Finite states in four dimensional quantized gravity

TL;DR

This paper introduces an algorithm to explicitly construct finite quantum states in four-dimensional quantum gravity using Ashtekar variables, extending the Kodama state to matter-coupled models and exploring their properties.

Contribution

It presents a novel algorithm for constructing finite quantum states in full gravity, generalizes the Kodama state to include matter, and introduces the semiclassical-quantum correspondence concept.

Findings

Generalized Kodama states include matter fields

Demonstrates equivalence of quantization procedures for these states

Provides a framework for nonperturbative quantum gravity states

Abstract

This is the first in a series of papers outlining an algorithm to explicitly construct finite quantum states of the full theory of gravity in Ashtekar variables. The algorithm is based upon extending some properties of a special state, the Kodama state for pure gravity with cosmological term, to matter-coupled models. We then illustrate a presciption for nonperturbatively constructing the generalized Kodama states, in preparation for subsequent works in this series. We also introduce the concept of the semiclassical-quantum correspondence (SQC). We express the quantum constraints of the full theory as a system of equations to be solved for the constituents of the `phase' of the wavefunction. Additionally, we provide a variety of representations of the generalized Kodama states including a generalization of the topological instanton term to include matter fields, for which we present arguments for the field-theoretical analogue of cohomology on infinite dimensional spaces. We demonstrate that the Dirac, reduced phase space and geometric quantization procedures are all equivalent for these generalized Kodama states as a natural consequence of the SQC. We relegate the method of the solution to the constraints and other associated ramifications of the generalized Kodama states to separate works.