### 4.8 Anisotropy: Connection variables

A densitized triad corresponding to a diagonal homogeneous metric has real components with
if [48]. Connection components are and are
conjugate to the , . In terms of triad variables we now have spin connection
components
and the Hamiltonian constraint (in the absence of matter)
Unlike in isotropic models, we now have inverse powers of even in the vacuum case through the spin
connection, unless we are in the Bianchi I model. This is a consequence of the fact that not just extrinsic
curvature, which in the isotropic case is related to the matter Hamiltonian through the Friedmann equation,
leads to divergences but also intrinsic curvature. These divergences are cut off by quantum geometry effects
as before such that also the dynamical behavior changes. This can again be dealt with by effective equations
where inverse powers of triad components are replaced by bounded functions [62]. However, even with those
modifications, expressions for curvature are not necessarily bounded unlike in the isotropic case.
This comes from the presence of different classical scales, , such that more complicated
expressions as in are possible, while in the isotropic model there is only one scale and
curvature can only be an inverse power of , which is then regulated by effective expressions like
.