Skip to main content
Log in

On the deformability of Heisenberg algebras

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Based on the vanishing of the second Hochschild cohomology group of the Weyl algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum mechanics. For the case of aq-oscillator there exists a deforming map to the classical algebra. It is shown that the differential calculus on quantum planes with involution, i.e., if one works in position-momentum realization, can be mapped on aq-difference calculus on a commutative real space. Although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not possess a proper group of global transformations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Awata, H., Noumi, M., Odake, S.: Lett. Math. Phys.30, 35 (1994)

    Article  Google Scholar 

  2. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Ann. Phys.111, 61 (1978)

    Article  Google Scholar 

  3. Bonechi, F., Giachetti, R., Sorace, E., Tarlini, M.: Commun. Math. Phys.169, 627 (1995)

    Article  Google Scholar 

  4. Cartan, H., Eilenberg, S.: Homological Algebra. Princeton, NJ: Princeton Univ. Press, 1956

    Google Scholar 

  5. Castellani, L.: Phys. Lett. B327, 22 (1994)

    Article  Google Scholar 

  6. Chodos, A., Caldi, D.G.: J. Phys. A24, 5505 (1991)

    Google Scholar 

  7. Curtright, T.I., Zachos, C.K.: Phys. Lett. B243, 237 (1990)

    Article  Google Scholar 

  8. Drinfeld, V.G.: Proc. Int. Cong. Math., Berkeley, 1986, p. 789

  9. Drinfeld, V.G.: Leningr. Math. J.2, 829 (1990)

    Google Scholar 

  10. du Cloux, F.: Asterisque (Soc. Math. France)124–125, 129 (1985)

    Google Scholar 

  11. Feinsilver, P.: Monatsh. Math.104, 89 (1987)

    Article  Google Scholar 

  12. Farlie, D.B., Zachos, C.K.: Phys. Lett. B256, 43 (1991)

    Article  Google Scholar 

  13. Fiore, G.: J. Math. Phys.36, 4363 (1995)

    Article  Google Scholar 

  14. Hayashi, T.: Commun. Math. Phys.127, 129 (1990)

    Article  Google Scholar 

  15. Gerstenhaber, M.: Ann. Math.79, 59 (1964)

    Google Scholar 

  16. Hebecker, A., Weich, W.: Lett. Math. Phys.26, 245 (1992)

    Article  Google Scholar 

  17. Jimbo, M.: Lett. Math. Phys.10, 63 (1985)

    Article  Google Scholar 

  18. Kassel, C.: Quantum Groups. Springer GTM155, New York: Springer Verlag 1995

    Google Scholar 

  19. Kulish, P.P., Damaskinsky, E.V.: J. Phys. A23, L415 (1990)

    Google Scholar 

  20. Ogievetsky, O.: Lett. Math. Phys.24, 121 (1992)

    Article  Google Scholar 

  21. Ogievetsky, O., Zumino, B.: Lett. Math. Phys.25, 121 (1992)

    Article  Google Scholar 

  22. Ogievetsky, O., Schmidke, W.B., Wess, J., Zumino, B.: Commun. Math. Phys.150, 495 (1992)

    Google Scholar 

  23. Pillin, M.: J. Math. Phys.35, 2804 (1994)

    Article  Google Scholar 

  24. Pillin, M., Schmidke, W.B., Wess, J.: Nucl. Phys. B403, 223 (1993)

    Article  Google Scholar 

  25. Rieffel, M.A.: Commun. Math. Phys.122, 531 (1989)

    Article  Google Scholar 

  26. Rosenberg, A.L.: Commun. Math. Phys.144, 41 (1992)

    Article  Google Scholar 

  27. Schwenk, J., Wess, J.: Phys. Lett. B291, 273 (1992)

    Article  Google Scholar 

  28. Schwenk, J.:q-deformed Fourier theory. Preprint MPI-PhT 94-36, hep-th-9406168

  29. Wess, J., Zumino, B.: Nucl. Phys. B (Proc. Suppl.)18B, 302 (1991)

    Article  Google Scholar 

  30. Zachos, C.: Contemp. Math.134, 351 (1992)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by H. Araki

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pillin, M. On the deformability of Heisenberg algebras. Commun.Math. Phys. 180, 23–38 (1996). https://doi.org/10.1007/BF02101181

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02101181

Keywords

Navigation