The linearization of the Kodama state

TL;DR

The paper analyzes whether the linearization of the Kodama state around de Sitter spacetime is normalizable with respect to the physical inner product of linearized gravity. By constructing the linearized theory in Ashtekar variables and solving the linearized constraints, it shows that the Lorentzian linearized Kodama state is not normalizable due to an indefinite quadratic form in each momentum mode, while the Euclidean linearized state is delta-functional normalizable since it is a pure phase. The results raise questions about the viability of the full Kodama state as a physical state for quantum gravity with a positive cosmological constant, suggesting that Euclidean formulations may fare better and that matter coupling or alternative inner products might influence the outcome. The work highlights the need for further study of the physical inner product, spin-foam projections, and the connection between full and linearized theories to assess the Kodama state's physical relevance.

Abstract

We study the question of whether the linearization of the Kodama state around classical deSitter spacetime is normalizable in the inner product of the theory of linearized gravitons on deSitter spacetime. We find the answer is no in the Lorentzian theory. However, in the Euclidean theory the corresponding linearized Kodama state is delta-functional normalizable. We discuss whether this result invalidates the conjecture that the full Kodama state is a good physical state for quantum gravity with positive cosmological constant.