Il n’est rien de plus précieux que le temps, puisque c’est le prix de l’éternité.

(There is nothing more precious than time, for it is the price of eternity.)

LOUIS BOURDALOUE

Sermon sur la perte de temps

In the classical situation, we always have trajectories on superspace running into singular submanifolds where some or all densitized triad components vanish. In semiclassical regimes one can think of physical solutions as wave packets following these trajectories in internal time, but at smaller triad components spreading and deformations from a Gaussian become stronger. Moreover, discreteness becomes essential and properties of difference equations need to be taken into account in order to see what is happening at the singular submanifolds.

The simplest situation is given by isotropic models where superspace is one dimensional with coordinate . Minisuperspace is thus disconnected classically with two sides separated by the classical singularity at . At this point, classical energy densities diverge and there is no well-defined initial value problem to evolve further. (Sometimes, formal extensions of solutions beyond a classical singularity exist [122], but they are never unique and unrelated to the solution preceding the singularity. This shows that a resolution of singularities has not only to provide a new region, but also an evolution there uniquely from initial values at one side.) A Wheeler-DeWitt quantization would similarly lead to diverging matter Hamiltonian operators and the initial value problem for the wave function generically breaks down. In isotropic loop quantum cosmology we have already seen that the matter Hamiltonian does not have diverging contributions from inverse metric components even at the classical singularity. Nevertheless, the evolution could break down if highest order coefficients in the difference equation become zero. This indeed happens with the non-symmetric constraint (49) or (51), but in these cases can be seen not to lead to any problems: some coefficients can become zero such that the wave function at remains undetermined by initial conditions, but the wave function at the other side of the classical singularity is still determined uniquely. There is no breakdown of evolution, and thus no singularity [39]. As one can see, this relies on crucial properties of the loop representation with well-defined inverse metric components and a difference rather than differential equation [43].

Also the structure of difference equations is important, depending on some choices. Most important is the factor ordering or symmetrization chosen. As just discussed, the ordering used earlier leads to non-singular evolution but with the wave function at the classical singularity itself remaining undetermined. In anisotropic models one can symmetrize the constraint and obtain a difference equation, such as (52), whose leading order coefficients never vanish. Evolution then never stops and even the wave function at the classical singularity is determined. In the isotropic case, direct symmetrization would lead to a break-down of evolution, which thus provides an example for singular quantum evolution and demonstrates the non-triviality of continuing evolution: The leading order coefficient would then be , which vanishes if and only if . Thus, in the backward evolution remains undetermined, just as is undetermined in the non-symmetric ordering. However, now would be needed to evolve further. Since it is not determined by initial data, one would need to prescribe this value, or else the evolution stops. There is thus a new region at negative , but evolution does not continue uniquely between the two sides. In such a case, even though curvature is bounded, the quantum system would be singular. Similar behavior happens in other orderings such as when triads are ordered to the left. Note that also in the full theory one cannot order triads to the left since otherwise the constraint would not be densely defined [194].

The breakdown of the symmetric ordering in isotropic models is special and related to the fact that all directions degenerate. The breakdown does not happen for a symmetric ordering in anisotropic or even inhomogeneous systems. One can avert it in isotropic cases by multiplying the constraint with before symmetrizing, so that the additional factor of leads to non-zero coefficients as in (50).

This is the general scheme, which also applies in more complicated cases. The prime example for the general homogeneous behavior is given by the Kasner evolution of the Bianchi I model. Here, the approach to the singularity is not isotropic but given in such a way that two of the three diagonal metric components become zero while the third one diverges. This would lead to a different picture than described before since the classical singularity then lies at the infinite boundary of metric or co-triad minisuperspace. Also unlike in the isotropic case, densities or curvature potentials are not necessarily bounded in general as functions on minisuperspace, and the classical dynamical approach is important. In densitized triad variables, however, we have a situation as before since here all components approach zero, although at different rates. Now the classical singularity is in the interior of minisuperspace and one can study the evolution as before, again right through the classical singularity. Note that densitized triad variables were required for a background independent quantization, and now independently for non-singular evolution.

Other homogeneous models are more complicated since for them Kasner motion takes place with a potential given by curvature components. Approximate Kasner epochs arise when the potential is negligible, intermitted by reflections at the potential walls where the direction of Kasner motion in the anisotropy plane changes. Still, since in each Kasner epoch the densitized triad components decrease, the classical singularity remains in the interior and is penetrated by the discrete quantum evolution.

One can use this for indications as to the general inhomogeneous behavior by making use of the BKL scenario. If this can be justified, in each spatial point the evolution of geometry is given by a homogeneous model. For the quantum formulation this indicates that also here classical singularities are removed. However, it is by no means clear whether the BKL scenario applies at the quantum level since even classically it is not generally established. If the scenario is not realized (or if some matter systems can change the local behavior), diverging are possible and the behavior would qualitatively be very different. One thus has to study the inhomogeneous quantum evolution directly as done before for homogeneous cases.

In the 1-dimensional models described here classical singularities arise when becomes zero. Since this is now a field, it depends on the point on the spatial manifold where the slice hits the classical singularity. At each such place, midisuperspace opens up to a new region not reached by the classical evolution, where the sign of changes and thus the local orientation of the triad. Again, the kinematics automatically provides us with these new regions just as needed, and quantum evolution continues. Also, the scheme is realized much more non-trivially, and now even the non-symmetric ordering is not allowed. This is a consequence of the fact that for a single edge label does not imply that neighboring volume eigenvalues vanish. There is thus no obvious decoupling in a non-singular manner, and it shows how less symmetric situations put more stringent restrictions on the allowed dynamics. Still, the availability of other possibilities, maybe with leading coefficients which can vanish and result in decoupling, needs to be analyzed. Most importantly, the symmetric version still leads to non-singular evolution even in those inhomogeneous cases which have local gravitational degrees of freedom [57].

There is thus a general scheme for the removal of singularities: In the classical situation, one has singular boundaries of superspace which cannot be penetrated. Densitized triad variables then lead to new regions, given by a change in the orientation factor which, however, does not help classically since singularities remain as interior boundaries. For the quantum situation one has to look at the constraint equation and see whether or not it uniquely allows to continue a wave function to the other side (which does not require time parameters even though they may be helpful if available). This usually depends on factor ordering and other choices that arise in the construction of constraint operators and play a role also for the anomaly issue. One can thus fix ambiguities by selecting a non-singular constraint if possible. However, the existence of non-singular versions, as realized in a natural fashion in homogeneous models, is a highly non-trivial and by no means automatic property of the theory showing its overall consistency.

In inhomogeneous models the issue is more complicated. We thus have a situation where the theory, which so far is well-defined, can be tested by trying to extend results to more general cases. It should also be noted that different models should not require different quantization choices unless symmetry itself is clearly responsible (as happens with the orientation factor in the symmetric ordering for an isotropic model, or when non-zero spin connection components receive covariant meaning in models), but that there should rather be a common scheme leading to non-singular behavior. This puts further strong conditions on the construction, and is possible only if one knows how models and the full theory are related.

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