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.
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