  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. This is achieved by introducing a geometric M{\o}ller
operator which implements a unitary isomorphism between the spaces of $L^2$
-initial data of particular symmetric systems we call weakly-hyperbolic and
which are coupled with admissible boundary conditions. In particular, we show
that for Dirac fields with MIT boundary conditions, this isomorphism can be
lifted to a $*$-isomorphism between the algebras of Dirac fields and that any
Hadamard state can be pulled back along this $*$-isomorphism preserving the
singular structure of its two-point distribution.
