  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.
