5.1 Quantization of one-Killing vector reductions
The quantization of this class of midisuperspaces has been considered by some authors but it is fair to
say at this point that only a very partial knowledge has been achieved. A preliminary analysis within the
LQG framework (with complex variables) was carried out in [119]. The fact that these models can be
interpreted as 2+1 dimensional gravity coupled to some matter fields suggests that a quantization of this
lower dimensional system essentially solves the problem for one-Killing vector reductions. In fact, some
claims about the perturbative renormalizability of 2+1 gravity coupled to scalar fields have appeared
in the literature [47] although, in our opinion, it is rather unclear how these results can be
extrapolated to symmetric gravity in 3+1 dimensions. The reason for this scepticism is the non-trivial
structure of the Hamiltonian for these systems when the right asymptotic behavior is incorporated
(taking into account that, as shown by Ashtekar and Varadarajan in [15], the Hamiltonian that
corresponds to the generator of time translations at spatial infinity is actually bounded from
above). In any case, the physical consequences of these perturbative analyses have not been
explored.
