gauge fixing which leads to a violation of the Local Gauss Law described by the so-called Gauss operator. We discuss the local quantizations of Yang-Mills theories in terms of the possible Gauss operators, which are argued to have a more physical meaning than the gauge fixings. We focus the attention on the local quantizations which leave the global gauge group and a subgroup of local gauge transformations unbroken, as the Feynman quantization of quantum Electrodynamics, and show that in the non-abelian case such properties cannot be realized together with Lorentz covariance; thus, quite generally, one cannot have the structural properties of the Feynman quantization of Quantum Electrodynamics. By relaxing the condition of Lorentz covariance, we obtain a classification of Gauss operators, which satisfy gauge covariant conservation laws and generate non-trivial residual subgroups of local gauge transformations.