Groupoid approach to noncommutative quantization of gravity

We propose a new scheme for quantizing gravity based on a noncommutative geometry. Our geometry corresponds to a noncommutative algebra A = G(c)(proportional to) (G,C) of smooth compactly supported complex functions (with convolution as multiplication) on the groupoid G=E triangle Gamma being the semidirect product of a structured space E of constant dimension (or a smooth manifold) and a group Gamma. In the classical case E is the total space of the frame bundle and Gamma is the Lorentz group. The differential geometry is developed in terms of a L(A)-submodule V of derivations of A and a noncommutative counterpart of Einstein's equation is defined. A pair (A, (V) over tilde), where (V) over tilde is a subset of derivations of A satisfying the noncommutative Einstein's equation, is called an Einstein pair. We introduce the representation of A in a suitable Hilbert space, by completing A with respect to the corresponding norm change it into a C*-algebra, and perform quantization with the help of the standard C*-algebraic method. Hermitian elements of this algebra are interpreted as quantum gravity observables. We introduce dynamical equation of quantum gravity which, together with the noncommutative counterpart of Einstein's equation. forms a noncommutative dynamical system. For a weak gravitational field this dynamical system splits into ordinary Einstein's equation of general relativity and Schrodinger's equation (in Heisenberg's picture of quantum mechanics. Some interpretative questions are considered. (C) 1997 American Institute of Physics.