The nature of the classical canonical phase-space variables for gravitysuggests that the associated quantum field operators should obey affinecommutation relations rather than canonical commutation relations. Prior to theintroduction of constraints, a primary kinematical representation is derived inthe form of a reproducing kernel and its associated reproducing kernel Hilbertspace. Constraints are introduced following the projection operator methodwhich involves no gauge fixing, no complicated moduli space, nor any auxiliaryfields. The result, which is only qualitatively sketched in the present paper,involves another reproducing kernel with which inner products are defined forthe physical Hilbert space and which is obtained through a reduction of theoriginal reproducing kernel. Several of the steps involved in this generalanalysis are illustrated by means of analogous steps applied to one-dimensionalquantum mechanical models. These toy models help in motivating andunderstanding the analysis in the case of gravity.
展开▼
