Instanton representation of Plebanski gravity: XVIII. Quantization and proposed resolution of the Kodama state

TL;DR

This paper develops a quantum framework for Plebanski gravity using the instanton representation, providing explicit solutions to the constraints and addressing the long-standing issue of the Kodama state's normalizability.

Contribution

It constructs a Hilbert space of quantum states for Plebanski gravity, solves the Hamiltonian constraint explicitly, and proposes a resolution to the Kodama state's normalizability problem.

Findings

Explicit Hilbert space of states solving GR constraints

Closed-form solutions to the Hamiltonian constraint

Resolution of the Kodama state's normalizability issue

Abstract

In this paper we have constructed a Hilbert space of states solving the initial value constraints of GR in the instanton representation of Plebanski gravity. The states are labelled by two free functions of position constructed from the eigenvalues of the CDJ matrix. This comprises the physical degrees of freedom of GR with a semiclassical limit corresponding to spacetimes of Petrov Type I, D and O. The Hamiltonian constraint in this representation is a hypergeometric differential equation on the states, for which we have provided a closed form solution. Additionally, we have clarified the role of the Kodama state within this Hilbert space structure, which provides a resolution to the issue of its normalizability raised by various authors.