Noncanonical quantization of gravity. I. Foundations of affine quantum gravity

The nature of the classical canonical phase-space variables for gravity sug gests that the associated quantum field operators should obey affine commut ation relations rather than canonical commutation relations. Prior to the i ntroduction of constraints, a primary kinematical representation is derived in the form of a reproducing kernel and its associated reproducing kernel Hilbert space. Constraints are introduced following the projection operator method, which involves no gauge fixing, no complicated moduli space, nor a ny auxiliary fields. The result, which is only qualitatively sketched in th e present paper, involves another reproducing kernel with which inner produ cts are defined for the physical Hilbert space and which is obtained throug h a reduction of the original reproducing kernel. Several of the steps invo lved in this general analysis are illustrated by means of analogous steps a pplied to one-dimensional quantum mechanical models. These toy models help in motivating and understanding the analysis in the case of gravity. (C) 19 99 American Institute of Physics. [S0022-2488(99)02011-3].