## Details

Original language | English |
---|---|

Article number | 104408 |

Journal | Journal of geometry and physics |

Volume | 171 |

Early online date | 22 Oct 2021 |

Publication status | Published - Jan 2022 |

## Abstract

We construct a metric structure on a configuration space of gauge connections and show that it naturally produces a candidate for a non-perturbative, 3+1 dimensional Yang-Mills-Dirac quantum field theory on a curved background. The metric structure is an infinite-dimensional Bott-Dirac operator and the fermionic sector of the emerging quantum field theory is generated by the infinite-dimensional Clifford algebra required to construct this operator. The Bott-Dirac operator interacts with the HD(M) algebra, which is a non-commutative algebra generated by holonomy-diffeomorphisms on the underlying manifold, i.e. parallel-transforms along flows of vector fields. This algebra combined with the Bott-Dirac operator encode the canonical commutation and anti-commutation relations of the quantised bosonic and fermionic fields. The square of the Bott-Dirac operator produces both the Yang-Mills Hamilton operator and the Dirac Hamilton operator as well as a topological Yang-Mills term alongside higher-derivative terms and a metric invariant.

## Keywords

- Gauge theory, Non-commutative geometry, Non-perturbative quantum field theory, Quantum field theory, Unification

## ASJC Scopus subject areas

- Mathematics(all)
**Mathematical Physics****Physics and Astronomy(all)**- Mathematics(all)
**Geometry and Topology**

## Cite this

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**The metric nature of matter.**/ Aastrup, Johannes; Grimstrup, Jesper Møller.

In: Journal of geometry and physics, Vol. 171, 104408, 01.2022.

Research output: Contribution to journal › Article › Research › peer review

*Journal of geometry and physics*, vol. 171, 104408. https://doi.org/10.1016/j.geomphys.2021.104408

*Journal of geometry and physics*,

*171*, Article 104408. https://doi.org/10.1016/j.geomphys.2021.104408

}

TY - JOUR

T1 - The metric nature of matter

AU - Aastrup, Johannes

AU - Grimstrup, Jesper Møller

N1 - Funding Information: This work is financially supported by Ilyas Khan , St. Edmunds College, Cambridge, United Kingdom, by entrepreneur Kasper Bloch Gevaldig , Copenhagen, Denmark, and by Regnestuen Haukohl & Køppen , Copenhagen, Denmark. Funding Information: We would like to thank Peter W. Michor for helpful information on the infinite-dimensional Levi-Civita connection. JMG would like to express his gratitude to Ilyas Khan, United Kingdom, to entrepreneur Kasper Bloch Gevaldig, Copenhagen, Denmark, and to the engineering company Regnestuen Haukohl & Køppen, Denmark, for their generous financial support. JMG would also like to express his gratitude to the following sponsors: Frank Jumppanen Andersen, Anders Arnfred, Ria Blanken, Bart De Boeck, Niels Peter Dahl, Niels Giroud, Claus Hansen, Tanina & Theo Jenk, Simon Kitson, Troels Fjordbak Larsen, Hans-Jørgen Mogensen, Tero Pulkkinen, Christopher Skak and Rolf Sleimann for their financial support, as well as all the backers of the 2016 Indiegogo crowdfunding campaign. JMG would also like to express his gratitude to the Institute of Analysis at the Gottfried Wilhelm Leibniz University in Hannover, Germany, for kind hospitality during numerous visits.

PY - 2022/1

Y1 - 2022/1

N2 - We construct a metric structure on a configuration space of gauge connections and show that it naturally produces a candidate for a non-perturbative, 3+1 dimensional Yang-Mills-Dirac quantum field theory on a curved background. The metric structure is an infinite-dimensional Bott-Dirac operator and the fermionic sector of the emerging quantum field theory is generated by the infinite-dimensional Clifford algebra required to construct this operator. The Bott-Dirac operator interacts with the HD(M) algebra, which is a non-commutative algebra generated by holonomy-diffeomorphisms on the underlying manifold, i.e. parallel-transforms along flows of vector fields. This algebra combined with the Bott-Dirac operator encode the canonical commutation and anti-commutation relations of the quantised bosonic and fermionic fields. The square of the Bott-Dirac operator produces both the Yang-Mills Hamilton operator and the Dirac Hamilton operator as well as a topological Yang-Mills term alongside higher-derivative terms and a metric invariant.

AB - We construct a metric structure on a configuration space of gauge connections and show that it naturally produces a candidate for a non-perturbative, 3+1 dimensional Yang-Mills-Dirac quantum field theory on a curved background. The metric structure is an infinite-dimensional Bott-Dirac operator and the fermionic sector of the emerging quantum field theory is generated by the infinite-dimensional Clifford algebra required to construct this operator. The Bott-Dirac operator interacts with the HD(M) algebra, which is a non-commutative algebra generated by holonomy-diffeomorphisms on the underlying manifold, i.e. parallel-transforms along flows of vector fields. This algebra combined with the Bott-Dirac operator encode the canonical commutation and anti-commutation relations of the quantised bosonic and fermionic fields. The square of the Bott-Dirac operator produces both the Yang-Mills Hamilton operator and the Dirac Hamilton operator as well as a topological Yang-Mills term alongside higher-derivative terms and a metric invariant.

KW - Gauge theory

KW - Non-commutative geometry

KW - Non-perturbative quantum field theory

KW - Quantum field theory

KW - Unification

UR - http://www.scopus.com/inward/record.url?scp=85118234454&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2021.104408

DO - 10.1016/j.geomphys.2021.104408

M3 - Article

AN - SCOPUS:85118234454

VL - 171

JO - Journal of geometry and physics

JF - Journal of geometry and physics

SN - 0393-0440

M1 - 104408

ER -