Path Integral Quantisation of Finite Noncommutative Geometries

TL;DR

This paper develops a path integral approach to quantising gravity using spectral action, summing over Dirac operators, and applies it to finite noncommutative geometries, revealing how graviton excitations affect distances.

Contribution

It introduces a novel path integral formalism for quantum gravity based on spectral action and demonstrates it on simple finite noncommutative geometries.

Findings

Distances shrink with increasing graviton excitations

Adding fermions reduces gravitational effects

Propagators and Greens functions for gravitons are calculated

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 two-point 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.