Generalizing the Kodama State II: Properties and Physical Interpretation

Abstract

In this second part of a two paper series we discuss the properties and physical interpretation of the generalized Kodama states. We first show that the states are the three dimensional boundary degrees of freedom of two familiar 4-dimensional topological invariants: the second Chern class and the Euler class. Using this, we show that the states have the familiar interpretation as WKB states, in this case corresponding not only to de Sitter space, but also to first order perturbations therein. In an appropriate spatial topology, the de Sitter solution has pure Chern-Simons functional form, and is the unique state in the class that is identically diffeomorphism and SU(2) gauge invariant. The q-deformed loop transform of this state yields evidence of a cosmological horizon when the deformation parameter is a root of untiy. We then discuss the behavior of the states under discrete symmetries, showing that the states violate $P$ and $T$ due to the presence of the Immirzi parameter, but they are $CPT$ invariant. We conclude with an interesting connection between the physical inner product and the Macdowell Mansouri formulation of gravity.