2.4 Midisuperspaces

The type of phenomena that can be described by a minisuperspace is rather limited because the metrics in these models effectively depend on a finite number of parameters. A less drastic simplification would consist in allowing some functional freedom but not the most general one. This is in essence the definition of a midisuperspace. More specifically the idea is to impose again symmetry requirements to restrict the set Riem (Σ ) used in the superspace construction in such a way that the allowed metrics are parametrized by functions rather than by numerical parameters. By doing this the hope is to increase the number of degrees of freedom of the models and eventually have local degrees of freedom. Notice that, as we will discuss below, the presence of fields at this stage does not preclude the possibility of having a finite-dimensional reduced phase space.

This can be accomplished, in particular, by restricting ourselves to metrics having a “low” number of spatial Killing vector fields. As we will see in the following, the case in which spacetime metrics are required to have two commuting Killing vector fields is specially appealing because some of these models are solvable both at the classical and quantum levels while, on the other hand, it is possible to keep several interesting features of full GR, such as an infinite number of degrees of freedom and diffeomorphism invariance. The Einstein–Rosen (ER) waves [84Jump To The Next Citation Point, 29Jump To The Next Citation Point] were the first symmetry reduction of this type that was considered from the Hamiltonian point of view with the purpose of studying its quantization [141Jump To The Next Citation Point]. As a matter of fact, Kuchaƙ introduced the term midisuperspace precisely to refer to this system [141Jump To The Next Citation Point, 142]. Other configurations of this type are the well-known Gowdy spacetimes [102Jump To The Next Citation Point, 103Jump To The Next Citation Point] that have been used as toy models in quantum gravity due to their possible cosmological interpretation.

A different type of systems that has been extensively studied and deserves close investigation is the spherically-symmetric ones (in vacuum or coupled to matter). These are, in a sense, midway between the Bianchi models and the midisuperspaces with an infinite number of physical degrees of freedom such as ER waves. General spherically-symmetric–spacetime metrics depend on functions of a radial coordinate and time, so these models are field theories. On the other hand, in vacuum, the space of physically-different spherical solutions to the Einstein field equations is finite dimensional (as shown by Birkhoff’s theorem). This means that the process of finding the reduced phase space (or, alternatively, gauge fixing) is non-trivial. The situation usually changes when matter is coupled owing to the presence of an infinite number of matter degrees of freedom in the matter sector. The different approaches to the canonical quantization of these types of models is the central topic of this Living Review.

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