The problem of quantizing general relativity (GR) is a very hard one. To this day, and despite continuing efforts, there is not a completely satisfactory theory of quantum gravity. In order to acquire the necessary intuition to deal with the very many issues whose resolution is required, it is often useful to concentrate on simplified models that exhibit only some of the difficulties present in the full theory (and hopefully in a milder guise).
A natural way to get simplified versions of a physical system is to introduce symmetries. They often allow us to get particular solutions in situations where a complete resolution is out of reach. Usually this happens because the effective dimensionality of the problem is reduced by the symmetry requirements. In the case of classical (i.e., non-quantum) systems, symmetry is often used at the level of the equations of motion. Actually, a good way to start exploring the concrete physics of a given model is to look for symmetric configurations. The effectiveness of this approach is reinforced by the fact that, in many instances, they are good approximations to real physical situations. For example, the waves produced on a water pond by a falling stone are very well described by rotationally-invariant functions satisfying the two-dimensional wave equation. This is so because the initial disturbance producing the wave is rotationally symmetric to a good approximation. This very same philosophy is used in many branches of physics. In GR, for instance, the most important and useful metrics solving the Einstein field equations exhibit some type of symmetry – just think of the Schwarzschild, Kerr, or Friedman spacetimes. Actually only a few closed form solutions to the Einstein field equations with no Killing fields are known .
There is a vast literature devoted to the classical aspects of the symmetry reductions that covers topics ranging from purely mathematical issues to physical applications in the fields of cosmology and black hole physics. These simplified systems also provide interesting quantum theories that are easier to handle than full gravity. There are two main types of models that can loosely be defined as those with a finite number of degrees of freedom (minisuperspaces) and those that require the introduction of infinitely many of them (midisuperspaces). The purpose of this Living Review is to explore the quantization of the latter, hence, we will only discuss those classical aspects that are of direct relevance to their quantization (for example the Hamiltonian description).
The paper is organized as follows. After this introduction we will review the history of midisuperspaces in Section 2. To this end we will give a general overview of symmetry reductions of GR. A very important idea that plays a central role in this subject is the principle of symmetric criticality. It provides a very useful simplification – especially when considering the Hamiltonian framework – because it allows us to derive everything from a symmetry-reduced variational principle obtained by restricting the Einstein–Hilbert action to the family of symmetric solutions of interest. Though not every reduced system is of this type, this happens to be the case for all the models that we will consider in the paper. After a discussion on some aspects concerning the mathematical description of superspace we will comment on the differences between minisuperspaces and midisuperspaces.
General issues concerning quantization will be addressed in Section 3, where we will quickly review – with the idea of dealing with the quantization of symmetry reductions – the different approaches to the quantization of constrained systems, i.e., reduced phase-space quantization, Dirac quantization, quantization of fully and partially gauge-fixed models and path-integral methods. We will end this section with a discussion of the differences between the “quantizing first and then reducing” and the “reducing first and then quantizing” points of view.
Section 4 is devoted to the discussion of some relevant classical aspects of midisuperspaces. We will consider, in particular, one-Killing vector reductions, two-Killing vector reductions and spherically-symmetric models and leave to Section 5 the main subject of the paper: the quantization of midisuperspaces. There we will review first the one-Killing vector case and then go to the more important – and developed – two-Killing vector reductions for which we will separately consider the quantization of Einstein–Rosen waves, Gowdy cosmologies and other related models. A similar discussion will be presented for spherically-symmetric midisuperspaces. We will look at both metric and loop quantum gravity (LQG) inspired quantizations for the different models. We conclude the paper with our conclusions and a discussion of the open problems.
We want to say some words about the philosophy of the paper. As the reader will see there is an unusually low number of formulas. This is so because we have chosen to highlight the main ideas and emphasize the connection between the different approaches and models. We provide a sizable bibliography at the end of the paper; technical details can be found there. We feel that the proper way to master the subject is to read the original papers, so we believe that we will have reached our goal if the present work becomes a useful guide to understanding the literature on the subject. We have tried to give proper credit to all the researchers who have made significant contributions to the quantization of midisuperspaces but of course some omissions are unavoidable. We will gladly correct them in coming updates of this Living Review.
Living Rev. Relativity 13, (2010), 6
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