Details
| Originalsprache | Englisch |
|---|---|
| Aufsatznummer | e70023 |
| Fachzeitschrift | Fortschritte der Physik |
| Jahrgang | 73 |
| Ausgabenummer | 8 |
| Publikationsstatus | Veröffentlicht - 18 Aug. 2025 |
Abstract
In this paper a candidate for a spectral triple on a quotient space of gauge connections modulo gauge transformations is construted and it is shown that it is related to a Kasparov type bi-module over two canonical algebras: the (Formula presented.) -algebra, which is a non-commutative (Formula presented.) -algebra generated by parallel transports along flows of vectorfields, and an exterior algebra on a space of gauge transformations. The latter algebra is related to the ghost sector in a BRST quantisation scheme. It is Previously shown that the key elements of bosonic and fermionic quantum field theory on a curved background emerge from a spectral triple of this type. In this paper it is shown that a dynamical metric on the underlying manifold also emerges from the construction. We first rigourously construct a Dirac type operator on the a quotient space of gauge connections modulo gauge transformations, and the commutator between this Dirac type operator and the (Formula presented.) -algebra is discussed. To do this a gauge-covariant metric on the configuration space is first constructed and it is used to construct the triple. The key step is that the volume of the quotient space to required to be finite, which amounts to an ultra-violet regularisation. Since the metric on the configuration space is dynamical with respect to the time-evolution generated by the Dirac type operator, The regularisation is interpreted as a physical feature (as opposed to static regularisations, which are always computational artefacts). Finally, a Bott–Dirac operator that connects our construction with quantum Yang-Mills theory is constructed.
ASJC Scopus Sachgebiete
- Physik und Astronomie (insg.)
- Allgemeine Physik und Astronomie
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in: Fortschritte der Physik, Jahrgang 73, Nr. 8, e70023, 18.08.2025.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - The Noncommutative Geometry of Configuration Spaces and the Problem of Reconciling General Relativity With Quantum Field Theory
AU - Aastrup, Johannes
AU - Grimstrup, Jesper Møller
N1 - Publisher Copyright: © 2025 Wiley-VCH GmbH.
PY - 2025/8/18
Y1 - 2025/8/18
N2 - In this paper a candidate for a spectral triple on a quotient space of gauge connections modulo gauge transformations is construted and it is shown that it is related to a Kasparov type bi-module over two canonical algebras: the (Formula presented.) -algebra, which is a non-commutative (Formula presented.) -algebra generated by parallel transports along flows of vectorfields, and an exterior algebra on a space of gauge transformations. The latter algebra is related to the ghost sector in a BRST quantisation scheme. It is Previously shown that the key elements of bosonic and fermionic quantum field theory on a curved background emerge from a spectral triple of this type. In this paper it is shown that a dynamical metric on the underlying manifold also emerges from the construction. We first rigourously construct a Dirac type operator on the a quotient space of gauge connections modulo gauge transformations, and the commutator between this Dirac type operator and the (Formula presented.) -algebra is discussed. To do this a gauge-covariant metric on the configuration space is first constructed and it is used to construct the triple. The key step is that the volume of the quotient space to required to be finite, which amounts to an ultra-violet regularisation. Since the metric on the configuration space is dynamical with respect to the time-evolution generated by the Dirac type operator, The regularisation is interpreted as a physical feature (as opposed to static regularisations, which are always computational artefacts). Finally, a Bott–Dirac operator that connects our construction with quantum Yang-Mills theory is constructed.
AB - In this paper a candidate for a spectral triple on a quotient space of gauge connections modulo gauge transformations is construted and it is shown that it is related to a Kasparov type bi-module over two canonical algebras: the (Formula presented.) -algebra, which is a non-commutative (Formula presented.) -algebra generated by parallel transports along flows of vectorfields, and an exterior algebra on a space of gauge transformations. The latter algebra is related to the ghost sector in a BRST quantisation scheme. It is Previously shown that the key elements of bosonic and fermionic quantum field theory on a curved background emerge from a spectral triple of this type. In this paper it is shown that a dynamical metric on the underlying manifold also emerges from the construction. We first rigourously construct a Dirac type operator on the a quotient space of gauge connections modulo gauge transformations, and the commutator between this Dirac type operator and the (Formula presented.) -algebra is discussed. To do this a gauge-covariant metric on the configuration space is first constructed and it is used to construct the triple. The key step is that the volume of the quotient space to required to be finite, which amounts to an ultra-violet regularisation. Since the metric on the configuration space is dynamical with respect to the time-evolution generated by the Dirac type operator, The regularisation is interpreted as a physical feature (as opposed to static regularisations, which are always computational artefacts). Finally, a Bott–Dirac operator that connects our construction with quantum Yang-Mills theory is constructed.
KW - configuration spaces
KW - Kasparov bi-modules
KW - noncommutative geometry
KW - quantum field theory
UR - http://www.scopus.com/inward/record.url?scp=105012118360&partnerID=8YFLogxK
U2 - 10.1002/prop.70023
DO - 10.1002/prop.70023
M3 - Article
AN - SCOPUS:105012118360
VL - 73
JO - Fortschritte der Physik
JF - Fortschritte der Physik
SN - 0015-8208
IS - 8
M1 - e70023
ER -