Manto: “Ich harre, um mich kreist die Zeit.”

(Manto: “I stand still, around me circles time.”)

Goethe

Faust

In isotropic models the gravitational part of the constraint corresponds to an ordinary difference operator, which can be interpreted as generating evolution in internal time. Thus one needs to specify only initial conditions to solve the equation. The number of conditions is large since, first, the procedure to construct the constraint operator usually results in higher-order equations and, second, this equation relates values of a wave function defined on an uncountable set. (The number of solutions for non-equidistant difference equations as they arise from refinement models has not been studied in detail and remains poorly understood.) In general, one has to choose a function on a real interval unless further conditions are used.

This can be achieved, for instance, by using observables that can reduce the kinematical framework back to wave functions defined on a countable discrete lattice [302]. Similar restrictions can come from semiclassical properties or the physical inner product [243], all of which has not yet been studied in generality.

The situation in homogeneous models is similar, but now one has several gravitational degrees of freedom only one of which is interpreted as internal time. One has a partial difference equation for a wave function on a minisuperspace with boundary, and initial as well as boundary conditions are required [56]. Boundary conditions are imposed only at nonsingular parts of minisuperspace, such as in LRS models (57). They must not be imposed at places of classical singularities, of course, where instead the evolution must continue just as it does at any regular part.

In inhomogeneous models, then, there are not only many independent kinematical variables but also many difference equations for only one wave function on midisuperspace. These difference equations are similar to those in homogeneous models, but they are coupled in complicated ways. Since one has several choices in the general construction of the constraint, there are different possibilities for the way difference equations arise and are coupled. Not all of them are expected to be consistent, i.e., in many cases some of the difference equations will not be compatible such that there would be no non-zero solution at all. This is related to the anomaly issue since the commutation behavior of difference operators is important for properties and the existence of common solutions.

The difference equations of loop quantum cosmology thus provide a general existence problem in analogy to the classical initial-value problem of general relativity. It plays a direct role in the singularity issue, which can be cast in the form of quantum hyperbolicity [71]: if a wave function on all of minisuperspace is completely determined by initial and boundary values, it can be evolved, in particular, through a classical singularity, which thus does not pose a boundary to quantum evolution. Here, special properties of difference as opposed to differential equations play an important role: “mantic” states such as in isotropic models can occur, whose coefficient in a state drops out of the recurrence relation instead of causing its breakdown. This formulation of the singularity issue is different from the usual one in the classical theory, making use of geodesic completeness. But it is closely related to the alternative classical formulation of generalized hyperbolicity [129, 304, 305] according to which a spacetime is nonsingular if it gives rise to well-posed evolution for field theories. By avoiding the use of geodesics and directly using potentially physical field theories, such a formulation is more suitable to be compared to quantum theories such as loop quantum cosmology with its principle of quantum hyperbolicity.

So far, the evolution operator in inhomogeneous models has not been studied in detail, and solutions in this case remain poorly understood. Thus, information on the quantum hyperbolicity problem is far from complete. The difficulty of this issue can be illustrated by the expectations in spherical symmetry, in which there is only one classical physical degree of freedom. If this is to be reproduced for semiclassical solutions of the quantum constraint, there must be a subtle elimination of infinitely many kinematical degrees of freedom such that in the end only one physical degree of freedom remains. Thus, from the many parameters needed in general to specify a solution to a set of difference equations, only one can remain when compatibility relations between the coupled difference equations and semiclassicality conditions are taken into account.

How much this cancellation depends on semiclassicality and asymptotic infinity conditions remains to be seen. Some influence is to be expected since classical behavior should have a bearing on the correct reproduction of classical degrees of freedom. However, it may also turn out that the number of solutions to the quantum constraint is more sensitive to quantum effects. It is already known from isotropic models that the constraint equation can imply additional conditions for solutions beyond the higher-order difference equation, as we will discuss in Section 5.19. This usually arises at the location of classical singularities where the order of the difference equation can change. Since quantum behavior at classical singularities is important for all these issues, the number of solutions can be different from the classically-expected freedom, even when combined with possible semiclassical requirements far away from the singularity.

A further illustration of the importance of such conditions is provided by the Schwarzschild interior, an anisotropic model [14]. As observed in [123], conditions on the wave function arising at the classical singularity imply that any physical solution is symmetric under orientation reversal (which is not guaranteed by other properties of the theory). Since the orientation changes during singularity traversal in loop quantum cosmology, this implies that the states before and after the Schwarzschild singularity are identical to each other, just as it is expected for a solution to be matched to a static exterior geometry. When there is collapsing matter, such a symmetry is no longer expected and the anisotropic but homogeneous model used to describe the Schwarzschild interior can no longer be used. In such a situation one would have to use spherically-symmetric states, which have a more complicated behavior. These relations provide interesting indications for an overall consistent general behavior.

We will now first discuss requirements for semiclassical regimes, followed by more information on possibly-arising additional conditions for solutions given in Section 5.19.

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