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

Christian Gérard, Michał Wrochna
March 27, 2014
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 $R^d$. 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 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.
open access link Commun. Math. Phys. 337(1), 253-320 (2015)
%%% contains utf-8, see: http://inspirehep.net/info/faq/general#utf8 %%% add \usepackage[utf8]{inputenc} to your latex preamble @article{Gerard:2014jba, author = "Gérard, Christian and Wrochna, Michal", title = "{Hadamard States for the Linearized Yang–Mills Equation on Curved Spacetime}", journal = "Commun. Math. Phys.", volume = "337", year = "2015", number = "1", pages = "253-320", doi = "10.1007/s00220-015-2305-0", eprint = "1403.7153", archivePrefix = "arXiv", primaryClass = "math-ph", SLACcitation = "%%CITATION = ARXIV:1403.7153;%%" }

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