The Quantum Wheeler-DeWitt operator can be derived from an affine commutation relation via the affine group representation formalism for gravity, wherein a family of gauge-diffeomorphism invariant affine coherent states are constructed from a fiducia
l state. In this article, the role of the fiducial state is played by a regularized Gaussian peaked on densitized triad configurations corresponding to 3-metrics of constant spatial scalar curvature. The affine group manifold consists of points in the upper half plane, wherein each point is labeled by two local gravitational degrees of freedom from the Yamabe construction. From this viewpoint, here we show that the translational subgroup of affine coherent states constitute a set of exact solutions of the Wheeler-DeWitt equation. The affine translational parameter $b$ admits a physical interpretation analogous to a continuous plane wave energy spectrum, where the curvature constant $k$ plays the role of the energy. This result shows that the affine translational subgroup generates transformations in the curvature constant $k$ from the Yamabe problem, while $k$ is inert under the kinematic symmetries of gravity.
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-Simo
ns 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 $frac{Delta{V}}{<{V}>}Delta {Q}geq 2pi Lambda L^2_{Planck}$ (wherein $V$ is the total volume) may apply to all physical states of quantum gravity.
