Note on Gauss operators and quantizations of Yang-Mills theories

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 that leave the global gauge group and a subgroup of local gauge transformations unbroken, such 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 nontrivial residual subgroups of local gauge transformations.