One can restrict the ambiguities to some degree by modeling the expression on that of the full theory. This means that one does not simply replace by an almost periodic function, but uses holonomies tracing out closed loops formed by symmetry generators [50, 54]. Moreover, the procedure can be embedded in a general scheme that encompasses different models and the full theory [292, 50, 68], further reducing ambiguities. In particular models with non-zero intrinsic curvature on their symmetry orbits, such as the closed isotropic model, can then be included in the construction. (There are different ways to do this consistently, with essentially identical results; compare [111] with [28] for the closed model and [300] with [285] for .) One issue to keep in mind is the fact that “holonomies” are treated differently in models and the full theory. In the latter case they are ordinary holonomies along edges, which can be shrunk and then approximate connection components. In models, on the other hand, one sometimes uses direct exponentials of connection components without integration. In such a case, connection components are reproduced in the corrections only when they are small. Alternatively, a scale-dependent can provide the suppression even if connection components remain large in semiclassical regimes. The requirement that this happens in an acceptable way provides restrictions on refinement models, especially if one goes beyond isotropy. Selecting refinement models, on the other hand, leads to important feedback for the full theory. The difference between the two ways of dealing with holonomies can be understood in inhomogeneous models, where they are both realized for different connection components.

In the flat case the construction is easiest, related to the Abelian nature of the symmetry group. One can directly use the exponentials in (44), viewed as 3-dimensional holonomies along integral curves, and mimic the full constraint where one follows a loop to get curvature components of the connection . Respecting the symmetry, this can be done in the model with a square loop in two independent directions and . This yields the product , which appears in a trace as in Equation (15), together with a commutator , using the remaining direction . The latter, following the general scheme of the full theory reviewed in Section 3.6, quantizes the contribution to the constraint, instead of directly using the simpler .

Taking the trace one obtains a diagonal operator

in terms of the volume operator, as well as the multiplication operator

as the only term resulting from in . In the triad representation, where instead of working with functions one works with the coefficients in an expansion , this operator is the square of a difference operator. The constraint equation thus takes the form of a difference equation for the wave function , which can be viewed as an evolution equation in internal time . Thus, discrete spatial geometry implies a discrete internal time [51]. The equation above results in the most direct path from a non-symmetric constraint operator with gravitational part acting as Operators of this form are derived in [54, 16] for a spatially-flat model (), in [111, 28, 286] for positive spatial curvature (), and in [300, 285] for negative spatial curvature (), which is not included in the forms of Equations (53) and (54). Note, however, that not all these articles used the same quantization scheme and thus operators even for one model appear different in details, although the main properties are the same. Some of the differences of quantization schemes will be discussed below. The main option in alternative quantizations relates to a possible scale dependence , which we leave open at this stage.One can symmetrize this operator and obtain a difference equation with different coefficients, which we do here after multiplying the operator with for reasons that will be discussed in the context of singularities in Section 5.16. The resulting difference equation is

where .Since , the difference equation is of higher order, even formulated on an uncountable set, and thus has many independent solutions if is constant. Most of them, however, oscillate on small scales, i.e., between and with small integer . Others oscillate only on larger scales and can be viewed as approximating continuum solutions. For non-constant , we have a difference equation of non-constant step size, where it is more complicated to analyze the general form of solutions. (In isotropic models, however, such equations can always be transformed to equidistant form up to factor ordering [75].) As there are quantization choices, the behavior of all the solutions leads to possibilities for selection criteria of different versions of the constraint. Most importantly, one chooses the routing of edges to construct the square holonomy, again the spin of a representation to take the trace [170, 299], and factor-ordering choices between quantizations of and . All these choices also appear in the full theory, such that one can draw conclusions for preferred cases there.

http://www.livingreviews.org/lrr-2008-4 |
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |