### 5.7 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 and connection with real numbers and (where coordinate volume has been absorbed as described in Appendix B.1) which are conjugate to each other, , and internal directions as in isotropic models [48]. 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 from gauge invariant variables and (except for remaining discrete gauge transformations changing the signs of two of the and together). In a general homogenous connection, gauge-dependent and gauge-invariant parameters are mixed together in , which both react differently to a change in . This makes it more difficult to discuss the structure of relevant function spaces without assuming diagonalization.

As mentioned, the variables and are not completely gauge invariant since a gauge transformation can flip the sign of two components and while keeping the third fixed. There is thus a discrete gauge group left, and only the total sign 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 expanded in an orthonormal basis

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

Densitized triad components are quantized by

which directly give the volume operator with spectrum

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