Path integral quantisation of finite noncommutative geometries. (English) Zbl 1011.83011
Summary: 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 \(M_2(\mathbb C)\). 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.
MSC:
| 83C45 | Quantization of the gravitational field |
| 81S40 | Path integrals in quantum mechanics |
| 81T75 | Noncommutative geometry methods in quantum field theory |
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