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5.16 Initial/boundary value problems

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. One thus 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 ym defined on an uncountable set. In general, one thus 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 [202]. Similar restrictions can come from semiclassical properties or the physical inner product [162Jump To The Next Citation Point], 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 [48Jump To The Next Citation Point]. Boundary conditions are imposed only at non-singular parts of minisuperspace such as m = 0 in LRS models (51View Equation). They must not be imposed at places of classical singularities, of course, where instead the evolution must continue just as 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 of a similar type as 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 how 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.

So far, the evolution operator in inhomogeneous models has not been studied in detail, and solutions in this case remain poorly understood. The difficulty of this issue can be illustrated by the expectations in spherical symmetry where 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 5.18. This usually arises at the place of classical singularities where the order of the difference equation can change. Since the quantum behavior at classical singularities is important here, the number of solutions can be different from the classically expected freedom, even when combined with possible semiclassical requirements far away from the singularity. We will now first discuss these requirements in semiclassical regimes, followed by more information on possibly arising additional conditions for solutions.

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