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

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;%%"
}

Keywords:

Hadamard states