Hadamard states for spacetimes with timelike boundaries

Abstract

We formulate and prove existence of pure quasifree Hadamard states for the Klein--Gordon field with Dirichlet boundary condition on a globally hyperbolic spacetime with timelike boundary. The microlocal condition is imposed on the one-particle Hilbert-space-valued distribution $p\circ G_D$ rather than on the two-point function. This is expressed by a $b$-wave-front set condition on the compressed future cone. The analysis on the manifold with corners $\bar{M}\times \bar{M}$ is then almost completely avoided. We prove that our version of the Hadamard condition is universal in the class of quasi-free Hadamard states. We also prove a propagation theorem for positive-energy microlocal splittings which uses only a microlocal splitting in the small $b$-calculus and well-posedness of the Dirichlet problem, and in particular requires no propagation theorem for reflected or gliding rays; combined with an explicit computation in the ultrastatic case and a Fulling--Narcowich--Wald deformation argument this yields the construction of a pure quasi-free Dirichlet--Hadamard state.