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 -