Locality and General Vacua in Quantum Field Theory

TL;DR

This paper advances the local description of quantum field theories by extending GBQFT to include non-Kähler vacua and timelike hypersurfaces, integrating algebraic and Euclidean approaches for a more general framework.

Contribution

It introduces a method to incorporate non-Kähler polarizations into GBQFT, enabling local descriptions on timelike hypersurfaces in Lorentzian spacetimes.

Findings

Generalization of vacuum state representation to non-Kähler vacua.

Implementation of algebraic QFT principles in GBQFT.

Application of GNS construction to twisted star-structures.

Abstract

We extend the framework of general boundary quantum field theory (GBQFT) to achieve a fully local description of realistic quantum field theories. This requires the quantization of non-K\"ahler polarizations which occur generically on timelike hypersurfaces in Lorentzian spacetimes as has been shown recently. We achieve this in two ways: On the one hand we replace Hilbert space states by observables localized on hypersurfaces, in the spirit of algebraic quantum field theory. On the other hand we apply the GNS construction to twisted star-structures to obtain Hilbert spaces, motivated by the notion of reflection positivity of the Euclidean approach to quantum field theory. As one consequence, the well-known representation of a vacuum state in terms of a sea of particle pairs in the Hilbert space of another vacuum admits a vast generalization to non-K\"ahler vacua, particularly relevant on timelike hypersurfaces.