### 5.7 Homogeneity

A Hamiltonian formulation is available for all homogeneous models of Bianchi class A [161], which have
structure constants fulfilling . The structure constants also determine left-invariant
1-forms in terms of which one can write a homogeneous connection as (see
Appendix B.1), where all freedom is contained in the -independent . A homogeneous
densitized triad can be written in a dual form with coefficients conjugate to . As in
isotropic models, one absorbs powers of the coordinate volume to obtain variables and
.
The kinematics is the same for all class A models, except possibly for slight differences in the
diffeomorphism constraint [29, 44]. 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 spin and
edge length both appear similarly as parameters. Spins are important to specify the contraction
and thus enter, e.g., 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; keeping both and independent leaves too many parameters.
Therefore, it is more difficult to determine what the analog of the Bohr compactification is in this
case.