As we have seen, the quantization of midisuperspace models has been – and still is – a very useful test bed to understand many different methods and techniques proposed for the quantization of GR. In this respect the models described in this Living Review will certainly be of use in the near future, in particular to explore new avenues to the quantization of gravity both in the geometrodynamical approach and in LQG. These models have also been used with some success to extract qualitative physical results that are expected to be present in full quantum gravity (black hole mass quantization, Hawking radiation, unitarity of quantum evolution, microcausality, singularity resolution, semiclassical limit, …). Certainly a complete full quantization of the symmetry reductions described here will shed light on many methods and approaches and may be useful to arrive at a fully functioning theory of quantum gravity.

Among the open problems that we think should be considered we provide the following list:

- From a purely mathematical point of view it would be interesting to study how midisuperspace models sit inside full superspace (both in metric and connection variables) in the spirit of analysis carried out by Jantzen for minisuperspaces [128].
- Advance in the study of one-Killing reductions (or equivalently 2+1 gravity coupled to matter fields). It is important to find out if these systems can be treated perturbatively after carefully introducing the necessary boundary terms in the action. From the non-perturbative point of view it is necessary to obtain the concrete form of the bounded Hamiltonians for open spatial topologies and understand their physical implications.
- Give a complete and unified Hamiltonian treatment of all the midisuperspaces that admit a general two-dimensional spatial isometry group, generalizing in this way the two-Killing vector reductions of GR considered so far. Special attention should be paid to the analysis of non-compact spatial topologies and general polarization.
- Obtain physical predictions from these models.
- It is important to attempt the quantization of two-Killing vector reductions coupled to different types of matter fields beyond massless scalars. Though there are some papers dealing with scalar and electromagnetic fields, there is still a lot of work to do. It must be said, however, that it may actually be very difficult to get exact solutions to them.
- Using the non-perturbative quantizations provided by Korotkin, Nicolai and Samtleben [182] to obtain physical predictions and check if expected quantum gravitational phenomena do actually occur. Among these we would suggest the study of microcausality in these models. Another interesting issue may be the coupling of matter fields – in particular massless scalars. If they can be described by using Fock spaces, their particle-like quanta might be used as quantum test particles to explore quantum geometry and the emergence of a classical spacetime.
- It would be interesting to provide consistent and complete quantizations inspired in LQG for Einstein–Rosen waves and Gowdy models – for which the available results are rather incomplete. In particular, it is important go beyond the current hybrid formulations to purely polymeric ones.
- Complete the quantization of gravitational plane waves.
- Advance in the geometrodynamical quantization of spherically-symmetric gravitational systems coupled to matter. In particular, in those situations where the reduced phase spaces are infinite dimensional. It would be desirable to reach a level of mathematical rigor on par with the one customary within LQG.
- Complete the program started by Gambini, Pullin and collaborators to understand spherically-symmetric gravitational systems (with and without matter) in LQG.
- Answer the following questions in the midisuperspace setting: Does LQG predict the mass quantization of black holes? Is it possible to describe Hawking radiation in this framework?
- Understand the fate of classical singularities both cosmological and in black holes. In the case of black holes there are claims of singularity resolution both in the geometrodynamical approach and in LQG. One should understand the possible relationship between both approaches.

We expect that a lot of progress on these issues will happen in the near future. We will report on them in future updates of this Living Review.

Living Rev. Relativity 13, (2010), 6
http://www.livingreviews.org/lrr-2010-6 |
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