Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 073, 83 pages      arXiv:2009.12342      https://doi.org/10.3842/SIGMA.2021.073

Locality and General Vacua in Quantum Field Theory

Daniele Colosi a and Robert Oeckl b
a) Escuela Nacional de Estudios Superiores, Unidad Morelia, Universidad Nacional Autónoma de México, C.P. 58190, Morelia, Michoacán, Mexico
b) Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, C.P. 58190, Morelia, Michoacán, Mexico

Received September 28, 2020, in final form July 13, 2021; Published online July 25, 2021

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-Kähler 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-Kähler vacua, particularly relevant on timelike hypersurfaces.

Key words: quantum field theory; general boundary formulation; quantization; LSZ reduction formula; symplectic geometry; Feynman path integral; reflection positivity.

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