Foundations of Physics

, Volume 44, Issue 11, pp 1168–1187 | Cite as

A Minimal Framework for Non-Commutative Quantum Mechanics

  • D. J. Hurley
  • M. A. Vandyck


Deformation quantisation is applied to ordinary Quantum Mechanics by introducing the star product in a configuration space combining a Riemannian structure with a Poisson one. A Hilbert space compatible with such a configuration space is designed. The dynamics is expressed by a Hermitian Hamiltonian containing a scalar potential and a one-form potential. As a simple illustration, it is shown how a particular type of non-commutativity of the star product is interpretable as generating the Zeeman effect of ordinary Quantum Mechanics.


Non-commutative quantum mechanics Deformation quantisation Konstevich product 

Mathematics Subject Classification

53D55 53D17 81R60 81S10 


  1. 1.
    Weyl, H.: The Theory of Groups and Quantum Mechanics. Dover, New York (1931)zbMATHGoogle Scholar
  2. 2.
    Wigner, E.P.: Quantum corrections for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)CrossRefADSGoogle Scholar
  3. 3.
    Moyal, J.E.: Quantum mechanics as a statistical theory. Proc. Camb. Phil. Soc. 45, 99–124 (1949)MathSciNetCrossRefzbMATHADSGoogle Scholar
  4. 4.
    Dito, G., Sternheimer, D.: Deformation quantization: genesis, developments and metamorphoses. In: Halbout, G. (ed.) Deformation Quantization. IRMA Lectures in Mathematical and Theoretical Physics 1, pp. 9–54. Walter de Gruyter, Berlin (2002)Google Scholar
  5. 5.
    Konstevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003)MathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Felder, G., Shoikhet, B.: Deformation quantization with traces. Lett. Math. Phys. 53, 75–86 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hurley, D., Vandyck, M.: \({\mathfrak{D}}\)-Differentiation and the structure of Quantum Mechanics. Found. Phys. 39, 433–473 (2009)MathSciNetCrossRefzbMATHADSGoogle Scholar
  8. 8.
    Hurley, D., Vandyck, M.: \({\mathfrak{D}}\)-Differentiation and the structure of Quantum Mechanics Part II: accelerated observers and fictitious forces. Found. Phys. 41, 667–685 (2011)MathSciNetCrossRefzbMATHADSGoogle Scholar
  9. 9.
    Kupriyanov, V., Vassilievich, D.: Star products made (somewhat) easier. Eur. Phys. J. C 58, 627–637 (2008)CrossRefzbMATHADSGoogle Scholar
  10. 10.
    Zotov, A.: On relation between Weyl and Konstevich quantum products. Direct evaluation up to the \(\hbar ^3\)-order. Mod. Phys. Lett. A 16, 615–625 (2001)MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Fedosov, B.: A simple geometrical construction of deformation quantization. J. Differ. Geom. 40, 213–218 (1994)MathSciNetzbMATHGoogle Scholar
  12. 12.
    McCurdy, S., Zumino, B.: Covariant star product for exterior differential forms on symplectic manifolds. AIP Conf. Proc. 1200, 204–214 (2010)CrossRefADSGoogle Scholar
  13. 13.
    Chaichian, M., Oksanen, M., Tureanu, A., Zet, G.: Covariant star product on symplectic and Poisson spacetime manifolds. Int. J. Mod. Phys. A 25, 3765–3796 (2010)MathSciNetCrossRefzbMATHADSGoogle Scholar
  14. 14.
    Hawkins, E.: Noncommutative rigidity. Commun. Math. Phys. 246, 211–235 (2004)MathSciNetCrossRefzbMATHADSGoogle Scholar
  15. 15.
    Hawkins, E.: The structure of noncommutative deformations. J. Differ. Geom. 77, 385–424 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Pinzul, A., Stern, A.: Gauge theory of the star product. Nuclear Phys. B 791, 284–297 (2008)MathSciNetCrossRefzbMATHADSGoogle Scholar
  17. 17.
    Kupriyanov, V.: A hydrogen atom on curved noncommutative space. J. Phys. A 46, 1–7 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Abraham, R., Marsden, J., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Springer, New York (1988)CrossRefzbMATHGoogle Scholar
  19. 19.
    Grosse, H., Wulkenhaar, R.: Renormalization of noncommutative Quantum Field Theory. In: Khalkhali, M., Marcolli, M. (eds.) Noncommutative Geometry. World Scientific, Singapore (2008)Google Scholar
  20. 20.
    Bagchi, B., Fring, A.: Minimal length in quantum mechanics and non-Hermitian Hamiltonian systems. Phys. Lett. A 373, 4307–4310 (2009)MathSciNetCrossRefzbMATHADSGoogle Scholar
  21. 21.
    Göckeler, M., Schücker, T.: Differential Geometry, Gauge Theories, and Gravity. Cambridge University Press, Cambridge (1990)Google Scholar
  22. 22.
    Gasiorowicz, S.: Quantum Mechanics. Wiley, New York (1974)Google Scholar
  23. 23.
    Kupriyanov, V.: Quantum mechanics with coordinate dependent noncommutativity. J. Math. Phys. 54, 112105–112124 (2013)MathSciNetCrossRefADSGoogle Scholar
  24. 24.
    Fring, A., Gouba, L., Scholtz, F.G.: Strings from position-dependent noncommutativity. J. Phys. A 43, 345401–345410 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsNational University of IrelandCorkIreland
  2. 2.Department of PhysicsNational University of IrelandCorkIreland

Personalised recommendations