  We try to answer the question whether the quantized free scalar field on a
spatially flat Friedmann-Robertson-Walker (FRW) spacetime is a matter model
that can induce a Chaplygin gas equation of state. For this purpose we first
describe how one can obtain every possible homogeneous and isotropic Hadamard
(HIH) state once any such state is given. We also identify a condition on the
scale factor sufficient to entail the existence of a simple HIH state $-$ this
state is constructed explicitly and can thence be used as a starting point for
constructing all HIH states. Furthermore, we employ these results to show that
on an FRW spacetime with non-positive constant scalar curvature there is, with
one exception, no Chaplygin gas equation of state compatible with any HIH
state. Finally, we argue that the semi-classical Einstein equation and the
Chaplygin gas equation of state can presumably not be consistently solved for
the quantized free scalar field.
