The space of continuous states of perturbative interacting quantum field
theories in globally hyperbolic curved spacetimes is determined. Following
Brunetti and Fredenhagen, we first define an abstract algebra of observables
which contains the Wick-polynomials of the free field as well as their
time-ordered products, and hence, by the well-known rules of perturbative
quantum field theory, also the observables (up to finite order) of interest for
the interacting quantum field theory. We then determine the space of continuous
states on this algebra. Our result is that this space consists precisely of
those states whose truncated n-point functions of the free field are smooth for
all n not equal to two, and whose two-point function has the singularity of a
Hadamard fundamental form. A crucial role in our analysis is played by the
positivity property of states. On the technical side, our proof involves
functional analytic methods, in particular the methods of microlocal analysis.Comment: 24 pages, Latex file, no figure