  We construct Hadamard states for the Yang-Mills equation linearized around a
smooth, space-compact background solution. We assume the spacetime is globally
hyperbolic and its Cauchy surface is compact or equal $\rr^d$. We first
consider the case when the spacetime is ultra-static, but the background
solution depends on time. By methods of pseudodifferential calculus we
construct a parametrix for the associated vectorial Klein-Gordon equation. We
then obtain Hadamard two-point functions in the gauge theory, acting on Cauchy
data. A key role is played by classes of pseudodifferential operators that
contain microlocal or spectral type low-energy cutoffs. The general problem is
reduced to the ultra-static spacetime case using an extension of the
deformation argument of Fulling, Narcowich and Wald. As an aside, we derive a
correspondence between Hadamard states and parametrices for the Cauchy problem
in ordinary quantum field theory.
