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

SIGMA 6 (2010), 083, 37 pages      arXiv:1003.5322
Contribution to the Special Issue “Noncommutative Spaces and Fields”

The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space

Everton M.C. Abreu a, b, Albert C.R. Mendes c, Wilson Oliveira c and Adriano O. Zangirolami c
a) Grupo de Física Teórica e Matemática Física, Departamento de Física, Universidade Federal Rural do Rio de Janeiro, BR 465-07, 23890-971, Seropédica, RJ, Brazil
b) Centro Brasileiro de Pesquisas Físicas (CBPF), Rua Xavier Sigaud 150, Urca, 22290-180, RJ, Brazil
c) Departamento de Física, ICE, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil

Received March 28, 2010, in final form October 02, 2010; Published online October 10, 2010

This work is an effort in order to compose a pedestrian review of the recently elaborated Doplicher, Fredenhagen, Roberts and Amorim (DFRA) noncommutative (NC) space which is a minimal extension of the DFR space. In this DRFA space, the object of noncommutativity (θμν) is a variable of the NC system and has a canonical conjugate momentum. Namely, for instance, in NC quantum mechanics we will show that θij (i,j=1,2,3) is an operator in Hilbert space and we will explore the consequences of this so-called ''operationalization''. The DFRA formalism is constructed in an extended space-time with independent degrees of freedom associated with the object of noncommutativity θμν. We will study the symmetry properties of an extended x+θ space-time, given by the group P', which has the Poincaré group P as a subgroup. The Noether formalism adapted to such extended x+θ (D=4+6) space-time is depicted. A consistent algebra involving the enlarged set of canonical operators is described, which permits one to construct theories that are dynamically invariant under the action of the rotation group. In this framework it is also possible to give dynamics to the NC operator sector, resulting in new features. A consistent classical mechanics formulation is analyzed in such a way that, under quantization, it furnishes a NC quantum theory with interesting results. The Dirac formalism for constrained Hamiltonian systems is considered and the object of noncommutativity θij plays a fundamental role as an independent quantity. Next, we explain the dynamical spacetime symmetries in NC relativistic theories by using the DFRA algebra. It is also explained about the generalized Dirac equation issue, that the fermionic field depends not only on the ordinary coordinates but on θμν as well. The dynamical symmetry content of such fermionic theory is discussed, and we show that its action is invariant under P'. In the last part of this work we analyze the complex scalar fields using this new framework. As said above, in a first quantized formalism, θμν and its canonical momentum πμν are seen as operators living in some Hilbert space. In a second quantized formalism perspective, we show an explicit form for the extended Poincaré generators and the same algebra is generated via generalized Heisenberg relations. We also consider a source term and construct the general solution for the complex scalar fields using the Green function technique.

Key words: noncommutativity; quantum mechanics; gauge theories.

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  1. Snyder H.S., Quantized space-time, Phys. Rev. 71 (1947), 38-41.
  2. Yang C.N., On quantized space-time, Phys. Rev. 72 (1947), 874-874.
  3. Green M., Schwarz J.H., Witten E., Superstring theory, Cambridge University Press, Cambridge, 1987.
    Polchinski J., String theory, Vols. 1 and 2, Cambridge University Press, Cambridge, 1998.
    Szabo R.J., An introduction to string theory and D-brane dynamics, Imperial College Press, London, 2004.
  4. Deriglazov A.A., Quantum mechanics on noncommutative plane and sphere from constrained systems, Phys. Lett. B 530 (2002), 235-243, hep-th/0201034.
  5. Deriglazov A.A., Noncommutative relativistic particle on the electromagnetic background, Phys. Lett. B 555 (2003), 83-88, hep-th/0208201.
    Deriglazov A.A., Poincaré covariant mechanics on noncommutative space, J. High Energy Phys. 2003 (2003), no. 3, 021, 9 pages, hep-th/0211105.
  6. Deriglazov A.A., Noncommutative version of an arbitrary nondegenerate mechanics, hep-th/0208072.
  7. Chakraborty B., Gangopadhyay S., Saha A., Seiberg-Witten map and Galilean symmetry violation in a non-commutative planar system, Phys. Rev. D 70 (2004), 107707, 4 pages, hep-th/0312292.
    Scholtz F.G., Chakraborty B., Gangopadhyay S., Hazra A.G., Dual families of noncommutative quantum systems, Phys. Rev. D 71 (2005), 085005, 11 pages, hep-th/0502143.
    Gangopadhyay S., Scholtz F.G., Path-integral action of a particle in the noncommutative plane, Phys. Rev. Lett. 102 (2009), 241602, 4 pages, arXiv:0904.0379.
    Banerjee R., Lee C., Yang H.S., Seiberg-Witten-type maps for currents and energy momentum tensors in noncommutative gauge theories, Phys. Rev. D 70 (2004), 065015, 7 pages, hep-th/0312103.
    Banerjee R., Kumar K., Seiberg-Witten maps and commutator anomalies in noncommutative electrodynamics, Phys. Rev. D 72 (2005), 085012, 10 pages, hep-th/0505245.
  8. Alvarez-Gaumé L., Meyer F., Vázquez-Mozo M.A., Comments on noncommutative gravity, Nuclear Phys. B 753 (2006), 92-117, hep-th/0605113.
    Calmet X., Kobakhidze A., Noncommutative general relativity, Phys. Rev. D 72 (2005), 045010, 5 pages, hep-th/0506157.
    Harikumar E., Rivelles V.O., Noncommutative gravity, Classical Quantum Gravity 23 (2006), 7551-7560, hep-th/0607115.
    Douglas M.R., Nekrasov N.A., Noncommutative field theory, Rev. Modern Phys. 73 (2001), 977-1029, hep-th/0106048.
    Szabo R.J., Quantum gravity, field theory and signatures of noncommutative spacetime, Gen. Relativity Gravitation 42 (2010), 1-29, arXiv:0906.2913.
    Szabo R.J., Symmetry, gravity and noncommutativity, Classical Quantum Gravity 23 (2006), R199-R242, hep-th/0606233.
    Müller-Hoissen F., Noncommutative geometries and gravity, in Recent Developments in Gravitation and Cosmology, AIP Conf. Proc., Vol. 977, Amer. Inst. Phys., Melville, NY, 2008, 12-29, arXiv:0710.4418.
    Freidel L., Livine E.R., 3D quantum gravity and effective noncommutative quantum field theory, Phys. Rev. Lett. 96 (2006), 221301, 4 pages, hep-th/0512113.
    Rivelles V., Noncommutative field theories and gravity, Phys. Lett. B 558 (2003), 191-196, hep-th/0212262.
    Steinacker H., Emergent gravity from noncommutative gauge theory, J. High Energy Phys. 2007 (2007), no. 12, 049, 36 pages, arXiv:0708.2426.
    Steinacker H., Emergent gravity and noncommutative branes from Yang-Mills matrix models, Nuclear Phys. B 810 (2009), 1-39, arXiv:0806.2032.
    Banerjee R., Yang H.S., Exact Seiberg-Witten map, induced gravity and topological invariants in non-commutative field theories, Nuclear Phys. B 708 (2005), 434-450, hep-th/0404064.
    Banerjee R., Chakraborty B., Ghosh S., Mukherjee P., Samanta S., Topics in noncommutative geometry inspired physics, Found. Phys. 39 (2009), 1297-1345, arXiv:0909.1000.
    Banerjee R., Mukherjee P., Samanta S., Lie algebraic noncommutative gravity, Phys. Rev. D 75 (2007), 125020, 7 pages, hep-th/0703128.
  9. Szabo R.J., Quantum field theory on noncommutative spaces, Phys. Rep. 378 (2003), 207-299, hep-th/0109162.
  10. Doplicher S., Fredenhagen K., Roberts J.E., Spacetime quantization induced by classical gravity, Phys. Lett. B 331 (1994), 39-44.
    Doplicher S., Fredenhagen K., Roberts J.E., The quantum structure of spacetime at the Planck scale and quantum fields, Comm. Math. Phys. 172 (1995), 187-220, hep-th/0303037.
  11. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  12. Connes A., Rieffel M.A., Yang-Mills for noncommutative two-tori, in Operator Algebras and Mathematical Physics (Iowa City, Iowa, 1985), Contemp. Math., Vol. 62, Amer. Math. Soc., Providence, RI, 1987, 237-266.
  13. Chamseddine A.H., Felder G., Fröhlich J., Gravity in non-commutative geometry, Comm. Math. Phys. 155 (1993), 205-218, hep-th/9209044.
    Kalau W., Walze M., Gravity, non-commutative geometry and the Wodzicki residue, J. Geom. Phys. 16 (1995), 327-344, gr-qc/9312031.
    Kastler D., The Dirac operator and gravitation, Comm. Math. Phys. 166 (1995), 633-644.
    Chamseddine A.H., Fröhlich J., Grandjean O., The gravitational sector in the Connes-Lott formulation of the standard model, J. Math. Phys. 36 (1995), 6255-6275, hep-th/9503093.
    Chamseddine A.H., Connes A., The spectral action principle, Comm. Math. Phys. 186 (1997), 731-750, hep-th/9606001.
  14. Douglas M.R., Hull C., D-branes and the noncommutative torus, J. High Energy Phys. 1998 (1998), 008, no. 2, 5 pages, hep-th/9711165.
  15. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, hep-th/9908142.
  16. Jaeckel J., Khoze V.V., Ringwald A., Telltale traces of U(1) fields in noncommutative standard model extensions, J. High Energy Phys. 2006 (2006), no. 2, 028, 21 pages, hep-ph/0508075.
  17. Carlson C.E., Carone C.D., Zobin N., Noncommutative gauge theory without Lorentz violation, Phys. Rev. D 66 (2002), 075001, 8 pages, hep-th/0206035.
  18. Banerjee R., Chakraborty B., Kumar K., Noncommutative gauge theories and Lorentz symmetry, Phys. Rev. D 70 (2004), 125004, 12 pages, hep-th/0408197.
    Iorio A., Comment on "Noncommutative gauge theories and Lorentz symmetry", Phys. Rev. D 77 (2008), 048701, 5 pages.
    Banerjee R., Chakraborty B., Kumar K., Reply to "Comment on `Noncommutative gauge theories and Lorentz symmetry"', Phys. Rev. D 77 (2008), 048702, 3 pages.
  19. Kase H., Morita K., Okumura Y., Umezawa E., Lorentz-invariant non-commutative space-time based on DFR algebra, Progr. Theoret. Phys. 109 (2003), 663-685, hep-th/0212176.
    Morita K., Okumura Y., Umezawa E., Lorentz invariance and the unitarity problem in non-commutative field theory, Progr. Theoret. Phys. 110 (2003), 989-1001.
  20. Haghighat M., Ettefaghi M.M., Parton model in Lorentz invariant non-commutative space, Phys. Rev. D 70 (2004), 034017, 6 pages, hep-ph/0405270.
  21. Carone C.D., Kwee H.J., Unusual high-energy phenomenology of Lorentz-invariant noncommutative field theories, Phys. Rev. D 73 (2006), 096005, 11 pages, hep-ph/0603137.
  22. Ettefaghi M.M., Haghighat M., Lorentz conserving noncommutative standard model, Phys. Rev. D 75 (2007), 125002, 11 pages, hep-ph/0703313.
  23. Saxell S., On general properties of Lorentz-invariant formulation of noncommutative quantum field theory, Phys. Lett. B 666 (2008), 486-490, arXiv:0804.3341.
  24. Aschieri P., Lizzi F., Vitale P., Twisting all the way: from classical mechanics to quantum fields, Phys. Rev. D 77 (2008), 025037, 16 pages, arXiv:0708.3002.
    Chaichian M., Salminem T., Tureanu A., Nishijima K., Noncommutative quantum field theory: a confrontation of symmetries, J. High Energy Phys. 2008 (2008), no. 6, 078, 20 pages, arXiv:0805.3500.
    Banerjee R., Kumar K., Deformed relativistic and nonrelativistic symmetries on canonical noncommutative spaces, Phys. Rev. D 75 (2007), 045008, 5 pages, hep-th/0604162.
  25. Chaichian M., Nishijima K., Tureanu A., An interpretation of noncommutative field theory in terms of a quantum shift, Phys. Lett. B 633 (2006), 129-133, hep-th/0511094.
    Gracia-Bondía J.M., Lizzi F., Ruiz Ruiz F., Vitale P., Noncommutative spacetime symmetries: Twist versus covariance, Phys. Rev. D 74 (2006), 025014, 8 pages, hep-th/0604206.
  26. Wess J., Deformed coordinate spaces; derivatives, in Proceedings of the BW2003 Workshop on Mathematical, Theoretical and Phenomenological Challenges Beyond the Standard Model: Perspectives of Balkans Collaboration (2003, Vrnjacka Banja, Serbia), Vrnjacka Banja, 2003, 122-128, hep-th/0408080.
    Dimitrijevic M., Wess J., Deformed bialgebra of diffeomorphims, hep-th/0411224.
    Aschieri P., Blohmann C., Dimitrijevic M., Meyer F., Schupp P., Wess J., A gravity theory on noncommutative spaces, Classical Quantum Gravity 22 (2005), 3511-3532, hep-th/0504183.
    Koch F., Tsouchnika E., Construction of θ-Poincaré algebras and their invariants on Mθ, Nuclear Phys. B 717 (2005), 387-403, hep-th/0409012.
  27. Chaichian M., Kulish P.P., Nishijima K., Tureanu A., On a Lorentz-invariant interpretation of noncommutative space-time and its implications on noncommutative QFT, Phys. Lett. B 604 (2004), 98-102, hep-th/0408069.
    Chaichian M., Presnajder P., Tureanu A., New concept of relativistic invariance in noncommutative space-time: twisted Poincaré symmetry and its implications, Phys. Rev. Lett. 94 (2005), 151602, 4 pages, hep-th/0409096.
  28. Duval C., Horváthy P.A., The exotic Galilei group and the "Peierls substitution", Phys. Lett. B 479 (2000), 284-290, hep-th/0002233.
  29. Chaichian M., Sheikh-Jabbari M.M., Tureanu A., Hydrogen atom spectrum and the Lamb shift in noncommutative QED, Phys. Rev. Lett. 86 (2001), 2716-2719, hep-th/0010175.
  30. Chaichian M., Demichec A., Presnajder P., Sheikh-Jabbari M.M., Tureanu A., Aharonov-Bohm effect in noncommutative spaces, Phys. Lett. B 527 (2002), 149-154, hep-th/0012175.
  31. Gamboa J., Loewe M., Rojas J.C., Noncommutative quantum mechanics, Phys. Rev. D 64 (2001), 067901, 3 pages, hep-th/0010220.
  32. Nair V.P., Polychronakos A.P., Quantum mechanics on the noncommutative plane and sphere, Phys. Lett. B 505 (2001), 267-274, hep-th/0011172.
  33. Banerjee R., A novel approach to noncommutativity in planar quantum mechanics, Modern Phys. Lett. A 17 (2002), 631-645, hep-th/0106280.
  34. Bellucci S., Nersessian A., Phases in noncommutative quantum mechanics on (pseudo)sphere, Phys. Lett. B 542 (2002), 295-300, hep-th/0205024.
  35. Ho P.-M., Kao H.-C., Noncommutative quantum mechanics from noncommutative quantum field theory, Phys. Rev. Lett. 88 (2002), 151602, 4 pages, hep-th/0110191.
  36. Smailagic A., Spallucci E., Feynman path integral on the non-commutative plane, J. Phys. A: Math. Gen. 36 (2003), L467-L471, hep-th/0307217.
    Smailagic A., Spallucci E., UV divergence free QFT on noncommutative plane, J. Phys. A: Math. Gen. 36 (2003), L517-L521, hep-th/0308193.
  37. Jonke L., Meljanac S., Representations of non-commutative quantum mechanics and symmetries, Eur. Phys. J. C 29 (2003), 433-439, hep-th/0210042.
  38. Kokado A., Okamura T., Saito T., Noncommutative quantum mechanics and Seiberg-Witten map, Phys. Rev. D 69 (2004), 125007, 6 pages, hep-th/0401180.
  39. Kijanka A., Kosinski P., On noncommutative isotropic harmonic oscillator, Phys. Rev. D 70 (2004), 127702, 3 pages, hep-th/0407246.
  40. Dadic I., Jonke L., Meljanac S., Harmonic oscillator on noncommutative spaces, Acta Phys. Slov. 55 (2005), 149-164, hep-th/0301066.
  41. Bellucci S., Yeranyan A., Noncommutative quantum scattering in a central field, Phys. Lett. B 609 (2005), 418-423, hep-th/0412305.
  42. Calmet X., Space-time symmetries of noncommutative spaces, Phys. Rev. D 71 (2005), 085012, 4 pages, hep-th/0411147.
    Calmet X., Selvaggi M., Quantum mechanics on noncommutative spacetime, Phys. Rev. D 74 (2006), 037901, 4 pages, hep-th/0608035.
  43. Scholtz F.G., Chakraborty B., Govaerts J., Vaidya S., Spectrum of the non-commutative spherical well, J. Phys. A: Math. Theor. 40 (2007), 14581-14592, arXiv:0709.3357.
  44. Rosenbaum M., Vergara J.D., Juarez L.R., Noncommutative field theory from quantum mechanical space-space noncommutativity, Phys. Lett. A 367 (2007), 1-10, arXiv:0709.3499.
  45. Iorio A., Sýkora T., On the space-time symmetries of noncommutative gauge theories, Internat. J. Modern Phys. A 17 (2002), 2369-2376, hep-th/0111049.
  46. Amorim R., Tensor operators in noncommutative quantum mechanics, Phys. Rev. Lett. 101 (2008), 081602, 4 pages, arXiv:0804.4400.
  47. Amorim R., Dynamical symmetries in noncommutative theories, Phys. Rev. D 78 (2008), 105003, 7 pages, arXiv:0808.3062.
  48. Amorim R., Fermions and noncommutative theories, J. Math. Phys. 50 (2009), 022303, 7 pages, arXiv:0808.3903.
  49. Amorim R., Tensor coordinates in noncommutative mechanics, J. Math. Phys. 50 (2009), 052103, 7 pages, arXiv:0804.4405.
  50. Amorim R., Abreu E.M.C., Quantum complex scalar fields and noncommutativity, Phys. Rev. D 80 (2009), 105010, 6 pages, arXiv:0909.0465.
    Amorim R., Abreu E.M.C., Guzman Ramirez W., Noncommutative relativistic particles, Phys. Rev. D 81 (2010), 105005, 7 pages, arXiv:1001.2178.
  51. Cohen-Tannoudji C., Diu B., Laloe F., Quantum mechanics, John Wiley & Sons, New York, 1997.
  52. Dirac P.A.M., Lectures on quantum mechanics, Yeshiva University, New York, 1964.
    Sundermeyer K., Constrained dynamics. With applications to Yang-Mills theory, general relativity, classical spin, dual string model, Lecture Notes in Physics, Vol. 169, Springer-Verlag, Berlin - New York, 1982.
    Henneaux M., Teitelboim C., Quantization of gauge systems, Princeton University Press, Princeton, NJ, 1992.
  53. Abreu E.M.C., Mendes A.C.R., Oliveira W., Zangirolami A.O., Compactification of the noncommutative DFRA space, work in progress.
  54. Oksanen M.A., Noncommutative gravitation as a gauge theory of twisted Poincaré symmetry, M.Sc. Thesis, 2008.
  55. Abe Y., Correspondence between Poincaré symmetry of commutative QFT and twisted Poincaré symmetry of noncommutative QFT, Phys. Rev. D 77 (2008), 125009, 9 pages, arXiv:0709.1010.
    Banerjee R., Samanta S., Gauge symmetries on θ-deformed spaces, J. High Energy Phys. 2007 (2007), no. 2, 046, 17 pages, hep-th/0611249.
    Banerjee R., Samanta S., Gauge generators, transformations and identities on a noncommutative space, Eur. Phys. J. C 51 (2007), 207-215, hep-th/0608214.
    Tureanu A., Twisted Poincaré symmetry and some implications on noncommutative quantum field theory, Prog. Theor. Phys. Suppl. 171 (2007), 34-41, arXiv:0706.0334.

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