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7.4 Minisuperspace approximation

Most physical applications in quantum gravity are obtained in mini- or midisuperspace truncations by focusing only on degrees of freedom relevant for a given situation of interest. Other degrees of freedom and their interactions with the remaining ones are ignored so as to simplify the complicated full dynamics. Their role, in particular for the evolution, however, is not always clear, and so one should check what happens if they are gradually tuned in.

There are examples, in the spirit of [210], where minisuperspace results are markedly different from less symmetric ones. In those analyses, however, already the classical reduction is unstable, or classical backreaction is important, and thus solutions that start almost symmetric move away rapidly from the symmetric sub-manifold of the full phase space. The failure of a minisuperspace quantization in those cases can already be decided classically and is not a quantum gravity issue. Even a violation of uncertainty relations, which occurs in any reduction at the quantum level, is not automatically dangerous, but only if corresponding classical models are unstable.

As for the general approach to a classical singularity, the anisotropic behavior, and not so much inhomogeneities, is considered to be essential. Isotropy can indeed be misleading, but the anisotropic behavior is more characteristic. In fact, relevant features of full calculations on a single vertex [120Jump To The Next Citation Point] agree with the anisotropic [5682], but not the isotropic behavior [54]. Also, patching together homogeneous models to form an inhomogeneous space reproduces some full results even at a quantitative level [100Jump To The Next Citation Point]. The main differences and simplifications of models can be traced back to an effective Abelianization of the full SU(2)-gauge transformations, which is not introduced by hand in this case but by a consequence of symmetries. It is also one of the reasons why geometrical configurations in models are usually easier to interpret than in the full theory. Most importantly, it implies strong conceptual simplifications since it allows a triad representation in which the dynamics can be understood more intuitively than in a connection representation. Explicit results in models have thus been facilitated by this property of basic variables, and therefore a comparison with analogous situations in the full theory is most interesting in this context, and most important as a test of models.

If one is using a quantization of a classically-reduced system, it can only be considered a model for full quantum gravity. Relations between different models and the full theory are important in order to specify to what degree such models approximate the full situation, and where additional correction terms from the ignored degrees of freedom have to be taken into account. This is under systematic investigation in loop quantum cosmology.

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