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5.8 Diagonalization

The situation simplifies if one considers diagonal models, which is usually also done in classical considerations since it does not lead to much loss of information. In a metric formulation, one requires the metric and its time derivative to be diagonal, which is equivalent to a homogeneous densitized triad pIi = p(I)ΛIi and connection φiI = c(I)ΛiI with real numbers cI and pI (where coordinate volume has been absorbed as described in Appendix B.1), which are conjugate to each other, {cI,pJ} = 8πγG δJI, and internal directions Λi I as in isotropic models [56Jump To The Next Citation Point]. In fact, the kinematics becomes similar to isotropic models, except that there are now three independent copies. The reason for the simplification is that we are able to separate off the gauge degrees of freedom in i Λ I from gauge-invariant variables cI and I p (except for remaining discrete gauge transformations changing the signs of two of the pI and cI together). In a general homogeneous connection, gauge-dependent and gauge-invariant parameters are mixed together in φi I, which both react differently to a change in μI. This makes it more difficult to discuss the structure of relevant function spaces without assuming diagonalization.

As mentioned, the variables pI and cI are not completely gauge invariant since a gauge transformation can flip the sign of two components pI and cI while keeping the third fixed. There is thus a discrete gauge group left, and only the total sign 1 2 3 sign(p p p ) is gauge invariant in addition to the absolute values.

Quantization can now proceed simply by using as Hilbert space the triple product of the isotropic Hilbert space, given by square-integrable functions on the Bohr compactification of the real line. This results in states |ψ ⟩ = ∑ ψ |μ ,μ ,μ ⟩ μ1,μ2,μ3 μ1,μ2,μ3 1 2 3 expanded in an orthonormal basis

i(μ c+μ c +μ c)∕2 ⟨c1,c2,c3|μ1,μ2,μ3 ⟩ = e 11 2 2 33 .

Gauge invariance under discrete gauge transformations requires ψμ1,μ2,μ3 to be symmetric under a flip of two signs in μI. Without loss of generality one can thus assume that ψ is defined for all real μ3 but only non-negative μ 1 and μ 2.

Densitized triad components are quantized by

I 1- 2 ˆp |μ1, μ2,μ3⟩ = 2 μIγℓP|μ1,μ2,μ3 ⟩,

which give the volume operator ∘ -------- ˆV = |ˆp1pˆ2pˆ3| directly with spectrum

∘ --------- V μ1,μ2,μ3 = (1γℓ2P)3∕2 |μ1μ2μ3|. 2

Moreover, after dividing out the remaining discrete gauge freedom the only independent sign in the triad components is given by the orientation 1 2 3 sign (ˆp ˆp ˆp ), which again leads to a doubling of the metric minisuperspace with a degenerate subset in the interior, where one of the I p vanishes.

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