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5.15 Dynamics: General construction

Not all steps in the construction of the full constraint can be taken over immediately to a model since symmetry requirements have to be respected. It is thus important to have a more general construction scheme that shows how generic different steps are, and whether or not crucial input in a given symmetric situation is needed.

We have already observed one such issue, which is the appearance of holonomies and also simple exponentials of connection components without integration. This is a consequence of different transformation properties of different connection components in a reduced context. Components along remaining inhomogeneous directions, such as Ax for Einstein–Rosen waves, play the role of connection components in the model, giving rise to ordinary holonomies. Other components, such as Az and A ϕ in Einstein–Rosen waves or all components in homogeneous models, transform as scalars and thus only appear in exponentials without integration. In the overall picture, we have the full theory with only holonomies, homogeneous models with only exponentials, and inhomogeneous models in between where both holonomies and exponentials appear.

Another crucial issue is that of intrinsic curvature encoded in the spin connection. In the full theory, the spin connection does not have any covariant meaning and in fact can be made to vanish locally. In symmetric models, however, some spin connection components can become covariantly well defined, since not all coordinate transformations are allowed within a model. In isotropic models, for instance, the spin connection is simply given by a constant proportional to the curvature parameter. Of particular importance is the spin connection when one considers semiclassical regimes because intrinsic curvature does not need to become small there, in contrast to extrinsic curvature. Since the Ashtekar connection mixes the spin connection and extrinsic curvature, its semiclassical properties can be rather complicated in symmetric models.

The full constraint is based on holonomies around closed loops in order to approximate Ashtekar-curvature components when the loop becomes small in a continuum limit. For homogeneous directions, however, one cannot shrink the loop and instead must work with exponentials of the components. One thus approximates the classical components only when arguments of the exponential are small. If these arguments were always connection components, one would not obtain the right semiclassical properties because those components can remain large. Thus, one must base the construction for homogeneous directions in models on extrinsic curvature components, i.e., subtract off the spin connection from the Ashtekar connection. For inhomogeneous directions, on the other hand, this is not possible since one needs a connection in order to define a holonomy.

At first sight this procedure may seem rather ad hoc and even goes half a step back to ADM variables since extrinsic curvature components are used. However, there are several places where this procedure turns out to be necessary for a variety of independent reasons. We have already seen in Section 5.11 that inhomogeneous models can lead to a complicated volume operator when one insists on using all Ashtekar connection components. When one allows for extrinsic curvature components in the way just described, on the other hand, the volume operator becomes straightforward. This appeared after performing a canonical transformation, which rests non-trivially on the form of inhomogeneous spin connections and extrinsic curvature tensors.

Moreover, in addition to the semiclassical limit used above as justification, one also has to discuss local stability of the resulting evolution equation [79Jump To The Next Citation Point]: since higher-order difference equations have additional solutions, one must ensure that they do not become dominant in order not to spoil the continuum limit. This is satisfied with the above construction, while it is generically violated if one were to use only connection components.

There is thus a common construction scheme available based on holonomies and exponentials. As already discussed, this is responsible for correction terms in a continuum limit, but also gives rise to the constraint equation being a difference equation in a triad representation, whenever it exists. In homogeneous models the structure of the resulting difference equation is clear, but there are different open possibilities in inhomogeneous models. This is intimately related to the issue of anomalies, which also appears only in inhomogeneous models.

Sometimes it is possible to use implications of refinements of the underlying discreteness to suppress large spin-connection components, and thus retain all Ashtekar-connection components even in homogeneous directions. However, the only non-trivial model so far in which this has been constructed is the closed isotropic model. In other models it is more complicated to compute the required holonomies in terms of components of invariant connections, and it is also unclear whether or not specifics of a given model are used too much in such constructions. Fortunately, in the closed model the result does not differ essentially from what is obtained by the general construction scheme.

With a fixed choice, one has to solve a set of coupled difference equations for a wave function on superspace. The basic question then is always what kind of initial or boundary-value problem has to be used in order to ensure the existence of solutions with suitable properties, e.g., in a semiclassical regime. Once this is specified one can already discuss the singularity problem since one needs to find out if initial conditions in one semiclassical regime together with boundary conditions away from classical singularities suffice for a unique solution on all of superspace. A secondary question is how this equation can be interpreted as an evolution equation for the wave function in an internal time. This is not strictly necessary for all purposes and can be complicated owing to the problem of time in general. Nevertheless, when available, an evolution interpretation can be helpful for interpretations.

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