  We address the Hamiltonian formulation of classical gauge field theories
while putting forward results some of which are not entirely new, though they
do not appear to be well known. We refer in particular to the fact that neither
the canonical energy momentum vector $(P^\mu )$ nor the gauge invariant energy
momentum vector $(P_{\textrm{inv}} ^\mu )$ do generate space-time translations
of the gauge field by means of the Poisson brackets: In a general gauge, one
has to consider the so-called kinematical energy momentum vector and, in a
specific gauge (like the radiation gauge in electrodynamics), one has to
consider the Dirac brackets rather than the Poisson brackets. Similar arguments
apply to rotations and to Lorentz boosts and are of direct relevance to the
"nucleon spin crisis" since the spin of the proton involves a contribution
which is due to the angular momentum vector of gluons and thereby requires a
proper treatment of the latter. We conclude with some comments on the
relationships between the different approaches to quantization (canonical
quantization based on the classical Hamiltonian formulation, Gupta-Bleuler,
path integrals, BRST, covariant canonical approaches).
