Communications in Mathematical Physics

, Volume 337, Issue 1, pp 253–320

Hadamard States for the Linearized Yang–Mills Equation on Curved Spacetime



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 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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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité Paris-Sud XI91405France
  2. 2.UMR 5582 CNRS, Institut Fourier, Université Joseph Fourier (Grenoble 1)Saint-Martin d’Hères CedexFrance

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