The Hilbert space of quantum gravity

In a recent paper one of us (MC) suggested a new approach to the quantization of the gravitational field-a presentation of gravity as the Hilbert space of the quantized system. This approach is developed and extended in the present paper. States in the Hilbert space are defined in terms of the representation of the Weyl tensor in the Newman-Penrose formulation of general relativity. The determination of first-order states in perturbation theory is shown to reduce to a solution of a set of equations for the coefficients of the expansion of one complex function in the matrix elements of the irreducible representations of the group SU ₂. For exact (nonperturbative) solutions it is shown that in the stationary case each solution of the field equations corresponds to a unique state in the Hilbert space. When gravitational radiation is present, an additional parameter is introduced. The classification of states in the Hilbert space is given.