  The Maxwell field with a general gauge fixing (GF) term is nontrivial, not
only the longitudinal and temporal modes are mixed up in the field equations,
but also unwanted consequences might arise from the GF term. We derive the
complete set of solutions in de Sitter space, and implement the covariant
canonical quantization which restricts the residual gauge transformation down
to a quantum residual gauge transformation. Then, in the Gupta-Bleuler (GB)
physical state, we calculate the stress tensor which is amazingly independent
of the gauge fixing constant and is also invariant under the quantum residual
gauge transformation.
  The transverse components are simply the same as those in the Minkowski
spacetime, and the transverse vacuum stress tensor has only one UV divergent
term ($\propto k^4$), which becomes zero by the 0th-order adiabatic
regularization. The longitudinal-temporal stress tensor in the GB state is zero
due to a cancelation between the longitudinal and temporal parts. More
interesting is the stress tensor of the GF term. Its particle contribution is
zero due to the cancelation in the GB state, and its vacuum contribution is
twice that of a minimally-coupling massless scalar field, containing $k^4$ and
$k^2$ divergences. After the 2nd-order adiabatic regularization, the GF vacuum
stress tensor becomes zero too, so that there is no need to introduce a ghost
field, and the zero GF vacuum stress tensor can not be a possible candidate for
the cosmological constant. Thus, all the physics predicted by the Maxwell field
with the GF term will be the same as that without the GF term. We also carry
out analogous calculation in the Minkowski spacetime, and the stress tensor is
similar to, but simpler than that in de Sitter space.
