Groupoid approach to noncommutative quantization of gravity

Abstract

We propose a new scheme for quantizing gravity based on a noncommutative geometry. Our geometry corresponds to a noncommutative algebra A=Gc∞(G,C) of smooth compactly supported complex functions (with convolution as multiplication) on the groupoid G=E◃Γ being the semidirect product of a structured space E of constant dimension (or a smooth manifold) and a group Γ. In the classical case E is the total space of the frame bundle and Γ is the Lorentz group. The differential geometry is developed in terms of a Z(A)-submodule V of derivations of A and a noncommutative counterpart of Einstein’s equation is defined. A pair (A,Ṽ), where Ṽ 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 Schrödinger’s equation (in Heisenberg’s picture) of quantum mechanics. Some interpretative questions are considered.