an:06424652
Zbl 1314.83025
G??rard, C.; Wrochna, M.
Hadamard states for the linearized Yang-Mills equation on curved spacetime
EN
Commun. Math. Phys. 337, No. 1, 253-320 (2015).
00342810
2015
j
83C47 81T13 70S15 83C05 35L05 81T20
Hadamard states; linearized Yang-Mills equation; curved spacetime; Klein-Gordon equation; Cauchy data
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 \textit{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.
Zbl 0495.35054