  We demonstrate that reality conditions for the Ashtekar connection imply a
non-trivial measure for the inner product of gravitational states in the
polarization where the Ashtekar connection is diagonal, and we express the
measure as the determinant of a certain first-order differential operator. This
result opens the possibility to perform a non-perturbative analysis of the
quantum gravity scalar product. In this polarization, the Chern-Simons-Kodama
state, which solves the constraints of quantum gravity for a certain factor
ordering, and which has de Sitter space as a semiclassical limit, is
perturbatively non-normalizable with respect to the naive ve inner product. Our
work reopens the question of whether this state might be normalizable when the
correct non-perturbative inner product and choice of integration contour are
taken into account. As a first step, we perform a semi-classical treatment of
the measure by evaluating it on the round three-sphere, viewed as a closed
spatial slice of de Sitter. The result is a simple, albeit divergent, infinite
product that might serve as a regulator for a more complete treatment of the
problem. Additionally, our results suggest deep connections between the problem
of computing the norm of the CSK state in quantum gravity and computing the
Chern-Simons partition function for a complex group.
