A systematic approach to the solution of the constraints of quantum gravity: The full theory

Abstract

This is the third paper in a series outlining an algorithm to construct finite states of quantum gravity in Ashtekar variables. In this paper we treat the case of the Klein--Gordon field quantized with gravity on the same footing. We address the full theory, outlining the solution to the constraints and the construction of the corresponding wavefunction of the universe. The basic method for the full theory is to expand the constraints relative to the solution for the pure Kodama state and rewrite them in the form of a generalized nonlinear group transformation of the CDJ matrix, viewed as a nine-dimensional vector. We then outline a prescription for finding the fixed point of the flow, and the corresponding generalized Kodama state for the full theory is constructed. The final solution is expressed in an asymptotic series in powers of model-specific matter inputs, suppressed by a small dimensionless constant, relative to the pure Kodama state. We discuss this expansion from different perspectives. Lastly, we explicitly show how the solution to the quantized constraints establishes a wavefunction of the universe with a predetermined semiclassical limit built in as a boundary condition on quantized gravity in the full theory.