Affine group representation formalism for four dimensional, Lorentzian, quantum gravity

TL;DR

This paper reformulates four-dimensional Lorentzian quantum gravity with a cosmological constant using affine group representations, enabling new solutions and insights into the structure of quantum states and constraints.

Contribution

It demonstrates the applicability of affine algebra quantization to Lorentzian quantum gravity with a non-zero cosmological constant, providing a new framework for constructing solutions.

Findings

Affine algebra quantization applies to Lorentzian quantum gravity with cosmological constant.

Unitary affine group representations yield a natural Hilbert space and physical states.

A fundamental uncertainty relation involving volume and Chern-Simons functional is proposed.

Abstract

Within the context of the Ashtekar variables, the Hamiltonian constraint of four-dimensional pure General Relativity with cosmological constant, $\Lambda$, is reexpressed as an affine algebra with the commutator of the imaginary part of the Chern-Simons functional, $Q$, and the positive-definite volume element. This demonstrates that the affine algebra quantization program of Klauder can indeed be applicable to the full Lorentzian signature theory of quantum gravity with non-vanishing cosmological constant; and it facilitates the construction of solutions to all of the constraints. Unitary, irreducible representations of the affine group exhibit a natural Hilbert space structure, and coherent states and other physical states can be generated from a fiducial state. It is also intriguing that formulation of the Hamiltonian constraint or Wheeler-DeWitt equation as an affine algebra requires a non-vanishing cosmological constant; and a fundamental uncertainty relation of the form $\frac{\Delta{V}}{<{V}>}\Delta {Q}\geq 2\pi \Lambda L^2_{Planck}$ (wherein $V$ is the total volume) may apply to all physical states of quantum gravity.