Superspace plays the role of the configuration space for general relativity in the traditional metric
representation. The associated cotangent bundle, when properly defined, is the phase space for the
Hamiltonian formulation of the theory. As a Hamiltonian formulation is the starting point for the
quantization of any mechanical or field system, the role of superspace and the need to understand its
mathematical structure cannot be overemphasized. A secondary role of superspace is that of providing
“variables for the wave function” in a functional Schrödinger representation for quantum gravity.
However, it should be noted at this point that even in the quantization of the simplest field
theories – such as scalar fields – it is necessary to suitably enlarge this configuration space and
allow for distributional, non-smooth objects to arrive at a consistent model (see, for example,
[12]). How – and if – this can be done in the geometrodynamical setting is an interesting, if
hard, question. This is directly related to the Wheeler–DeWitt approach to the quantization of
GR [79
].
The precise definition of the geometry of a three manifold requires some discussion (see [87, 98
] and
references therein for a nice introduction to the subject). Here we will content ourselves with a quick review
of the most important issues. It is important to remark at this point that the non-generic character of
geometries with non-trivial isometry groups has a very clear reflection in superspace: they correspond to
singularities.
The geometry of a four-dimensional manifold in relativists’ parlance refers to equivalence classes
of suitably smooth Lorentzian metrics defined on it. Two metrics are declared equivalent if
they are connected by a diffeomorphism. Though one might naively think that this is just a
mathematically-sensible requirement, in fact, it is quite natural from a physical point of view. The reason is
that ultimately the geometry must be probed by physical means. This, in turn, demands an
operational definition of the (possibly idealized) physical processes allowing us to explore – actually
measure – it. This is in the spirit of special and general relativity, where the definition of physical
magnitudes such as lengths, distances, velocities and the like requires the introduction of concrete
procedures to measure them by using basic tools such as clocks, rulers and light rays. Every
transformation of the manifold (and the objects defined on it) that does not affect the operational
definition of the measuring processes will be physically unobservable. Diffeomorphisms are such
transformations. Notice that this prevents us from identifying physical events with points in the spacetime
manifold as a diffeomorphism can take a given event from one point of the manifold to another
(see [163]).
The precise definition and description of the space of geometries requires the introduction of mathematical objects and structures at different levels:
After doing this one has to study the quotient . Naturally, superspace will inherit
some background properties from those carried by the different elements needed to properly define it.
The resulting space has a rich structure and interesting properties that we will very quickly
comment on here (the interested reader is referred to [98
] and the extensive bibliography cited
there).
An important issue is related to the appearance of singularities in this quotient space associated with
the fact that in many instances the spatial manifold allows for the existence of invariant metrics under
non-trivial symmetry groups (leading to a non-free action of the diffeomorphisms). This turns out to be a
problem that can be dealt with in the sense that the singularities are minimally resolved (see [87]). It is
important to mention at this point that the symmetry reductions that we will be considering here
consist precisely in restrictions to families of symmetric metrics that, consequently, sit at the
singularities of the full superspace. This fact, however, does not necessarily imply that the
reduced systems are pathological. In fact some of them are quite well behaved as we will show in
Section 4.
Finally, we point out that both the space of Riemannian metrics and the quotient space
mentioned above are endowed with natural topologies that actually turn them into very well-behaved
topological spaces (for instance, they are metrizable – and hence paracompact –, second countable and
connected). The space of metrics
can be described as a principal bundle with basis
and structure group given by
(that is, the proper subgroup
of those diffeomorphisms of
that fix a preferred point
and the tangent space
at this point). Finally, a family of ultralocal metrics is naturally defined in superspace [98].
Some of these properties are inherited by the spaces of symmetric geometries that we consider
here.
Other approaches to the quantization of GR, and in particular loop quantum gravity, rely on spaces of
connections rather than in spaces of metrics. Hence, in order to study symmetry reductions in these
frameworks, one should discuss the properties of such “connection superspaces” and then consider the
definition of symmetric connections and how they fit into these spaces. The technical treatment of the
spaces of Yang–Mills connections modulo gauge transformations has been developed in the late seventies by
Singer and other authors [198, 166]. These results have been used by Ashtekar, Lewandowski [11] and
others to give a description of the spaces of connections modulo gauge (encompassing diffeomorphisms) and
their extension to symmetry reductions have been explored by Bojowald [44] and collaborators as
a first step towards the study of symmetry reductions in LQG. These will be mentioned in
Section 5.
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Living Rev. Relativity 13, (2010), 6
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