Free Fermi and Bose Fields in TQFT and GBF

Abstract

We present a rigorous and functorial quantization scheme for linear fermionic and bosonic field theory targeting the topological quantum field theory (TQFT) that is part of the general boundary formulation (GBF). Motivated by geometric quantization, we generalize a previous axiomatic characterization of classical linear bosonic field theory to include the fermionic case. We proceed to describe the quantization scheme, combining a Fock space quantization for state spaces with the Feynman path integral for amplitudes. We show rigorously that the resulting quantum theory satisfies the axioms of the TQFT, in a version generalized to include fermionic theories. In the bosonic case we show the equivalence to a previously developed holomorphic quantization scheme. Remarkably, it turns out that consistency in the fermionic case requires state spaces to be Krein spaces rather than Hilbert spaces. This is also supported by arguments from geometric quantization and by the explicit example of the Dirac field theory. Contrary to intuition from the standard formulation of quantum theory, we show that this is compatible with a consistent probability interpretation in the GBF. Another surprise in the fermionic case is the emergence of an algebraic notion of time, already in the classical theory, but inherited by the quantum theory. As in earlier work we need to impose an integrability condition in the bosonic case for all TQFT axioms to hold, due to the gluing anomaly. In contrast, we are able to renormalize this gluing anomaly in the fermionic case.