Building a quantum spacetime

A quantum spacetime is constructed from the free data given at past null infinity. One starts with a field equation for a scalar function Z on the initial surface and then shows that the solution depends on four constants of integration. These constants become the spacetime points and the level surfaces of the scalar function, i.e., Z = const, become null hypersurfaces on the derived spacetime. A phase space together with a complex structure are constructed on past null infinity. This Hilbert space of incoming gravitons possesses a natural foliation which defines superselection sectors on the space of asymptotic quantum states. The dynamics of null surface quantization provides spacetime-valued quantum operators on the superselection sectors. It is shown that the spacetime points themselves become operators with nonvanishing commutation relations.