We recall some well- and some lesser known results about the Kodama state and the related $\theta$ ambiguity in defining canonical variables. Based on them, we make some comments highlighting that the Kodama state for real connection variables can be given a precise meaning and that it implements a vacuum peaked on a (in a suitable sense) maximally symmetric geometry. We also highlight the similarity of this construction with the isolated horizon boundary condition $F\propto \mathrm{\Sigma }$ and stress that it is, in agreement with earlier work, inadequate to define the notion of a quantum horizon.

• Received 25 March 2016

DOI:https://doi.org/10.1103/PhysRevD.93.124042