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

DOI10.1063/1.3298678zbMATH Open1309.81077arXiv1005.2361OpenAlexW3106514536MaRDI QIDQ5246523

Author name not available (Why is that?)

Publication date: 21 April 2015

Published in: Journal of Mathematical Physics (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/1005.2361

Mathematics Subject Classification ID

Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics (70H40) Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices (81Q35)

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