The quantum holonomydiffeomorphism algebra and quantum gravity
Abstract
We introduce the quantum holonomydiffeomorphism ∗algebra, which is generated by holonomydiffeomorphisms on a threedimensional manifold and translations on a space of SU(2)connections. We show that this algebra encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Furthermore, we show that semiclassical states exist on the holonomydiffeomorphism part of the algebra but that these states cannot be extended to the full algebra. Via a Diractype operator we derive a certain class of unbounded operators that act in the GNS construction of the semiclassical states. These unbounded operators are the type of operators, which we have previously shown to entail the spatial threedimensional Dirac operator and DiracHamiltonian in a semiclassical limit. Finally, we show that the structure of the Hamilton constraint emerges from a YangMillstype operator over the space of SU(2)connections.
 Publication:

International Journal of Modern Physics A
 Pub Date:
 March 2016
 DOI:
 10.1142/S0217751X16500482
 arXiv:
 arXiv:1404.1500
 Bibcode:
 2016IJMPA..3150048A
 Keywords:

 Quantum gravity;
 noncommutative geometry;
 unification;
 04.60.Ds;
 04.60.Pp;
 12.10.Dm;
 Canonical quantization;
 Loop quantum gravity quantum geometry spin foams;
 Unified theories and models of strong and electroweak interactions;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 20 pages, 4 figures