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Hadamard states for the linearized Yang-Mills equation on curved spacetime. (English) Zbl 1314.83025
Summary: 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 $$\mathbb R^d$$.
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 S. A. Fulling et al. [Ann. Phys. 136, 243–272 (1982; Zbl 0495.35054)].
As an aside, we derive a correspondence between Hadamard states and parametrices for the Cauchy problem in ordinary quantum field theory.

##### MSC:
 83C47 Methods of quantum field theory in general relativity and gravitational theory 81T13 Yang-Mills and other gauge theories in quantum field theory 70S15 Yang-Mills and other gauge theories in mechanics of particles and systems 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 35L05 Wave equation 81T20 Quantum field theory on curved space or space-time backgrounds
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