### 3.7 Open issues

For an anomaly-free quantization the constraint operators have to satisfy an algebra mimicking the
classical one. There are arguments that this is the case for the quantization as described above when each
loop contains exactly one vertex of a given graph [191], but the issue is still open. Moreover, the
operators are quite complicated and it is not easy to see if they have the correct expectation values in
appropriately defined semiclassical states.
Even if one regards the quantization and semiclassical issues as satisfactory, one has to face several
hurdles in evaluating the theory. There are interpretational issues of the wave function obtained as a
solution to the constraints, and also the problem of time or observables emerges [143]. There is a wild
mixture of conceptual and technical problems at different levels, not at least because the operators are quite
complicated. For instance, as seen in the rewriting procedure above, the volume operator plays an
important role even if one is not necessarily interested in the volume of regions. Since this
operator is complicated, without an explicitly known spectrum, it translates to complicated matrix
elements of the constraints and matter Hamiltonians. Loop quantum gravity should thus be
considered as a framework rather than a uniquely defined theory, which however has important rigid
aspects. This includes the basic representation of the holonomy-flux algebra and its general
consequences.

All this should not come as a surprise since even classical gravity, at this level of generality, is
complicated enough. Most solutions and results in general relativity are obtained with approximations or
assumptions, one of the most widely used being symmetry reduction. In fact, this allows access to the most
interesting gravitational phenomena such as cosmological expansion, black holes and gravitational waves.
Similarly, symmetry reduction is expected to simplify many problems of full quantum gravity by resulting in
simpler operators and by isolating conceptual problems such that not all of them need to be considered at
once.