arXiv subjects

Michal Wrochna

Publications and source records attributed to Michal Wrochna.

4 recordsLinked to original sources

arXiv2018

The massive Feynman propagator on asymptotically Minkowski spacetimes II

We consider the massive Klein-Gordon equation on short-range asymptotically Minkowski spacetimes. Extending our results in [GW1], we show that the Klein-Gordon operator with Feynman type boundary conditions at infinite times is invertible and that its inverse, called the Feynman inverse, satisfies the microlocal conditions of Feynman parametrices in the sense of Duistermaat and Hörmander. This supplements the recent work of Vasy with more explicit techniques.

math.AP↗

arXiv2017

Hadamard states for the Klein-Gordon equation on Lorentzian manifolds of bounded geometry

We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter and Kerr-Kruskal spacetimes. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseu-dodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state's two-point function having Cauchy data given by pseudodifferential operators. This allows us to conclude that there is a one-parameter family of elliptic pseudodifferential operators that encodes both the choice of (pure, regular) Hadamard state and the underlying spacetime metric.

math-ph↗

arXiv2017

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 $\rr^d$. 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.

math-ph↗

arXiv2016

Feynman propagators and Hadamard states from scattering data for the Klein-Gordon equation on asymptotically Minkowski spacetimes

We consider the massive Klein-Gordon equation on a class of asymptotically static spacetimes. We prove the existence and Hadamard property of the in and out states constructed by scattering theory methods. Assuming in addition that the metric approaches that of Minkowski space at infinity in a short-range way, jointly in time and space variables, we define Feynman scattering data and prove the Fredholm property of the Klein-Gordon operator with the associated Atiyah-Patodi-Singer boundary conditions. We then construct a parametrix (with compact remainder terms) for the Fredholm problem and prove that it is also a Feynman parametrix in the sense of Duistermaat and H{ö}rmander.

math-ph↗