Path Integral Quantisation of Finite Noncommutative Geometries

We present a path integral formalism for quantising the spectral action of noncommutative geometry, based on the principle of summing over all Dirac operators. The approach is demonstrated on two simple finite noncommutative geometries. 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 for the distances are evaluated, and we discover that distances can shrink with increasing graviton excitations. We find that adding fermions reduces the effects of the gravitational field. We also make a comparison with Rovelli’s canonical quantisation approach described in [?], and briefly discuss the quantisation of a Riemannian manifold.