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5.6 Homogeneity

A Hamiltonian formulation is available for all homogeneous models of Bianchi class A [111], which have structure constants CIJK fulfilling CIJI = 0. The structure constants also determine left-invariant 1-forms wI in terms of which one can write a homogeneous connection as Ai = ~fiwI a I a (see Appendix B.1) where all freedom is contained in the x-independent ~ i f I. A homogeneous densitized triad can be written in a dual form with coefficients I ~pi conjugate to i ~fI. As in isotropic models, one absorbs powers of the coordinate volume to obtain variables fiI and pI i.

The kinematics is the same for all class A models, except possibly for slight differences in the diffeomorphism constraint [2536Jump To The Next Citation Point]. Connection components define a distinguished triple of su(2) elements f~iIti, one for each independent direction of space. Holonomies in those directions are then obtained as h(mI)= exp(mIfiIti) (- SU(2) I with parameters mI for the edge lengths. Cylindrical functions depend on those holonomies, i.e., are countable superpositions of terms (m1) (m2) (m3) f(h1 ,h 2 ,h3 ). A basis can be written down as spin network states

(m1) (m2) (m3) (m1) A (m2) A (m3) A B B B f (h1 ,h2 ,h 3 ) = rj1(h 1 )B11rj2(h 2 )B22rj3(h 3 )B33K A11A22A33

where the matrix K specifies how the representation matrices are contracted to a gauge invariant function of fi I. There are uncountably many such states for different mI 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 K 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 jIdI. On the other hand, there is still a degeneracy of spin and edge length and keeping both jI and dI 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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