Path integral quantisation of finite noncommutative geometries
Abstract
We present a path integral formalism for quantising gravity in the form of the spectral action. Our basic principle is to sum over all Dirac operators. The approach is demonstrated on two simple finite noncommutative geometries: the twopoint space, and the matrix geometry M_{2}( C) . On the first, the graviton is described by a Higgs field, and on the second, it is described by a gauge field. We start with the partition function and calculate the propagator and Greens functions for the gravitons. The expectation values of distances are evaluated and we discover that distances shrink with increasing graviton excitations. We find that adding fermions reduces the effects of the gravitational field. A comparison is also made with Rovelli's canonical quantisation approach, and with his idea of spectral path integrals. We include a brief discussion on the quantisation of a Riemannian manifold.
 Publication:

Journal of Geometry and Physics
 Pub Date:
 December 2002
 DOI:
 10.1016/S03930440(01)00064X
 arXiv:
 arXiv:grqc/0007005
 Bibcode:
 2002JGP....44..115H
 Keywords:

 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory
 EPrint:
 12 pages, LaTeX2e, significantly revised. This is the published version with some minor additions