Dirac Operators on Configuration Spaces

Fermions with Half-integer Spin, Real Structure, and Yang–Mills Quantum Field Theory

authored by: Johannes Aastrup, Jesper Møller Grimstrup
Abstract: In this paper, the development of a spectral triple-like construction on a configuration space of gauge connections is continued. It has previously been shown that key elements of bosonic and fermionic quantum field theory emerge from such a geometrical framework. In this paper, a central problem concerning the inclusion of fermions with half-integer spin into this framework is solved. The tangent space of the configuration space is mapped into a similar space based on spinors, and this map is used to construct a Dirac operator on the configuration space. A real structure acting in a Hilbert space over the configuration space is also constructed. Finally, it is shown that the self-dual and anti-self-dual sectors of the Hamiltonian of a nonperturbative quantum Yang-Mills theory emerge from a unitary transformation of a Dirac equation on a configuration space of gauge fields. The dual and anti-dual sectors are shown to emerge in a two-by-two matrix structure.
Organisation(s): Faculty of Mathematics and Physics
External Organisation(s): Copenhagen
Type: Article
Journal: Fortschritte der Physik
ISSN: 0015-8208
Publication date: 03.02.2025
Publication status: E-pub ahead of print
Peer reviewed: Yes
ASJC Scopus subject areas: General Physics and Astronomy
Electronic version(s): https://doi.org/10.48550/arXiv.2410.07290 (Access: Open)
https://doi.org/10.1002/prop.202500003 (Access: Closed)

BibTeX

@article{bcb512d834dc4199b21cdd6d6b08af63,
title = "Dirac Operators on Configuration Spaces: Fermions with Half-integer Spin, Real Structure, and Yang–Mills Quantum Field Theory",
abstract = "In this paper, the development of a spectral triple-like construction on a configuration space of gauge connections is continued. It has previously been shown that key elements of bosonic and fermionic quantum field theory emerge from such a geometrical framework. In this paper, a central problem concerning the inclusion of fermions with half-integer spin into this framework is solved. The tangent space of the configuration space is mapped into a similar space based on spinors, and this map is used to construct a Dirac operator on the configuration space. A real structure acting in a Hilbert space over the configuration space is also constructed. Finally, it is shown that the self-dual and anti-self-dual sectors of the Hamiltonian of a nonperturbative quantum Yang-Mills theory emerge from a unitary transformation of a Dirac equation on a configuration space of gauge fields. The dual and anti-dual sectors are shown to emerge in a two-by-two matrix structure.",
keywords = "configuration space, noncommutative geometry, real structure, Yang–Mills theory",
author = "Johannes Aastrup and Grimstrup, {Jesper M{\o}ller}",
note = "Publisher Copyright: {\textcopyright} 2025 Wiley-VCH GmbH.",
year = "2025",
month = feb,
day = "3",
doi = "10.48550/arXiv.2410.07290",
language = "English",
journal = "Fortschritte der Physik",
issn = "0015-8208",
publisher = "Wiley-VCH Verlag",
}