  Affine quantum gravity involves (i) affine commutation relations to ensure
metric positivity, (ii) a regularized projection operator procedure to
accomodate first- and second-class quantum constraints, and (iii) a hard-core
interpretation of nonlinear interactions to understand and potentially overcome
nonrenormalizability. In this program, some of the less traditional
mathematical methods employed are (i) coherent state representations, (ii)
reproducing kernel Hilbert spaces, and (iii) functional integral
representations involving a continuous-time regularization. Of special
importance is the profoundly different integration measure used for the
Lagrange multiplier (shift and lapse) functions. These various concepts are
first introduced on elementary systems to help motivate their application to
affine quantum gravity.
