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.
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