Details
| Original language | English |
|---|---|
| Article number | 103466 |
| Number of pages | 19 |
| Journal | Journal of Geometry and Physics |
| Volume | 145 |
| Issue number | 145 |
| Early online date | 28 Jun 2019 |
| Publication status | Published - Nov 2019 |
Abstract
A general framework of non-perturbative quantum field theory on a curved background is proposed. A quantum field theory is in this setting characterised by an embedding of a space of field configurations into a Hilbert space over R ∞. This embedding, which is only local up to a scale that we interpret as the Planck scale, coincides in the local and flat limit with the plane wave expansion known from canonical quantisation. We identify a universal Bott–Dirac operator acting in the Hilbert space over R ∞ and show that it gives rise to the free Hamiltonian both in the case of a scalar field theory and in the case of a Yang–Mills theory. These theories come with a canonical fermionic sector for which the Bott–Dirac operator also provides the Hamiltonian. We prove that Hilbert space representations of algebras of observables exist non-perturbatively for a real scalar theory and for a gauge theory, both with or without the fermionic sectors, and show that the free theories are given by semi-finite spectral triples over the respective configuration spaces. Finally, we propose a class of quantum field theories whose interactions are generated by inner fluctuations of the Bott–Dirac operator.
Keywords
- Non-perturbative quantum field theory, Noncommutative geometry, Unified field theory
ASJC Scopus subject areas
- Mathematics(all)
- Mathematical Physics
- Physics and Astronomy(all)
- Mathematics(all)
- Geometry and Topology
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In: Journal of Geometry and Physics, Vol. 145, No. 145, 103466, 11.2019.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - On nonperturbative quantum field theory and noncommutative geometry
AU - Aastrup, Johannes
AU - Grimstrup, Jesper M.
N1 - Funding Information: We would like to thank Prof. Nigel Higson for bringing his work with Prof. Gennadi Kasparov on the Bott–Dirac operator to our attention. JMG would like to express his gratitude towards Ilyas Khan, United Kingdom, for his 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-Jørgen 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. Also, JMG would like to thank the mathematical Institute at the Leibniz University in Hannover for kind hospitality during numerous visits.
PY - 2019/11
Y1 - 2019/11
N2 - A general framework of non-perturbative quantum field theory on a curved background is proposed. A quantum field theory is in this setting characterised by an embedding of a space of field configurations into a Hilbert space over R ∞. This embedding, which is only local up to a scale that we interpret as the Planck scale, coincides in the local and flat limit with the plane wave expansion known from canonical quantisation. We identify a universal Bott–Dirac operator acting in the Hilbert space over R ∞ and show that it gives rise to the free Hamiltonian both in the case of a scalar field theory and in the case of a Yang–Mills theory. These theories come with a canonical fermionic sector for which the Bott–Dirac operator also provides the Hamiltonian. We prove that Hilbert space representations of algebras of observables exist non-perturbatively for a real scalar theory and for a gauge theory, both with or without the fermionic sectors, and show that the free theories are given by semi-finite spectral triples over the respective configuration spaces. Finally, we propose a class of quantum field theories whose interactions are generated by inner fluctuations of the Bott–Dirac operator.
AB - A general framework of non-perturbative quantum field theory on a curved background is proposed. A quantum field theory is in this setting characterised by an embedding of a space of field configurations into a Hilbert space over R ∞. This embedding, which is only local up to a scale that we interpret as the Planck scale, coincides in the local and flat limit with the plane wave expansion known from canonical quantisation. We identify a universal Bott–Dirac operator acting in the Hilbert space over R ∞ and show that it gives rise to the free Hamiltonian both in the case of a scalar field theory and in the case of a Yang–Mills theory. These theories come with a canonical fermionic sector for which the Bott–Dirac operator also provides the Hamiltonian. We prove that Hilbert space representations of algebras of observables exist non-perturbatively for a real scalar theory and for a gauge theory, both with or without the fermionic sectors, and show that the free theories are given by semi-finite spectral triples over the respective configuration spaces. Finally, we propose a class of quantum field theories whose interactions are generated by inner fluctuations of the Bott–Dirac operator.
KW - Non-perturbative quantum field theory
KW - Noncommutative geometry
KW - Unified field theory
UR - http://www.scopus.com/inward/record.url?scp=85069005130&partnerID=8YFLogxK
U2 - 10.1016/j.geomphys.2019.06.017
DO - 10.1016/j.geomphys.2019.06.017
M3 - Article
VL - 145
JO - Journal of Geometry and Physics
JF - Journal of Geometry and Physics
SN - 0393-0440
IS - 145
M1 - 103466
ER -