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The Maxwell field with a general gauge fixing (GF) term is nontrivial, notonly the longitudinal and temporal modes are mixed up in the field equations,but also unwanted consequences might arise from the GF term. We derive thecomplete set of solutions in de Sitter space, and implement the covariantcanonical quantization which restricts the residual gauge transformation downto a quantum residual gauge transformation. Then, in the Gupta-Bleuler (GB)physical state, we calculate the stress tensor which is amazingly independentof the gauge fixing constant and is also invariant under the quantum residualgauge transformation.  The transverse components are simply the same as those in the Minkowskispacetime, and the transverse vacuum stress tensor has only one UV divergentterm ($\propto k^4$), which becomes zero by the 0th-order adiabaticregularization. The longitudinal-temporal stress tensor in the GB state is zerodue to a cancelation between the longitudinal and temporal parts. Moreinteresting is the stress tensor of the GF term. Its particle contribution iszero due to the cancelation in the GB state, and its vacuum contribution istwice that of a minimally-coupling massless scalar field, containing $k^4$ and$k^2$ divergences. After the 2nd-order adiabatic regularization, the GF vacuumstress tensor becomes zero too, so that there is no need to introduce a ghostfield, and the zero GF vacuum stress tensor can not be a possible candidate forthe cosmological constant. Thus, all the physics predicted by the Maxwell fieldwith the GF term will be the same as that without the GF term. We also carryout analogous calculation in the Minkowski spacetime, and the stress tensor issimilar to, but simpler than that in de Sitter space.

Maxwell field with gauge fixing term in de Sitter space: exact solution and stress tensor