Because irrational numbers are always the result of calculations, never the result of direct measurement, might it not be possible in physics to abandon irrational numbers altogether and work only with the rational numbers? That is certainly possible, but it would be a revolutionary change. …
At some future time, when much more is known about space and time and the other magnitudes of physics, we may find that all of them are discrete.
An Introduction to the Philosophy of Science
So far we have mainly described the kinematical construction of symmetric models in loop quantum gravity up to the point where the Hamiltonian constraint appears. Since many dynamical issues in different models appear in a similar fashion, we discuss them in this section with a common background. The main feature is that dynamics is formulated by a difference equation, which by itself, compared to the usual appearance of differential equations, implies new properties of evolution. Depending on the model there are different classes, which even within a given model are subject to quantization choices. Yet, since there is a common construction procedure many characteristic features are very general.
Classically, curvature encodes the dynamics of geometry and does so in quantum gravity, too. On the other hand, quantum geometry is most intuitively understood in eigenstates of geometry, e.g., a triad representation if it exists, in which curvature is unsharp. Anyway, only solutions to the Hamiltonian constraint are relevant, which in general are peaked neither on spatial geometry nor on extrinsic curvature. The role of curvature thus has a different, less direct meaning in quantum gravity. Still, it is instructive to quantize classical expressions for curvature in special situations, such as in isotropy. Since the resulting operator is bounded, it has played an influential role in the development of statements regarding the fate of classical singularities.
However, one has to keep in mind that isotropy is a very special case, as emphasized before, and already anisotropic models shed quite a different light on curvature quantities . Isotropy is special because there is only one classical spatial length scale given by the scale factor , such that intrinsic curvature can only be a negative power, such as just for dimensional reasons. That quantum corrections to this expression are not obvious in this model or others is illustrated by comparing the intrinsic curvature term , which remains unchanged and thus unbounded in the purely isotropic quantization, with the term coming from a matter Hamiltonian, where the classical divergence of is cut off.
In an anisotropic model we do have different classical scales and thus dimensionally also terms like are possible. It is then not automatic that the quantization is bounded even if were to be bounded. As an example of such quantities consider the spatial curvature scalar given by with in Equation (40) through the spin-connection components. When quantized and then reduced to isotropy, one does obtain a cutoff to the intrinsic curvature term as mentioned in Section 4.12, but the anisotropic expression remains unbounded on minisuperspace. The limit to vanishing triad components is direction dependent and the isotropic case picks out a vanishing limit. However, in general this is not the limit taken along dynamical trajectories. Similarly, in the full theory one can show that inverse volume operators are not bounded even, in contrast to anisotropic models, in states where the volume eigenvalue vanishes . However, this is difficult to interpret since nothing is known about its relevance to dynamics; even the geometrical role of spin labels, and thus of the configurations considered, is unclear.
It is then quantum dynamics that is necessary to see what properties are relevant and how degenerate configurations are approached. This should allow one to check if the classical boundary a finite distance away is removed in quantum gravity. This can only happen if quantum gravity provides candidates for a region beyond the classical singularity, and means to probe how to evolve there. The most crucial aim is to prevent incompleteness of spacetime solutions or their quantum replacements. Even if curvature would be finite, by itself it would not be enough since one could not tell if the singularity persists as incompleteness. Only a demonstration of continuing evolution can ultimately show that singularities are absent.
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