Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

Abstract

The aim of this paper is to prove the existence of Hadamard states for Dirac fields coupled with MIT boundary conditions on any globally hyperbolic manifold with timelike boundary once a suitable propagation of singularities theorem is assumed. To this avail, we consider particular pairs of weakly-hyperbolic symmetric systems coupled with admissible boundary conditions. We then prove the existence of an isomorphism between the solution spaces to the Cauchy problems associated with these operators – this isomorphism is in fact unitary between the spaces of $L^2$-initial data. In particular, we show that for Dirac fields with MIT boundary conditions, this isomorphism can be lifted to a $\ast$-isomorphism between the algebras of Dirac fields and that any Hadamard state can be pulled back along this $\ast$-isomorphism preserving the singular structure of its two-point distribution.