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On Representations of the Quantum Holonomy Diffeomorphism Algebra

Aastrup, Johannes; Grimstrup, Jesper Møller

Fortschritte der Physik. Volume 67:Issue 4 (2019); pp n/a-n/a -- Wiley-VCH Verlag

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  • Title:
    On Representations of the Quantum Holonomy Diffeomorphism Algebra
  • Author: Aastrup, Johannes;
    Grimstrup, Jesper Møller
  • Found In: Fortschritte der Physik. Volume 67:Issue 4 (2019); pp n/a-n/a
  • Journal Title: Fortschritte der Physik
  • Subjects: Physics--Periodicals; mathematical physics--noncommutative geometry--non‐perturbative--quantum gravity; Dewey: 530.05
  • Rights: legaldeposit
  • Publication Details: Wiley-VCH Verlag
  • Abstract: Abstract:

    In this paper we establish the existence of the non‐perturbative theory of quantum gravity known as quantum holonomy theory by showing that a Hilbert space representation of theQHD(M)algebra, which is an algebra generated by holonomy‐diffeomorphisms and by translation operators on an underlying configuration space of connections, exist. We construct operators, which correspond to the Hamiltonian of general relativity and the Dirac Hamiltonian, and show that they give rise to their classical counterparts in a classical limit. We also find that the structure of an almost‐commutative spectral triple emerge in the same limit. The Hilbert space representation, that we find, is non‐local, which appears to rule out spacial singularities such as the big bang and black hole singularities. Finally, the framework also permits an interpretation in terms of non‐perturbative Yang‐Mills theory as well as other non‐perturbative quantum field theories. This paper is the first of two, where the second paper contains mathematical details and proofs.

    Abstract :

    In this paper the authors establish the existence of the non‐perturbative theory of quantum gravity known as quantum holonomy theory by showing that a Hilbert space representation of theQHD(M)algebra, which is an algebra generated by holonomy‐diffeomorphisms and by translation operators on an underlying configuration space of connections, exist. Operators are constructed, which correspond to the Hamiltonian of general relativity and the Dirac Hamiltonian, and show that they give rise to their classical counterparts in a classical limit. The structure of an almost‐commutative spectral triple emerge in the same limit. The Hilbert space representation, is non‐local, which appears to rule out spacial singularities such as the big bang and black hole singularities. Finally, the framework also permits an interpretation in terms of non‐perturbative Yang‐Mills theory as well as other non‐perturbative quantum field theories. This paper is the first of two, where the second paper will contain mathematical details and proofs.


  • Identifier: System Number: LDEAvdc_100085517461.0x000001; Journal ISSN: 0015-8208; 10.1002/prop.201800080
  • Publication Date: 2019
  • Physical Description: Electronic
  • Shelfmark(s): ELD Digital store

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