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3.8 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 quantization as described above, when each loop α contains exactly one vertex of a given graph [288], 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 the problem of time or observables emerges as well [209]. There is a wild mixture of conceptual and technical problems at different levels, not 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. By systematic perturbation expansions around symmetric models, the crucial physical issues facing loop quantum gravity can be analyzed without restricting the number of degrees of freedom.


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