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.
© Max Planck Society and the author(s)