Details
| Original language | English |
|---|---|
| Article number | 085016 |
| Journal | Classical and quantum gravity |
| Volume | 30 |
| Issue number | 8 |
| Publication status | Published - 5 Apr 2013 |
Abstract
A new approach to a unified theory of quantum gravity based on noncommutative geometry and canonical quantum gravity is presented. The approach is built around a *-algebra generated by local holonomy-diffeomorphisms on a 3-manifold and a quantized Dirac-type operator, the two capturing the kinematics of quantum gravity formulated in terms of Ashtekar variables. We prove that the separable part of the spectrum of the algebra is contained in the space of measurable connections modulo gauge transformations and we give limitations to the non-separable part. The construction of the Dirac-type operator - and thus the application of noncommutative geometry - is motivated by the requirement of diffeomorphism invariance. We conjecture that a semi-finite spectral triple, which is invariant under volume-preserving diffeomorphisms, arises from a GNS construction of a semi-classical state. Key elements of quantum field theory emerge from the construction in a semi-classical limit, as does an almost commutative algebra. Finally, we note that the spectrum of loop quantum gravity emerges from a discretization of our construction. Certain convergence issues are left unresolved. This paper is the first of two where the second paper [1] is concerned with mathematical details and proofs concerning the spectrum of the holonomy-diffeomorphism algebra.
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Physics and Astronomy (miscellaneous)
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In: Classical and quantum gravity, Vol. 30, No. 8, 085016, 05.04.2013.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - C*-algebras of holonomy-diffeomorphisms and quantum gravity
T2 - I
AU - Aastrup, Johannes
AU - Grimstrup, Jesper Møller
PY - 2013/4/5
Y1 - 2013/4/5
N2 - A new approach to a unified theory of quantum gravity based on noncommutative geometry and canonical quantum gravity is presented. The approach is built around a *-algebra generated by local holonomy-diffeomorphisms on a 3-manifold and a quantized Dirac-type operator, the two capturing the kinematics of quantum gravity formulated in terms of Ashtekar variables. We prove that the separable part of the spectrum of the algebra is contained in the space of measurable connections modulo gauge transformations and we give limitations to the non-separable part. The construction of the Dirac-type operator - and thus the application of noncommutative geometry - is motivated by the requirement of diffeomorphism invariance. We conjecture that a semi-finite spectral triple, which is invariant under volume-preserving diffeomorphisms, arises from a GNS construction of a semi-classical state. Key elements of quantum field theory emerge from the construction in a semi-classical limit, as does an almost commutative algebra. Finally, we note that the spectrum of loop quantum gravity emerges from a discretization of our construction. Certain convergence issues are left unresolved. This paper is the first of two where the second paper [1] is concerned with mathematical details and proofs concerning the spectrum of the holonomy-diffeomorphism algebra.
AB - A new approach to a unified theory of quantum gravity based on noncommutative geometry and canonical quantum gravity is presented. The approach is built around a *-algebra generated by local holonomy-diffeomorphisms on a 3-manifold and a quantized Dirac-type operator, the two capturing the kinematics of quantum gravity formulated in terms of Ashtekar variables. We prove that the separable part of the spectrum of the algebra is contained in the space of measurable connections modulo gauge transformations and we give limitations to the non-separable part. The construction of the Dirac-type operator - and thus the application of noncommutative geometry - is motivated by the requirement of diffeomorphism invariance. We conjecture that a semi-finite spectral triple, which is invariant under volume-preserving diffeomorphisms, arises from a GNS construction of a semi-classical state. Key elements of quantum field theory emerge from the construction in a semi-classical limit, as does an almost commutative algebra. Finally, we note that the spectrum of loop quantum gravity emerges from a discretization of our construction. Certain convergence issues are left unresolved. This paper is the first of two where the second paper [1] is concerned with mathematical details and proofs concerning the spectrum of the holonomy-diffeomorphism algebra.
UR - http://www.scopus.com/inward/record.url?scp=84876123773&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1209.5060
DO - 10.48550/arXiv.1209.5060
M3 - Article
AN - SCOPUS:84876123773
VL - 30
JO - Classical and quantum gravity
JF - Classical and quantum gravity
SN - 0264-9381
IS - 8
M1 - 085016
ER -