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 . As a matter of fact, Kuchař 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.
Living Rev. Relativity 13, (2010), 6
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