The kinematics is the same for all class A models, except possibly for slight differences in the diffeomorphism constraint [25, 36]. Connection components define a distinguished triple of su(2) elements , one for each independent direction of space. Holonomies in those directions are then obtained as with parameters for the edge lengths. Cylindrical functions depend on those holonomies, i.e., are countable superpositions of terms . A basis can be written down as spin network states
where the matrix specifies how the representation matrices are contracted to a gauge invariant function of . There are uncountably many such states for different and thus the Hilbert space is non-separable. In contrast to isotropic models, the general homogeneous theory is genuinely SU(2) and therefore not much simpler than the full theory for individual calculations.
As a consequence of homogeneity we observe the same degeneracy as in isotropic models where both spin and edge length appear similarly as parameters. Spins are important to specify the contraction and thus appear, e.g., in the volume spectrum. For this one needs to know the spins, and it is not sufficient to consider only products . On the other hand, there is still a degeneracy of spin and edge length and keeping both and independent leaves too many parameters. It is therefore more difficult to determine what the analog of the Bohr compactification is in this case.
© Max Planck Society and the author(s)