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7.5 Quantum geometry: from models to the full theory

By now, many models are available explicitly and can be compared with each other and the full theory. Original investigations were done in isotropic models, which in many respects are special, but important aspects of the loop quantization are now known to be realized in all models and sometimes the full theory, without contradictions so far. Thus there is a consistent picture of singularity-free dynamical behavior together with candidates for characteristic phenomenology.

There are certainly differences between models, which can be observed already for geometrical spectra such as area or volume. Akin to level splitting in atoms or molecules, spectra become more complicated when symmetry is relaxed [96Jump To The Next Citation Point10795]. In addition, the behavior of densities or curvatures on arbitrary geometrical configurations can be different in different models. In isotropic models, densities are bounded, which is a kinematical statement, but in this case important for a singularity-free evolution. It is important here since minisuperspace is just one-dimensional and so dynamical trajectories could not pass regions of unbounded curvature, should they exist. Anisotropic models are more characteristic for the approach to classical singularities, and here curvature expressions in general remain unbounded if all of minisuperspace is considered. Again, this is only kinematical, and here the dynamics tells us that evolution does not proceed along directions of unbounded curvature. This is similar in inhomogeneous models studied so far.

In the full theory the situation again becomes more complicated since here densities can be unbounded, even on degenerate configurations of vanishing-volume eigenvalue [120]. In this case, however, it is not known what the significance for evolution is, or even the geometrical meaning of the degenerate configurations.

As an analogy one can, as before, take the spectroscopy of atoms and level splitting. Essential properties, such as the stability of the hydrogen atom in quantum mechanics as opposed to the classical theory, are unchanged if complicated interactions are taken into account. It is important to observe, in this context, that stability can and does change if arbitrary interactions are considered, rather than realistic ones, which one has already fixed from other observations. Hydrogen then remains stable under those realistic interactions, but its properties would change drastically if any possible interaction term were considered. Similarly, it is not helpful to consider the behavior of densities on arbitrary geometries unless it is known which configurations are important for dynamics or at least their geometrical role is clear. Dynamics in the canonical picture is encoded in the Hamiltonian constraint, and including it (or suitable observables) in the analysis is analogous, in the picture of atomic spectra, to making use of realistic gravitational interaction terms. In the full theory, such an analysis is currently beyond reach, but it has been extensively studied in loop quantum cosmology. Since the non-singular behavior of models, whether or not curvature is bounded, is a consequence of basic effects and the representation derived from the full theory, it can be taken as reliable information on the behavior in quantum geometry.

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