Towards state locality in quantum field theory: free fermions

Abstract

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.Comment: 26 pages, LaTeX + AMS + Xy-pic; v2: abstract rewritten, references updated; v3: discussion extended, references adde

A local and operational framework for the foundations of physics

“…In particular, for theories of physical interest a sufficiently general implementation of state spaces on hypersurfaces that have boundaries has not yet been achieved. It has been suggested and in fact demonstrated for free fermions that this problem can be at least partially solved by working in the positive formalism instead [87]. On the other hand, the positive formalism equips quantum field theory with a notion of local measurement via probes (Subsection 6.9), going far beyond the limitations of the S-matrix.…”

“…However, a flaw of that version was that spaces of generalized boundary conditions had to be taken to be complex rather than real. A real and more definitive version was provided in the paper [87]. The latter paper also proposes for fermionic theories a partial solution of the problem of global vacua (see Subsection 8.2) by making use of the positive formalism.…”

Locality and General Vacua in Quantum Field Theory

“…In a different guise this takes the form of the Reeh-Schlieder theorem [45]. In the fermionic case these problems can be partially solved by going to a mixed state formalism and at the same time selectively dropping the polarization information [41]. In the bosonic case one could use "auxiliary" Kähler polarizations at the price of a direct physical interpretation of the respective state spaces.…”

Coherent States in Fermionic Fock-Krein Spaces and Their Amplitudes

“…where {ζ j } j∈I is an adapted orthonormal basis of L. (Compare formula (50) in [7]. The relative factor arises from the difference between using a real and a complex basis…”

“…(Compare formula (51) in [7]. Again, a relative factor arises from the difference between using a real and a complex basis.…”