Generalized Kodama partition functions: A preview into normalizability for the generalized Kodama states

TL;DR

This paper computes the partition function for generalized Kodama states in quantum gravity, demonstrating its renormalizability and the existence of a well-defined effective action, with implications for link invariants and matter separation.

Contribution

It introduces a method to compute and analyze the partition function of GKod, showing its renormalizability and fixed vertices, advancing understanding of quantum gravity states.

Findings

Partition function for GKod is renormalizable as a loop expansion.

The GKod partition function contains an infinite set of fixed 1PI vertices.

The phase of GKod is equivalent to a well-defined effective action.

Abstract

In this paper we outline the computation of the partition function for the generalized Kodama states (GKod) of quantum gravity using the background field method. We show that the coupling constant for GKod is the same dimensionless coupling constant that appears in the partition function of the pure Kodama state (Chern--Simons functional) and argue that the GKod partition function is renormalizable as a loop expansion in direct analogy to Chern--Simons perturbation theory. The GKod partition function contains an infinite set of 1PI vertices uniquely fixed, as a result of the semiclassical-quantum correspondence, by the first-order vertex. This implies the existence of a well-defined effective action for the partition function since the `phase' of the GKod, provided a finite state exists, is equivalent to this effective action. Additionally, the separation of the matter from the gravitational contributions bears a resemblance to the infinite dimensional analogue to Kaluza--Klein theory. Future directions of research include extension of the computations of this paper to the norm of the GKod as well as to examine the analogue of the Chern--Simons Jone's polynomials and link invariants using the GKod as a measure.