A possible mathematics for the unification of quantum mechanics and general relativity

The paper summarizes and generalizes a recently proposed mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics. The framework is based on Hilbert spaces H of functions of four space-time variables x, t, furnished with an additional indefinite inner product invariant under Poincaré transformations, and isomorphisms of these spaces that preserve the indefinite metric. The indefinite metric is responsible for breaking the symmetry between space and time variables and for selecting a family of Hilbert subspaces that are preserved under Galileo transformations. Within these subspaces the usual quantum mechanics with Shrödinger evolution and t as the evolution parameter is derived. Simultaneously, the Minkowski space-time is isometrically embedded into H, Poincaré transformations have unique extensions to isomorphisms of H and the embedding commutes with Poincaré transformations. The main new result is a proof that the framework accommodates arbitrary pseudo-Riemannian space-times furnished with the action of the diffeomorphism group.