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

SIGMA 3 (2007), 092, 14 pages      arXiv:0709.3698
Contribution to the Proceedings of the 3-rd Microconference Analytic and Algebraic Methods III

Miscellaneous Applications of Quons

Maurice R. Kibler
Université de Lyon, Institut de Physique Nucléaire, Université Lyon 1 and CNRS/IN2P3, 43 bd du 11 novembre 1918, F-69622 Villeurbanne Cedex, France

Received July 23, 2007, in final form September 21, 2007; Published online September 24, 2007

This paper deals with quon algebras or deformed oscillator algebras, for which the deformation parameter is a root of unity. We motivate why such algebras are interesting for fractional supersymmetric quantum mechanics, angular momentum theory and quantum information. More precisely, quon algebras are used for (i) a realization of a generalized Weyl-Heisenberg algebra from which it is possible to associate a fractional supersymmetric dynamical system, (ii) a polar decomposition of SU2 and (iii) a construction of mutually unbiased bases in Hilbert spaces of prime dimension. We also briefly discuss (symmetric informationally complete) positive operator valued measures in the spirit of (iii).

Key words: quon algebra; q-deformed oscillator algebra; fractional supersymmetric quantum mechanics; polar decompostion of SU2; mutually unbiased bases; positive operator valued measures.

pdf (274 kb)   ps (178 kb)   tex (18 kb)


  1. Arik M., Coon D.D., Hilbert spaces of analytic functions and generalized coherent states, J. Math. Phys. 17 (1976), 524-527.
  2. Kuryshkin M.V., Opérateurs quantiques généralisés de création et d'annihilation, Ann. Fond. Louis de Broglie 5 (1980), 111-125.
  3. Jannussis A.D., Papaloucas L.C., Siafarikas P.D., Eigenfunctions and eigenvalues of the q-differential operators, Hadronic J. 3 (1980), 1622-1632.
  4. Biedenharn L.C., The quantum group SUq(2) and a q-analogue of the boson operators, J. Phys. A: Math. Gen. 22 (1989), L873-L878.
  5. Sun C.-P., Fu H.-C., The q-deformed boson realisation of the quantum group SU(n)q and its representations, J. Phys. A: Math. Gen. 22 (1989), L983-L986.
  6. Macfarlane A.J., On q-analogues of the quantum harmonic oscillator and the quantum group SU(2)q, J. Phys. A: Math. Gen. 22 (1989), 4581-4588.
  7. Chaturvedi S., Kapoor A.K., Sandhya R., Srinivasan V., Simon R., Generalized commutation relations for a single-mode oscillator, Phys. Rev. A 43 (1991), 4555-4557.
  8. Greenberg O.W., Quons, an interpolation between Bose and Fermi oscillators, cond-mat/9301002.
  9. Daoud M., Hassouni Y., Kibler M., The k-fermions as objects interpolating between fermions and bosons, in Symmetries in Science X, Editors B. Gruber and M. Ramek, Plenum Press, New York, 1998, 63-67, quant-ph/9710016.
  10. Daoud M., Hassouni Y., Kibler M., Generalized supercoherent states, Phys. Atom. Nuclei 61 (1998), 1821-1824, quant-ph/9804046.
  11. Ge M.-L., Su G., The statistical distribution function of the q-deformed harmonic-oscillator, J. Phys. A: Math. Gen. 24 (1991), L721-L723.
  12. Martín-Delgado M.A., Planck distribution for a q-boson gas, J. Phys. A: Math. Gen. 24 (1991), L1285-L1291.
  13. Lee C.R., Yu J.-P., On q-deformed free-electron gases, Phys. Lett. A 164 (1992), 164-166.
  14. Su G., Ge M.-L., Thermodynamic characteristics of the q-deformed ideal Bose-gas, Phys. Lett. A 173 (1993), 17-20.
  15. Tuszy\'nski J.A., Rubin J.L, Meyer J., Kibler M., Statistical mechanics of a q-deformed boson gaz, Phys. Lett. A 175 (1993), 173-177.
  16. Man'ko V.I., Marmo G., Solimeno S., Zaccaria F., Correlation functions of quantum q-oscillators, Phys. Lett. A 176 (1993), 173-175, hep-th/9303008.
  17. Hsu R.-R., Lee C.-R., Statistical distribution of gases which obey q-deformed commutation relations, Phys. Lett. A 180 (1993), 314-316.
  18. Granovskii Ya.I., Zhedanov A.S., Production of q-bosons by a classical current - an exactly solvable model, Modern Phys. Lett. A 8 (1993), 1029-1035.
  19. Chaichian M., Felipe R.G., Montonen C., Statistics of q-oscillators, quons and relations to fractional statistics, J. Phys. A: Math. Gen. 26 (1993), 4017-4034, hep-th/9304111.
  20. Gupta R.K., Bach C.T., Rosu H., Planck distribution for a complex q-boson gas, J. Phys. A: Math. Gen. 27 (1994), 1427-1433.
  21. R.-Monteiro M.A., Roditi I., Rodrigues L.M.C.S., Gamma-point transition in quantum q-gases, Phys. Lett. A 188 (1994), 11-15.
  22. Gong R.-S., Thermodynamic characteristics of the (p,q)-deformed ideal Bose-gas, Phys. Lett. A 199 (1995), 81-85.
  23. Daoud M., Kibler M., Statistical-mechanics of qp-bosons in D-dimensions, Phys. Lett. A 206 (1995), 13-17, quant-ph/9512006.
  24. Witten E., Gauge-theories, vertex models, and quantum groups, Nuclear Phys. B 330 (1990), 285-346.
  25. Iwao S., Knot and conformal field-theory approach in molecular and nuclear-physics, Prog. Theor. Phys. 83 (1990), 363-367.
  26. Bonatsos D., Raychev P.P., Roussev R.P., Smirnov Yu.F., Description of rotational molecular-spectra by the quantum algebra SUq(2), Chem. Phys. Lett. 175 (1990), 300-306.
  27. Chang Z., Guo H.-Y., Yan H., The q-deformed oscillator model and the vibrational-spectra of diatomic-molecules, Phys. Lett. A 156 (1991), 192-196.
  28. Chang Z., Yan H., Diatomic-molecular spectrum in view of quantum group-theory, Phys. Rev. A 44 (1991), 7405-7413.
  29. Bonatsos D., Drenska S.B., Raychev P.P., Roussev R.P., Smirnov Yu.F., Description of superdeformed bands by the quantum algebra SUq(2), J. Phys. G: Nucl. Part. Phys. 17 (1991), L67-L73.
  30. Bonatsos D., Raychev P.P., Faessler A., Quantum algebraic description of vibrational molecular-spectra, Chem. Phys. Lett. 178 (1991), 221-226.
  31. Bonatsos D., Argyres E.N., Raychev P.P., SUq(1,1) description of vibrational molecular-spectra, J. Phys. A: Math. Gen. 24 (1991), L403-L408.
  32. Jenkovszky L., Kibler M., Mishchenko A., Two-parameter quantum-deformed dual amplitude, Modern Phys. Lett A 10 (1995), 51-60, hep-th/9407071.
  33. Barbier R., Kibler M., Application of a two-parameter quantum algebra to rotational spectroscopy of nuclei, Rep. Math. Phys. 38 (1996), 221-226, nucl-th/9602015.
  34. Kundu A., q-boson in quantum integrable systems, SIGMA 3 (2007), 040, 14 pages, nlin.SI/0701030.
  35. Rubakov V.A., Spiridonov V.P., Parasupersymmetric quantum-mechanics, Modern Phys. Lett. A 3 (1988), 1337-1347.
  36. Beckers J., Debergh N., Parastatistics and supersymmetry in quantum-mechanics, Nuclear Phys. B 340 (1990), 767-776.
  37. Debergh N., On parasupersymmetric Hamiltonians and vector-mesons in magnetic-fields, J. Phys. A: Math. Gen. 27 (1994), L213-L217.
  38. Khare A., Parasupersymmetry in quantum mechanics, J. Math. Phys. 34 (1993), 1277-1294.
  39. Filippov A.T., Isaev A.P., Kurdikov A.B., Paragrassmann extensions of the Virasoro algebra, Internat. J. Modern Phys. A 8 (1993), 4973-5003, hep-th/9212157.
  40. Durand S., Fractional superspace formulation of generalized mechanics, Modern Phys. Lett. A 8 (1993), 2323-2334, hep-th/9305130.
  41. Klishevich S., Plyushchay T., Supersymmetry of parafermions, Modern Phys. Lett. A 14 (1999), 2739-2752, hep-th/9905149.
  42. Daoud M., Kibler M., Fractional supersymmetric quantum mechanics as a set of replicas of ordinary supersymmetric quantum mechanics, Phys. Lett. A 321 (2004), 147-151, math-ph/0312019.
  43. Witten E., Dynamical breaking of supersymmetry, Nuclear Phys. B 188 (1981), 513-554.
  44. Daoud M., Kibler M.R., Fractional supersymmetry and hierarchy of shape invariant potentials, J. Math. Phys. 47 (2006), 122108, 11 pages, quant-ph/0609017.
  45. Kibler M., Daoud M., Variations on a theme of quons: I. A non standard basis for the Wigner-Racah algebra of the group SU(2), Recent Res. Devel. Quantum Chem. 2 (2001), 91-99, physics/9712034.
  46. Kibler M.R., Representation theory and Wigner-Racah algebra of the group SU(2) in a noncanonical basis, Collect. Czech. Chem. Commun. 70 (2005), 771-796, quant-ph/0504025.
  47. Kibler M.R., Angular momentum and mutually unbiased bases, Internat. J. Modern Phys. B 20 (2006), 1792-1801, quant-ph/0510124.
  48. Kibler M.R., Planat M., A SU(2) recipe for mutually unbiased bases, Internat. J. Modern Phys. B 20 (2006), 1802-1807, quant-ph/0601092.
  49. Albouy O., Kibler M.R., SU2 nonstandard bases: case of mutually unbiased bases, SIGMA 3 (2007), 076, 22 pages, quant-ph/0701230.
  50. Ivanovic I.D., Geometrical description of quantum state determination, J. Phys. A: Math. Gen. 14 (1981), 3241-3245.
  51. Stovícek P., Tolar J., Quantum mechanics in a discrete space-time, Rep. Math. Phys. 20 (1984), 157-170.
  52. Balian R., Itzykson C., Observations sur la mécanique quantique finie, C.R. Acad. Sci. (Paris) 303 (1986), 773-778.
  53. Wootters W.K., Fields B.D., Optimal state-determination by mutually unbiased measurements, Ann. Phys. (N.Y.) 191 (1989), 363-381.
  54. Barnum H., MUBs and spherical 2-designs, quant-ph/0205155.
  55. Bandyopadhyay S., Boykin P.O., Roychowdhury V., Vatan F., A new proof for the existence of mutually unbiased bases, Algorithmica 34 (2002), 512-528, quant-ph/0103162.
  56. Chaturvedi S., Aspects of mutually unbiased bases in odd-prime-power dimensions, Phys. Rev. A 65 (2002), 044301, 3 pages, quant-ph/0109003.
  57. Pittenger A.O., Rubin M.H., Mutually unbiased bases, generalized spin matrices and separability, Linear Algebra Appl. 390 (2004), 255-278, quant-ph/0308142.
  58. Klappenecker A., Rötteler M., Constructions of mutually unbiased bases, Lect. Notes Comput. Sci. 2948 (2004), 137-144, quant-ph/0309120.
  59. Bengtsson I., Three ways to look at mutually unbiased bases, quant-ph/0610216.
  60. Berndt B.C., Evans R.J., Williams K.S., Gauss and Jacobi Sums, Wiley, New York, 1998.
  61. Zauner G., Quantendesigns: Grundzüge einer nichtcommutativen Designtheorie, Diploma Thesis, University of Wien, 1999.
  62. Caves C.M., Fuchs C.A., Schack R., Unknown quantum states: the quantum de Finetti representation, J. Math. Phys. 43 (2002), 4537-4559, quant-ph/0104088.
  63. Renes J.M., Blume-Kohout R., Scott A.J., Caves C.M., Symmetric informationally complete quantum measurements, J. Math. Phys. 45 (2004), 2171-2180, quant-ph/0310075.
  64. Appleby D.M., Symmetric informationally complete-positive operator valued measures and the extended Clifford group, J. Math. Phys. 46 (2005), 052107, 29 pages, quant-ph/0412001.
  65. Grassl M., Tomography of quantum states in small dimensions, Elec. Notes Discrete Math. 20 (2005), 151-164.
  66. Klappenecker A., Rötteler M., Mutually unbiased bases are complex projective 2-designs, quant-ph/0502031.
  67. Klappenecker A., Rötteler M., Shparlinski I.E., Winterhof A., On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states, J. Math. Phys. 46 (2005), 082104, 17 pages, quant-ph/0503239.
  68. Weigert S., Simple minimal informationally complete measurements for qudits, Internat. J. Modern Phys. B 20 (2006), 1942-1955, quant-ph/0508003.
  69. Albouy A., Kibler M.R., A unified approach to SIC-POVMs and MUBs, arXiv:0704.0511.
  70. Znojil M., PT-symmetric regularizations in supersymmetric quantum mechanics, J. Phys. A: Math. Gen. 37 (2004), 10209-10222, hep-th/0404145.
  71. Hall J.L., Rao A., SIC-POVMs exist in all dimensions, arXiv:0707.3002.

Previous article   Next article   Contents of Volume 3 (2007)