Details
Original language | English |
---|---|
Article number | 2100106 |
Journal | Fortschritte der Physik |
Volume | 69 |
Issue number | 10 |
Early online date | 7 Sept 2021 |
Publication status | Published - 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
ASJC Scopus subject areas
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Fortschritte der Physik, Vol. 69, No. 10, 2100106, 06.10.2021.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Non-perturbative Quantum Field Theory and the Geometry of Functional Spaces
AU - Aastrup, Johannes
AU - Grimstrup, Jesper Møller
N1 - Funding Information: JMG would like to express his gratitude towards Ilyas Khan, United Kingdom, and towards the engineering company Regnestuen Haukohl & K?ppen, Denmark, and towards Kasper Bloch Gevaldig, Copenhagen, Denmark, for their generous financial support. JMG would also like to express his gratitude towards the following sponsors: Ria Blanken, Niels Peter Dahl, Simon Kitson, Rita and Hans-J?rgen Mogensen, Tero Pulkkinen and Christopher Skak for their financial support, as well as all the backers of the 2016 Indiegogo crowdfunding campaign, that has enabled this work. Finally, JMG would like to thank the mathematical Institute at the Leibniz University in Hannover for kind hospitality during numerous?visits. This work is financially supported by Ilyas Khan, St. Edmunds College, Cambridge, United Kingdom, and Kasper Bloch Gevaldig, Copenhagen, Denmark, and by Regnestuen Haukohl & K?ppen, Copenhagen, Denmark. Funding Information: This work is financially supported by Ilyas Khan, St. Edmunds College, Cambridge, United Kingdom, and Kasper Bloch Gevaldig, Copenhagen, Denmark, and by Regnestuen Haukohl & Køppen, Copenhagen, Denmark.
PY - 2021/10/6
Y1 - 2021/10/6
N2 - 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.
AB - 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.
KW - infinite-dimensional geometry
KW - non-perturbative quantum field theory
KW - noncommutative geometry
UR - http://www.scopus.com/inward/record.url?scp=85114308802&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1910.01841
DO - 10.48550/arXiv.1910.01841
M3 - Article
AN - SCOPUS:85114308802
VL - 69
JO - Fortschritte der Physik
JF - Fortschritte der Physik
SN - 0015-8208
IS - 10
M1 - 2100106
ER -