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On Representations of the Quantum Holonomy Diffeomorphism Algebra
Author(s) -
Aastrup Johannes,
Grimstrup Jesper Møller
Publication year - 2019
Publication title -
fortschritte der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.201800080
Subject(s) - holonomy , hilbert space , hamiltonian (control theory) , mathematics , quantum gravity , operator algebra , diffeomorphism , algebra over a field , pure mathematics , physics , mathematical physics , theoretical physics , quantum mechanics , quantum , mathematical optimization
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 the QHD ( 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.