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 -