  Hilbert spaces of states can be constructed in standard quantum field theory
only for infinitely extended spacelike hypersurfaces, precluding a more local
notion of state. In fact, the Reeh-Schlieder Theorem prohibits the localization
of states on pieces of hypersurfaces in the standard formalism. From the point
of view of geometric quantization the problem lies in the non-locality of the
complex structures associated to hypersurfaces in standard quantization. We
show that using a weakened version of the positive formalism puts this problem
into a new perspective. This is a local TQFT type formalism based on
super-operators and mixed state spaces rather than on amplitudes and pure state
spaces as the one of Atiyah-Segal. In particular, we show that in the case of
purely fermionic degrees of freedom the complex structure can be dispensed with
when the notion of state is suitably generalized. These generalized states do
localize on compact hypersurfaces with boundaries. For the simplest case of
free fermionic fields we embed this in a rigorous and functorial quantization
scheme yielding a local description of the quantum theory. Crucially, no
classical data is needed beyond the structures evident from a Lagrangian
setting. When the classical data is augmented with complex structures on
hypersurfaces, the quantum data correspondingly augment to the full positive
formalism. This scheme is applicable to field theory in curved spacetime, but
also to field theories without metric background.
