Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The fermionic Heisenberg group and its Q-representations

Abstract

A nonstandard way of representing canonical anticommutation relations is presented. It is connected with a generalization of the Heisenberg group to the case of graded phase space. We show how Grassmann harmonic analysis can be performed and what are the Q-representations of such a generalized Heisenberg group. As in the conventional case, the Schrödinger and Bargmann-Fock realizations are shown to exist. The Grassmann-Hermite polynomials via the generalized Bargmann transform are presented and new Grassmann-Laguerre polynomials are obtained.

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

References

  1. 1.

    Howe R., Bull. Amer. Math. Soc. (N.S.) 3, 821 (1980).

  2. 2.

    Taylor M. E., Noncommutative Harmonic Analysis. Amer. Math. Soc., Providence, Rhode Island, 1986.

  3. 3.

    Lion G. and Vergne M., The Weil Representation, Maslov Index and Theta Series, Birkhäuser, Boston, 1980.

  4. 4.

    Farant J. and Khélifa H., Deux cours d'analyse harmonique, Birkhäuser, Boston, 1987.

  5. 5.

    Tilgner H., J. Pure Appl. Algebra 10, 163 (1977).

  6. 6.

    Frydryszak A. and Jakóbczyk L., Lett. Math. Phys. 16, 101 (1988).

  7. 7.

    Frydryszak A., Lett. Math. Phys. 20, 159 (1990).

  8. 8.

    Berezin F. A., The Method of Second Quantization. Academic Press, New York, 1966.

  9. 9.

    Frydryszak, A., The classical and quantum pseudomechanics, Wrocŀaw University, 1981 (unpublished).

  10. 10.

    Frydryszak, A. and Jakóbczyk, L., Wrocŀaw University preprint ITP UWr 88/699 (March 1988).

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Frydryszak, A. The fermionic Heisenberg group and its Q-representations. Lett Math Phys 26, 105–114 (1992). https://doi.org/10.1007/BF00398807

Download citation

Mathematics Subject Classifications (1991)

  • 81R05
  • 22E70
  • 43A32