### 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 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 [84, 29] were the first symmetry reduction of this
type that was considered from the Hamiltonian point of view with the purpose of studying its
quantization [141]. As a matter of fact, KuchaĆ introduced the term midisuperspace precisely to
refer to this system [141, 142]. Other configurations of this type are the well-known Gowdy
spacetimes [102, 103] 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.