  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 fiducial 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.
