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5.7 Homogeneity

A Hamiltonian formulation is available for all homogeneous models of Bianchi class A [161], which have structure constants CIJK fulfilling CIJI = 0. The structure constants also determine left-invariant 1-forms ωI in terms of which one can write a homogeneous connection as Ai = &tidle;φiωI a I a (see Appendix B.1), where all freedom is contained in the x-independent &tidle;i φI. A homogeneous densitized triad can be written in a dual form with coefficients I &tidle;pi conjugate to i &tidle;φI. As in isotropic models, one absorbs powers of the coordinate volume to obtain variables φiI and pI i.

The kinematics is the same for all class A models, except possibly for slight differences in the diffeomorphism constraint [2944Jump To The Next Citation Point]. Connection components define a distinguished triple of SU(2) elements φ&tidle;iIτi, one for each independent direction of space. Holonomies in those directions are then obtained as h (μI)= exp(μIφiIτi) ∈ SU (2) I with parameters μI for the edge lengths. Cylindrical functions depend on those holonomies, i.e., are countable superpositions of terms (μ1) (μ2) (μ3) f(h1 ,h 2 ,h3 ). A basis can be written down as spin network states

(μ1) (μ2) (μ3) (μ1) A (μ2) A (μ3) A B B B f (h1 ,h2 ,h 3 ) = ρj1(h 1 )B11ρj2(h 2 )B22ρj3(h 3 )B33K A11A22A33

where the matrix K specifies how the representation matrices are contracted to a gauge invariant function of φi I. There are uncountably many such states for different μI 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 K 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 jIδI. On the other hand, there is still a degeneracy of spin and edge length; keeping both jI and δI independent leaves too many parameters. Therefore, it is more difficult to determine what the analog of the Bohr compactification is in this case.


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