A difference to homogeneous models, however, is that the internal directions of a connection and a triad do not need to be identical, which in homogeneous models with internal directions is the case as a consequence of the Gauss constraint . With inhomogeneous fields, now, the Gauss constraint reads
As a consequence, is not conjugate to anymore, and instead the momentum of is . This seems to make quantization more complicated since the momenta will be quantized to simple flux operators, but do not directly determine the geometry, such as the volume . For this, one would need to know the angle , which depends on both connections and triads. Moreover, it would not be obvious how to obtain a discrete volume spectrum since volume does not depend only on fluxes.
It turns out that there is a simple canonical transformation, which allows one to work with canonical variables and playing the role of momenta of and [109, 34, 35]. This seems to be undesirable, too, since now the connection variables that play an important role for holonomies are modified. That these canonical variables are very natural, however, follows after one considers the structure of spin connections and extrinsic curvature tensors in this model. The new canonical variables are then simply given by , , i.e., proportional to extrinsic curvature components. Thus, in the inhomogeneous model we simply replace connection components with extrinsic curvature in homogeneous directions (note that remains unchanged) while momenta remain elementary triad components. This is part of a broader scheme, which is also important for the Hamiltonian constraint operator (Section 5.15).
With the polarization condition the kinematics of the quantum theory simplify. Relevant holonomies are given by along edges in the one-dimensional manifold and
in vertices with real . Cylindrical functions depend on finitely many of these holonomies, whose edges and vertices form a graph in the one-dimensional manifold. Flux operators, i.e., quantized triad components, act simply by
Since triad components now have simple quantizations, one can directly combine them to get the volume operator and its spectrum
Commutators between holonomies and the volume operator will technically be similar to homogeneous models, except that there are more possibilities to combine different edges. Accordingly, one can easily compute all matrix elements of composite operators such as the Hamiltonian constraint. The result is only more cumbersome because there are more terms to keep track of. Again as in diagonal homogeneous cases, the triad representation exists and one can formulate the constraint equation there. Now, however, one has infinitely many coupled difference equations for the wave function since the lapse function is inhomogeneous, providing one difference equation for each vertex. With refinement, only and can possibly be scale dependent but not , which gives the representation for a holonomy along an inhomogeneous direction.
There are obvious differences to cases considered previously owing to inhomogeneity. For instance, each edge label can take positive or negative values, or go through zero during evolution, corresponding to the fact that a spatial slice does not need to intersect the classical singularity everywhere. Also the structure of coefficients of the difference equations, though qualitatively similar to homogeneous models, is changed crucially in inhomogeneous models, mainly due to the volume eigenvalues (70). Now, a single edge label , say, and thus can be zero without volume eigenvalues in neighboring vertices having zero volume.
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