  In this article, I present a volume average regularization for the second
functional derivative operator that appears in the metric-basis Wheeler-DeWitt
equation. Naively, the second functional derivative operator in the
Wheeler-DeWitt equation is infinite, since it contains terms with a factor of a
delta function or derivatives of the delta function. More precisely, the second
functional derivative contains terms that are only well defined as a
distribution---these terms only yield meaningful results when they appear
within an integral. The second functional derivative may, therefore, be
regularized by performing an integral average of the distributional terms over
some finite volume; I argue that such a regularization is appropriate if one
regards quantum general relativity (from which the Wheeler-DeWitt equation may
be derived) to be the low-energy effective field theory of a full theory of
quantum gravity. I also show that a volume average regularization can be viewed
as a natural generalization of the same-variable second partial derivative for
an ordinary multivariable function. Using the regularized second functional
derivative operator, I construct an approximate solution to the Wheeler-DeWitt
equation in the low-curvature, long-distance limit.

General Relativity and Quantum Cosmology