\[q_{c[a} q_{b]d} = \beta \theta F_{abcd} = \beta \theta R^{(3)}_{ab cd} - 2 \beta^2 \theta \epsilon_{cde} \sqrt{q} \nabla_{[a} K_{b]} {}^e - 2 \beta^3 \theta K_{c[a} K_{b]d} \]

with $\beta$ being the Barbero-Immirzi parameter. In case of a vanishing extrinsic curvature $K_{ab}$, this condition is nothing else than maximal symmetry for the spatial geometry. A vacuum based on variables with $\theta \neq 0$ is thus very interesting from the point of view of constructing semi-classical states, since it is peaked on a non-degenerate homogeneous geometry. $0 < \theta < \infty$ thus seems to interpolate between the Ashtekar-Lewandowski vacuum ($\theta=0$) and the recently introduced Dittrich-Geiller vacuum, which implements $F = 0$.

People familiar with isolated horizons and their application in LQG will also notice the strong similarity of the condition $P^{ai}=0$ with the isolated horizon boundary condition which is imposed in the black hole entropy computation in LQG (a pullback of $P^{ai} = 0$ to a 2-surface with $\theta$ depending on the total black hole area). It was shown before that computations along the original lines are actually applicable to general boundaries. Here, we also see that trying to impose the isolated horizon boundary condition directly on the full LQG Hilbert space is only enforcing a part of maximal symmetry condition, as opposed to selecting horizons (in contrast to what is often suggested). Thus, also less mainstream works along these lines, while very interesting, do not provide suitable definitions of quantum horizons, but rather quantum symmetric surfaces.