Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 005, 11 pages      arXiv:1011.5049
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

The Quantum 3D Superparticle

Luca Mezincescu a and Paul K. Townsend b
a) Department of Physics, University of Miami, Coral Gables, FL 33124, USA
b) Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

Received November 29, 2010, in final form January 05, 2011; Published online January 10, 2011

The minimal (N=1) superparticle in three spacetime dimensions (3D) is quantized. For non-zero mass it describes a spin-1/4 semion supermultiplet of ''relativistic helicities'' (−1/4,1/4). The addition of a parity-violating Lorentz-Wess-Zumino term shifts this to (β−1/4,β+1/4) for arbitrary β. For zero mass, in which case spin is not defined, the quantum superparticle describes a supermultiplet of one boson and one fermion.

Key words: superparticle; semion.

pdf (327 kb)   tex (15 kb)


  1. Binegar B., Relativistic field theories in three dimensions, J. Math. Phys. 23 (1982), 1511-1517.
  2. Deser S., Jackiw R., Statistics without spin: massless D=3 systems, Phys. Lett. B 263 (1991), 431-436.
  3. Sorokin D.P., Volkov D.V., (Anti)commuting spinors and supersymmetric dynamics of semions, Nuclear Phys. B 409 (1993), 547-564.
  4. Gorbunov I.V., Kuzenko S.M., Lyakhovich S.L., N=1, D=3 superanyons, OSp(2|2) and the deformed Heisenberg algebra, Phys. Rev. D 56 (1997), 3744-3755, hep-th/9702017.
  5. Horvathy P.A., Plyushchay M.S., Valenzuela M., Bosons, fermions and anyons in the plane, and supersymmetry, Ann. Physics 325 (2010), 1931-1975, arXiv:1001.0274.
  6. Volkov D.V., Quartions in relativistic field theory, JETP Lett. 49 (1989), 541-543.
  7. Sorokin D., The Heisenberg algebra and spin, 3.0.CO;2-J">Fortsch. Phys. 50 (2002), 724-728.
  8. Witten E., Supersymmetric index of three-dimensional gauge theory, in The Many Faces of the Superworld, Editor M. Shifman, World Sci. Publ., River Edge, NJ, 2000, 156-184, hep-th/9903005.
  9. Pedder C., Sonner J., Tong D., The berry phase of D0-branes, J. High Energy Phys. 2008 (2008), no. 3, 065, 14 pages, arXiv:0801.1813.
  10. Mezincescu L., Townsend P.K., Anyons from strings, Phys. Rev. Lett. 105 (2010), 191601, 4 pages, arXiv:1008.2334.
  11. Casalbuoni R., The classical mechanics for Bose-Fermi systems, Nuovo Cimento A 33 (1976), 389-431.
  12. Brink L., Schwarz J.H., Quantum superspace, Phys. Lett. B 100 (1981), 310-312.
  13. Martin J.L., Generalized classical dynamics and the 'classical analogue' of a Fermi oscillator, Proc. Roy. Soc. London. Ser. A 251 (1959), 536-542.
    Martin J.L., The Feynman principle for a Fermi system, Proc. Roy. Soc. London. Ser. A 251 (1959), 543-549.
  14. Schonfeld J.F., A mass term for three-dimensional gauge fields, Nuclear Phys. B 185 (1981), 157-171.
  15. Cortés J.L., Plyushchay M.S., Anyons as spinning particles, Internat. J. Modern Phys. A 11 (1996), 3331-3362, hep-th/9505117.
  16. Siegel W., Hidden local supersymmetry in the supersymmetric particle action, Phys. Lett. B 128 (1983), 397-399.
  17. Mezincescu L., Townsend P.K., Semionic supersymmetric solitons, J. Phys. A: Math. Theor. 43 (2010), 465401, 12 pages, arXiv:1008.2775.
  18. Townsend P.K., The story of M, in The Future of Theoretical Physics and Cosmology (Cambridge, 2002), Editors G.W. Gibbons, E.P.S. Shellard and S.J. Rankin, Cambridge University Press, Cambridge, 2003, 484-493, hep-th/0205309.
  19. Bergshoeff E.A., Hohm O., Rosseel J., Townsend P.K., On maximal massive 3D supergravity, Classical Quantum Gravity 27 (2010), 235012, 24 pages, arXiv:1007.4075.
  20. Plyushchay M.S., Operator quantization of massless superparticle, Internat. J. Modern Phys. A 6 (1991), 2497-2517.
  21. Klishevich S.M., Plyushchay M.S., Rausch de Traubenberg M., Fractional helicity, Lorentz symmetry breaking and anyons, Nuclear Phys. B 616 (2001), 419-436, hep-th/0101190.

Previous article   Next article   Contents of Volume 7 (2011)