Non-perturbative Quantum Field Theory and the Geometry of Functional Spaces

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Original languageEnglish
Article number2100106
JournalFortschritte der Physik
Volume69
Issue number10
Early online date7 Sept 2021
Publication statusPublished - 6 Oct 2021

Abstract

In this paper we construct a non-commutative geometry over a configuration space of gauge connections and show that it gives rise to a candidate for an interacting, non-perturbative quantum gauge theory coupled to a fermionic field on a curved background. The non-commutative geometry is given by an infinite-dimensional Bott-Dirac type operator, whose square gives the Hamilton operator, and which interacts with an algebra generated by holonomy-diffeomorphisms. The Bott-Dirac operator and the associated Hilbert space relies on a metric on the configuration space of connections, which effectively works as a covariant ultra-violet regulator. We show that the construction coincides with perturbative quantum field theory in a local limit. Questions concerning Lorentz invariance and the fermionic sector as well as the issue of existence are left open.

Keywords

    infinite-dimensional geometry, non-perturbative quantum field theory, noncommutative geometry

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Non-perturbative Quantum Field Theory and the Geometry of Functional Spaces. / Aastrup, Johannes; Grimstrup, Jesper Møller.
In: Fortschritte der Physik, Vol. 69, No. 10, 2100106, 06.10.2021.

Research output: Contribution to journalArticleResearchpeer review

Aastrup J, Grimstrup JM. Non-perturbative Quantum Field Theory and the Geometry of Functional Spaces. Fortschritte der Physik. 2021 Oct 6;69(10):2100106. Epub 2021 Sept 7. doi: 10.48550/arXiv.1910.01841, 10.1002/prop.202100106
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abstract = "In this paper we construct a non-commutative geometry over a configuration space of gauge connections and show that it gives rise to a candidate for an interacting, non-perturbative quantum gauge theory coupled to a fermionic field on a curved background. The non-commutative geometry is given by an infinite-dimensional Bott-Dirac type operator, whose square gives the Hamilton operator, and which interacts with an algebra generated by holonomy-diffeomorphisms. The Bott-Dirac operator and the associated Hilbert space relies on a metric on the configuration space of connections, which effectively works as a covariant ultra-violet regulator. We show that the construction coincides with perturbative quantum field theory in a local limit. Questions concerning Lorentz invariance and the fermionic sector as well as the issue of existence are left open.",
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