On the canonical approach to quantum gravity

General relativity has the property that, under the natural projection mapping, the image of the constraint surface in the phase space is a proper subset of the configuration space. This feature is not shared by other field theories of direct physical interest such as the Yang-Mills theory, nor by constrained systems which have been modeled after general relativity and analyzed in detail. Therefore, to gain insight into this feature, a new example with a finite number of degrees of freedom is introduced and quantized. The analysis suggests that, in the canonical approach, the reduced phase-space method is likely to yield an incomplete description of quantum gravity. In particular, contrary to the indication provided by this method, quantum gravity may admit states with negative energies.