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Pré-Publication, Document De Travail Année : 2014

Hadamard states for the linearized Yang-Mills equation on curved spacetime

Christian Gérard
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Michal Wrochna

Résumé

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.
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Dates et versions

hal-00967054 , version 1 (27-03-2014)
hal-00967054 , version 2 (06-08-2014)
hal-00967054 , version 3 (23-09-2014)
hal-00967054 , version 4 (02-03-2017)

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Christian Gérard, Michal Wrochna. Hadamard states for the linearized Yang-Mills equation on curved spacetime. 2014. ⟨hal-00967054v3⟩
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