Foundations of Physics

, Volume 41, Issue 1, pp 129–140 | Cite as

Geometry of the Unification of Quantum Mechanics and Relativity of a Single Particle

  • A. KryukovEmail author


The paper summarizes, generalizes and reveals the physical content of a recently proposed 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. 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 embedded into H as a set of point-localized states, Poincaré transformations obtain unique extensions to H and the embedding commutes with Poincaré transformations. Furthermore, the framework accommodates arbitrary pseudo-Riemannian space-times furnished with the action of the diffeomorphism group.


Quantum mechanics Special relativity General relativity Indefinite inner product space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Horwitz, L.P., On the definition and evolution of states in relativistic classical and quantum mechanics. Found. Phys. 22, 421 (1992) CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Newton, T.D., Wigner, E.P.: Localized states for elementary systems. Rev. Mod. Phys. 21, 400 (1949) zbMATHCrossRefADSGoogle Scholar
  3. 3.
    Hegerfeldt, G.C.: Remark on causality and particle localization. Phys. Rev. D 10, 3320 (1974) CrossRefADSGoogle Scholar
  4. 4.
    Stueckelberg, E.C.G.: Remarks on the creation of pairs of particles in the theory of relativity. Helv. Phys. Acta 14, 372, 585 (1941) MathSciNetGoogle Scholar
  5. 5.
    Stueckelberg, E.C.G.: Mechanics of a matter particle in the theory of relativity and in the theory of quanta. Helv. Phys. Acta 15, 23 (1942) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Horwitz, L.P., Piron, C.: Relativistic dynamics. Helv. Phys. Acta 46, 316 (1973) Google Scholar
  7. 7.
    Kryukov, A.: On the problem of emergence of classical space-time: the quantum-mechanical approach. Found. Phys. 34, 1225 (2004) zbMATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Kryukov, A.: Quantum mechanics on Hilbert manifolds: the principle of functional relativity. Found. Phys. 36, 175 (2006) zbMATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Kryukov, A.: On the measurement problem for a two-level quantum system. Found. Phys. 37, 3 (2007) zbMATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Kryukov, A.: Geometric derivation of quantum uncertainty. Phys. Lett. A 370, 419 (2007) MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Kryukov, A.: Linear algebra and differential geometry on abstract Hilbert space. Int. J. Math. Math. Sci. 14, 2241 (2005) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kryukov, A.: Nine theorems on the unification of quantum mechanics and relativity. J. Math. Phys. 49, 102108 (2008) CrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Gel’fand, I.M., Vilenkin, N.Y.: Generalized Functions, vol. 4. Academic Press, New York/London (1964) zbMATHGoogle Scholar
  14. 14.
    Kryukov, A.: to appear Google Scholar
  15. 15.
    Horwitz, L.P., Rabin, Y.: Relativistic diffraction. Lett. Nuovo Cimento 17, 501 (1976) CrossRefGoogle Scholar
  16. 16.
    Horwitz, L.P.: On the significance of a recent experiment demonstrating quantum interference in time. Phys. Lett. A 355, 1 (2006) CrossRefADSGoogle Scholar
  17. 17.
    Horwitz, L.P., Rotbart, F.: Nonrelativistic limit of relativistic quantum mechanics. Phys. Rev. D 24, 2127 (1981) CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wisconsin CollegesMadisonUSA

Personalised recommendations