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

TL;DR

This paper proposes a mathematical framework that unifies quantum mechanics and special relativity by embedding Minkowski space-time into a Hilbert space with an indefinite metric, accommodating arbitrary pseudo-Riemannian space-times.

Contribution

It introduces a novel Hilbert space formalism with an indefinite metric that unifies quantum mechanics and relativity, extending to arbitrary pseudo-Riemannian space-times.

Findings

Framework unifies special relativity and quantum mechanics.

Embedding of Minkowski space into Hilbert space preserves Poincaré symmetry.

Accommodates arbitrary pseudo-Riemannian space-times with diffeomorphism group action.

Abstract

This 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 Poincar\'e 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 Schr\"odinger evolution and t as the evolution parameter is derived. Simultaneously, the Minkowski space-time is isometrically embedded into H, Poincar\'e transformations have unique extensions to isomorphisms of H and the embedding commutes with Poincar\'e 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.