  Afshordi, Aslanbeigi and Sorkin have recently proposed a construction of a
distinguished "S-J state" for scalar field theory in (bounded regions of)
general curved spacetimes. We establish rigorously that the proposal is
well-defined on globally hyperbolic spacetimes or spacetime regions that can be
embedded as relatively compact subsets of other globally hyperbolic spacetimes,
and also show that, whenever the proposal is well-defined, it yields a pure
quasifree state. However, by explicitly considering portions of ultrastatic
spacetimes, we show that the S-J state is not in general a Hadamard state. In
the specific case where the Cauchy surface is a round 3-sphere, we prove that
the representation induced by the S-J state is generally not unitarily
equivalent to that of a Hadamard state, and indeed that the representations
induced by S-J states on nested regions of the ultrastatic spacetime also fail
to be unitarily equivalent in general. The implications of these results are
discussed.
