- Defining a group action on the solution space of the full model.
- Finding a suitable parametrization of the solutions invariant under the group action.
- Obtaining the field equations describing the symmetric configurations.

When these steps can be successfully carried out, the final outcome of this process is a set of equations for the symmetry-reduced system. There are two conceivable ways to get them. The direct one consists in particularizing the general field equations to the invariant solutions obtained in the second step (by using some of the parametrizations introduced there). A second more indirect way would rely on the use of a symmetry-reduced action principle. This may seem an unnecessary detour but, if we intend to quantize the reduced system, it becomes an unavoidable step as we need a Hamiltonian formulation to define the dynamics of the quantized model. Though one may naively expect that the reduced action can be obtained by just restricting the one describing the full (i.e., non-reduced) model to the parameterized symmetric configurations, there are subtleties that may actually prevent us from doing so. We will discuss these problems in Section 2.1.1 devoted to symmetric criticality.

The transit to the quantum version of symmetry reductions of classical theories (involving either mechanical systems or fields) is quite non-trivial. This is a very important topic that plays a central role in the present paper so we discuss it here in some detail. There are several questions to be addressed in this respect:

- Definition of the symmetric quantum states and/or quantum symmetry reductions.
- Evolution of the symmetric states under full dynamics and the reduced dynamics.
- Comparison between the two: can we derive one from the other?

The first of these issues is usually discussed as the problem of understanding the commutativity of symmetry reduction and quantization, i.e., to figure out if the result of “first quantizing and then reducing” is the same as the one of “first reducing and then quantizing”. The other two items are also important, for example, to assess to what extent the results obtained in quantum cosmology (in its different incarnations including LQC) can be taken as hard physical predictions of quantum gravity and not only as suggestive hints about the physics of the complete theory. Of course the usual problems encountered in the quantization of constrained systems will also be present here. We will return to these issues in Section 3.

The original formulation of the principle of symmetric criticality, telling us when symmetric extremals of a functional can be obtained as the ones corresponding to the symmetry reduction of it, was stated by Palais in a variety of different settings [186]. The adaptation of this principle to GR was discussed in detail by Fels and Torre [86] though its importance was recognized since the early seventies (see [126] for an excellent review).

As mentioned above, the classical reduction process for a field theory is performed in several steps [186, 203]. One starts by defining a group action on the space of fields of the model, find then a parametrization of the most general configuration invariant under the group action and, finally, obtaining the form of the equations of motion restricted to these symmetric field configurations (the reduced field equations). General solutions to these equations correspond to symmetric solutions of the full theory.

In the case of GR one can ask oneself if the reduced field equations can be obtained as the Euler–Lagrange equations derived from some reduced Lagrangian and also if this Lagrangian can be obtained by simply restricting the Einstein–Hilbert action to the class of metrics compatible with the chosen symmetries. Obviously this would be the simplest (and more desirable) situation but one cannot exclude, in principle, that the reduced field equations could come from an action that is not the symmetry-reduced one (or even that they cannot be derived from a well-defined action principle). If a Hamiltonian formulation can be obtained for a symmetry reduction of a physical system then it is possible to consider its quantization. This is the path followed in quantum cosmology and in the study of the midisuperspace models that are the subject of this review.

The parametrization of the invariant field configurations usually involves the introduction of a set of
arbitrary functions whose number is smaller than the number of original field components. Furthermore, a
judicious choice of coordinates adapted to the symmetry normally restricts the number of variables upon
which these functions depend. In some instances it is possible to work with a single independent variable.
This happens, for example, in Bianchi models where these unknown functions depend only on a “time
coordinate” that labels compact homogenous spatial slices of spacetime. Another instance of this behavior is
provided by static, spherical, vacuum spacetimes where the arbitrary functions appearing in the metric
depend on an area variable (usually proportional to r^{2}). In both cases the field equations reduce to
ordinary differential equations. This, in turn, shows that these particular symmetry reductions
of GR describe systems with a finite number of degrees of freedom, i.e., purely mechanical
models.

Necessary and sufficient conditions guaranteeing that the principle of symmetric criticality holds in GR are given in Theorem 5.2 of [86]. They are technical in nature but their role is to prevent the occurrence of the two conceivable scenarios in which the symmetric criticality principle may fail. The first has to do with the possibility that the surface terms coming from integration by parts after performing variations in the full action do not reduce to the ones corresponding to the reduced action (this is what happens for Bianchi B models). The second is related to the fact that considering only “symmetric variations” may not give all the field equations but only a subset of them. An important comment to make at this point is that it is always possible to check if the symmetric criticality principle holds just by considering the group action because it is not tied to the form of a specific Lagrangian. This remarkable fact allows us to check the validity of the principle for whole families of symmetric models irrespective of their dynamics. In fact, for the types of vacuum models that are the main subject of this Living Review, symmetric criticality can be shown to hold [86, 203] and, hence, we have a simple way to get a Hamiltonian for the reduced systems. In the spherically-symmetric case the result holds as a consequence of the compactedness of the group of symmetries [203] (in the case of the two-Killing symmetry reductions the validity of the principle is justified in the papers [86, 203]). If scalar fields are coupled to gravity the principle still holds, however the introduction of other matter fields must be treated with care because their presence may influence the action of the symmetry group [86].

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