  We investigate the relationship between the covariant and the
three-dimensional (equal-time) formulations of quantum kinetic theory. We show
that the three-dimensional approach can be obtained as the energy average of
the covariant formulation. We illustrate this statement in scalar and spinor
QED. For scalar QED we derive Lorentz covariant transport and constraint
equations directly from the Klein-Gordon equation rather than through the
previously used Feshbach-Villars representation. We then consider pair
production in a spatially homogeneous but time-dependent electric field and
show that the pair density is derived much more easily via the energy averaging
method than in the equal-time representation. Proceeding to spinor QED, we
derive the covariant version of the equal-time equation derived by
Bialynicki-Birula et al. We show that it must be supplemented by another
self-adjoint equation to obtain a complete description of the covariant spinor
Wigner operator. After spinor decomposition and energy average we study the
classical limit of the resulting three-dimensional kinetic equations. There are
only two independent spinor components in this limit, the mass density and the
spin density, and we derive also their covariant equations of motion. We then
show that the equal-time kinetic equation provides a complete description only
for constant external electromagnetic fields, but is in general incomplete. It
must be supplemented by additional constraints which we derive explicitly from
the covariant formulation.
