Canonical quantisation of a spherically symmetric, massless scalar field interacting with gravity in (2+1) dimensions

A spherically symmetric, massless scalar field is allowed to interact with gravity via Einstein's field equations in a three-dimensional spacetime. The imposition of spherical symmetry makes this a minisuperspace model. The model is quantised by splitting spacetime into space and time via the ADM procedure, and then imposing canonical communication relations. Since this is a minisuperspace model, a comparison is made of such models to models with no imposed symmetries in flat spacetime. This comparison is done to gain some insight into how the imposition of a symmetry at the classical level affects the resulting quantum theory. The two main points made about the model are as follows. One, the quantised model is found to violate the principle of microcausality. It is shown in this case that there is no resulting violation of causality. Two, the vacuum fluctuations of the metric are studied and it is found that these fluctuations become infinite even for a finite-sized smearing volume.