Towards state locality in quantum field theory: free fermions

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.