### 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
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 [56]. 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 homogeneous 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 give the volume operator directly with spectrum

Moreover, after dividing out the remaining discrete gauge freedom the only independent sign in the
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.