Some notes on the Kodama state, maximal symmetry, and the isolated horizon boundary condition

TL;DR

This paper clarifies the nature of the Kodama state in loop quantum gravity, showing it corresponds to a maximally symmetric vacuum and discussing its relation to isolated horizon conditions.

Contribution

It provides a precise interpretation of the Kodama state for real connection variables and compares it with isolated horizon boundary conditions.

Findings

Kodama state can be meaningfully defined for real connection variables.

The state corresponds to a vacuum peaked on maximally symmetric geometry.

Isolated horizon boundary condition $F \,\propto\, \Sigma$ is inadequate for defining quantum horizons.

Abstract

We recall some well and some less 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 $F \propto \Sigma$ and stress that it is, in agreement with earlier work, inadequate to define the notion of a quantum horizon.