Noncommutative geometry and cosmology

Physical Review D - Particles, Fields, Gravitation and Cosmology

Volumn 70, Issue 10, 2004, Pages

  Barbosa, G D ^a   Pinto Neto, N ^a  

^a CENTRO BRASILEIRO DE PESQUISAS FÍSICAS   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE; COSMOS; EVOLUTION; GEOMETRY; GRAVITY; QUANTUM MECHANICS;

EID: 42749099573     PISSN: 05562821     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevD.70.103512     Document Type: Article

References (76)

1. R. J. Szabo, Phys. Rep. 378, 207 (2003).
2. M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73, 977 (2002).
3. N. Seiberg and E. Witten, J. High Energy Phys. 09 (1999) 032.
4. S. Doplicher, K. Fredenhagen, and J. E. Roberts, Phys. Lett. B 331, 39 (1994); Commun. Math. Phys. 172, 187 (1995).
5. S. Doplicher, K. Fredenhagen, and J. E. Roberts, Phys. Lett. B 331, 39 (1994); Commun. Math. Phys. 172, 187 (1995).
6. S. Minwalla, M. Van Raamsdonk, and N. Seiberg, J. High Energy Phys. 02 (2000) 020.
7. S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. Lane, and T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001); C. E. Carlson, C. D. Carone, and R. F. Lebed, Phys. Lett. B 518, 201 (2001); 549, 337 (2002).
8. S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. Lane, and T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001); C. E. Carlson, C. D. Carone, and R. F. Lebed, Phys. Lett. B 518, 201 (2001); 549, 337 (2002).
9. S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. Lane, and T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001); C. E. Carlson, C. D. Carone, and R. F. Lebed, Phys. Lett. B 518, 201 (2001); 549, 337 (2002).
10. I. Bars, hep-th/0109132.
11. J. M. Romero and J. A. Santiago, hep-th/0310266.
12. R. Brandenberger and P.-M. Ho, Phys. Rev. D 66, 023517 (2002); Q.-G. Huang and M. Li, J. Cosmol. Astropart. Phys. 11 (2003) 001; J. High Energy Phys. 06 (2003) 014; astro-ph/0311378; H. Kim, G. S. Lee, and Y. S. Myung, hep-th/0402018; H. Kim, G. S. Lee, H. W. Lee, and Y. S. Myung, Phys. Rev. D 70, 043521 (2004); Y. S. Myung, Phys. Lett. B 601, 1 (2004); Dao-jun Liu and Xin-zhou Li, astro-ph/0402063; G. Calcagni, hep-th/0406006 [Phys. Rev. D (to be published)]; hep-ph/0406057; Rong-Gen Cai, Phys. Lett. B 593, 1 (2004); C.-S. Chu, B. R. Greene, and G. Shiu, Mod. Phys. Lett. A 16, 2231 (2001).
13. R. Brandenberger and P.-M. Ho, Phys. Rev. D 66, 023517 (2002); Q.-G. Huang and M. Li, J. Cosmol. Astropart. Phys. 11 (2003) 001; J. High Energy Phys. 06 (2003) 014; astro-ph/0311378; H. Kim, G. S. Lee, and Y. S. Myung, hep-th/0402018; H. Kim, G. S. Lee, H. W. Lee, and Y. S. Myung, Phys. Rev. D 70, 043521 (2004); Y. S. Myung, Phys. Lett. B 601, 1 (2004); Dao-jun Liu and Xin-zhou Li, astro-ph/0402063; G. Calcagni, hep-th/0406006 [Phys. Rev. D (to be published)]; hep-ph/0406057; Rong-Gen Cai, Phys. Lett. B 593, 1 (2004); C.-S. Chu, B. R. Greene, and G. Shiu, Mod. Phys. Lett. A 16, 2231 (2001).
14. R. Brandenberger and P.-M. Ho, Phys. Rev. D 66, 023517 (2002); Q.-G. Huang and M. Li, J. Cosmol. Astropart. Phys. 11 (2003) 001; J. High Energy Phys. 06 (2003) 014; astro-ph/0311378; H. Kim, G. S. Lee, and Y. S. Myung, hep-th/0402018; H. Kim, G. S. Lee, H. W. Lee, and Y. S. Myung, Phys. Rev. D 70, 043521 (2004); Y. S. Myung, Phys. Lett. B 601, 1 (2004); Dao-jun Liu and Xin-zhou Li, astro-ph/0402063; G. Calcagni, hep-th/0406006 [Phys. Rev. D (to be published)]; hep-ph/0406057; Rong-Gen Cai, Phys. Lett. B 593, 1 (2004); C.-S. Chu, B. R. Greene, and G. Shiu, Mod. Phys. Lett. A 16, 2231 (2001).
15. R. Brandenberger and P.-M. Ho, Phys. Rev. D 66, 023517 (2002); Q.-G. Huang and M. Li, J. Cosmol. Astropart. Phys. 11 (2003) 001; J. High Energy Phys. 06 (2003) 014; astro-ph/0311378; H. Kim, G. S. Lee, and Y. S. Myung, hep-th/0402018; H. Kim, G. S. Lee, H. W. Lee, and Y. S. Myung, Phys. Rev. D 70, 043521 (2004); Y. S. Myung, Phys. Lett. B 601, 1 (2004); Dao-jun Liu and Xin-zhou Li, astro-ph/0402063; G. Calcagni, hep-th/0406006 [Phys. Rev. D (to be published)]; hep-ph/0406057; Rong-Gen Cai, Phys. Lett. B 593, 1 (2004); C.-S. Chu, B. R. Greene, and G. Shiu, Mod. Phys. Lett. A 16, 2231 (2001).
16. R. Brandenberger and P.-M. Ho, Phys. Rev. D 66, 023517 (2002); Q.-G. Huang and M. Li, J. Cosmol. Astropart. Phys. 11 (2003) 001; J. High Energy Phys. 06 (2003) 014; astro-ph/0311378; H. Kim, G. S. Lee, and Y. S. Myung, hep-th/0402018; H. Kim, G. S. Lee, H. W. Lee, and Y. S. Myung, Phys. Rev. D 70, 043521 (2004); Y. S. Myung, Phys. Lett. B 601, 1 (2004); Dao-jun Liu and Xin-zhou Li, astro-ph/0402063; G. Calcagni, hep-th/0406006 [Phys. Rev. D (to be published)]; hep-ph/0406057; Rong-Gen Cai, Phys. Lett. B 593, 1 (2004); C.-S. Chu, B. R. Greene, and G. Shiu, Mod. Phys. Lett. A 16, 2231 (2001).
17. R. Brandenberger and P.-M. Ho, Phys. Rev. D 66, 023517 (2002); Q.-G. Huang and M. Li, J. Cosmol. Astropart. Phys. 11 (2003) 001; J. High Energy Phys. 06 (2003) 014; astro-ph/0311378; H. Kim, G. S. Lee, and Y. S. Myung, hep-th/0402018; H. Kim, G. S. Lee, H. W. Lee, and Y. S. Myung, Phys. Rev. D 70, 043521 (2004); Y. S. Myung, Phys. Lett. B 601, 1 (2004); Dao-jun Liu and Xin-zhou Li, astro-ph/0402063; G. Calcagni, hep-th/0406006 [Phys. Rev. D (to be published)]; hep-ph/0406057; Rong-Gen Cai, Phys. Lett. B 593, 1 (2004); C.-S. Chu, B. R. Greene, and G. Shiu, Mod. Phys. Lett. A 16, 2231 (2001).
18. R. Brandenberger and P.-M. Ho, Phys. Rev. D 66, 023517 (2002); Q.-G. Huang and M. Li, J. Cosmol. Astropart. Phys. 11 (2003) 001; J. High Energy Phys. 06 (2003) 014; astro-ph/0311378; H. Kim, G. S. Lee, and Y. S. Myung, hep-th/0402018; H. Kim, G. S. Lee, H. W. Lee, and Y. S. Myung, Phys. Rev. D 70, 043521 (2004); Y. S. Myung, Phys. Lett. B 601, 1 (2004); Dao-jun Liu and Xin-zhou Li, astro-ph/0402063; G. Calcagni, hep-th/0406006 [Phys. Rev. D (to be published)]; hep-ph/0406057; Rong-Gen Cai, Phys. Lett. B 593, 1 (2004); C.-S. Chu, B. R. Greene, and G. Shiu, Mod. Phys. Lett. A 16, 2231 (2001).
19. R. Brandenberger and P.-M. Ho, Phys. Rev. D 66, 023517 (2002); Q.-G. Huang and M. Li, J. Cosmol. Astropart. Phys. 11 (2003) 001; J. High Energy Phys. 06 (2003) 014; astro-ph/0311378; H. Kim, G. S. Lee, and Y. S. Myung, hep-th/0402018; H. Kim, G. S. Lee, H. W. Lee, and Y. S. Myung, Phys. Rev. D 70, 043521 (2004); Y. S. Myung, Phys. Lett. B 601, 1 (2004); Dao-jun Liu and Xin-zhou Li, astro-ph/0402063; G. Calcagni, hep-th/0406006 [Phys. Rev. D (to be published)]; hep-ph/0406057; Rong-Gen Cai, Phys. Lett. B 593, 1 (2004); C.-S. Chu, B. R. Greene, and G. Shiu, Mod. Phys. Lett. A 16, 2231 (2001).
20. M. Maceda, J. Madore, P. Manousselis, and G. Zoupanos, Eur. Phys. J. C 36, 529 (2004).
21. H. Garcia-Compeán, O. Obregón, and C. Ramírez, Phys. Rev. Lett. 88, 161301 (2002).
22. H. Everett, Rev. Mod. Phys. 29, 454 (1957).
23. R. P. Feynman, F. B. Morinigo, and W. G. Wagner, Feynman Lectures on Gravitation (Addison-Wesley, Reading, MA, 1995).
24. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1993).
25. G. 't Hooft, hep-th/0003005; Classical Quantum Gravity 16, 3263 (1999).
26. R. Kantowski and R. K. Sachs, J. Math. Phys. (N.Y.) 7, 443 (1966); G. F. R. Ellis, J. Math. Phys. (N.Y.) 8, 1171 (1967); C. B. Collins, J. Math. Phys. (N.Y.) 18, 2116 (1977); E. Weber, J. Math. Phys. (N.Y.) 25, 3279 (1984); M. P. Ryan and L. C. Shepley, Homogeneous Relativistic Cosmologies (Princeton University Press, Princeton, NJ, 1975).
27. R. Kantowski and R. K. Sachs, J. Math. Phys. (N.Y.) 7, 443 (1966); G. F. R. Ellis, J. Math. Phys. (N.Y.) 8, 1171 (1967); C. B. Collins, J. Math. Phys. (N.Y.) 18, 2116 (1977); E. Weber, J. Math. Phys. (N.Y.) 25, 3279 (1984); M. P. Ryan and L. C. Shepley, Homogeneous Relativistic Cosmologies (Princeton University Press, Princeton, NJ, 1975).
28. R. Kantowski and R. K. Sachs, J. Math. Phys. (N.Y.) 7, 443 (1966); G. F. R. Ellis, J. Math. Phys. (N.Y.) 8, 1171 (1967); C. B. Collins, J. Math. Phys. (N.Y.) 18, 2116 (1977); E. Weber, J. Math. Phys. (N.Y.) 25, 3279 (1984); M. P. Ryan and L. C. Shepley, Homogeneous Relativistic Cosmologies (Princeton University Press, Princeton, NJ, 1975).
29. R. Kantowski and R. K. Sachs, J. Math. Phys. (N.Y.) 7, 443 (1966); G. F. R. Ellis, J. Math. Phys. (N.Y.) 8, 1171 (1967); C. B. Collins, J. Math. Phys. (N.Y.) 18, 2116 (1977); E. Weber, J. Math. Phys. (N.Y.) 25, 3279 (1984); M. P. Ryan and L. C. Shepley, Homogeneous Relativistic Cosmologies (Princeton University Press, Princeton, NJ, 1975).
30. R. Kantowski and R. K. Sachs, J. Math. Phys. (N.Y.) 7, 443 (1966); G. F. R. Ellis, J. Math. Phys. (N.Y.) 8, 1171 (1967); C. B. Collins, J. Math. Phys. (N.Y.) 18, 2116 (1977); E. Weber, J. Math. Phys. (N.Y.) 25, 3279 (1984); M. P. Ryan and L. C. Shepley, Homogeneous Relativistic Cosmologies (Princeton University Press, Princeton, NJ, 1975).
31. H.-D. Conradi, Classical Quantum Gravity 12, 2423 (1995).
32. A. N. Makarenko, V. V. Obukhov, and K. E. Osetrin, gr-qc/0301124.
33. D. Solomons, P. Dunsby, and G. Ellis, gr-qc/0103087.
34. C. Simeone, Gen. Relativ. Gravit. 34, 1887 (2002); 32, 1835 (2000).
35. C. Simeone, Gen. Relativ. Gravit. 34, 1887 (2002); 32, 1835 (2000).
36. M. Cavaglià, V. Alfaro, and A. T. Filippov, Int. J. Mod. Phys. D 4, 661 (1995); O. Obregon and M. P. Ryan, Mod. Phys. Lett. A 13, 3251 (1998); L. M. Campbell and L. J. Garay, Phys. Lett. B 254, 49 (1991); M. Cavaglià, Mod. Phys. Lett. A 9, 1897 (1994).
37. M. Cavaglià, V. Alfaro, and A. T. Filippov, Int. J. Mod. Phys. D 4, 661 (1995); O. Obregon and M. P. Ryan, Mod. Phys. Lett. A 13, 3251 (1998); L. M. Campbell and L. J. Garay, Phys. Lett. B 254, 49 (1991); M. Cavaglià, Mod. Phys. Lett. A 9, 1897 (1994).
38. M. Cavaglià, V. Alfaro, and A. T. Filippov, Int. J. Mod. Phys. D 4, 661 (1995); O. Obregon and M. P. Ryan, Mod. Phys. Lett. A 13, 3251 (1998); L. M. Campbell and L. J. Garay, Phys. Lett. B 254, 49 (1991); M. Cavaglià, Mod. Phys. Lett. A 9, 1897 (1994).
39. M. Cavaglià, V. Alfaro, and A. T. Filippov, Int. J. Mod. Phys. D 4, 661 (1995); O. Obregon and M. P. Ryan, Mod. Phys. Lett. A 13, 3251 (1998); L. M. Campbell and L. J. Garay, Phys. Lett. B 254, 49 (1991); M. Cavaglià, Mod. Phys. Lett. A 9, 1897 (1994).
40. D. Bohm, Phys. Rev. 85, 166 (1952); 85, 180 (1952).
41. D. Bohm, Phys. Rev. 85, 166 (1952); 85, 180 (1952).
42. D. Bohm, B. J. Hiley, and P. N. Kaloyerou, Phys. Rep. 144, 349 (1987).
43. D. Bohm and B. J. Hiley, The Undivided Universe: An Ontological Interpretation of Quantum Theory (Routledge & Kegan Paul, London, 1993); V. Allori and N. Zanghi, quant-ph/0112008.
44. P. R. Holland, The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics (Cambridge University Press, Cambridge, 1993).
45. J. Acacio de Barros and N. Pinto-Neto, Int. J. Mod. Phys. D 7, 201 (1998).
46. J. Kowalski-Glikman and J. C. Vink, Classical Quantum Gravity 7, 901 (1990); Phys. Lett. A 162, 35 (1992); Y. V. Shtanov, Phys. Rev. D 54, 2564 (1996); R. Colistete, Jr., J. C. Fabris, and N. Pinto-Neto, Phys. Rev. D 62, 083507 (2000); F. Shojai and A. Shojai, J. High Energy Phys. 05 (2001) 037; Classical Quantum Gravity 21, 1 (2004); W.-H. Huang and I.-C. Wang, gr-qc/0309042.
47. J. Kowalski-Glikman and J. C. Vink, Classical Quantum Gravity 7, 901 (1990); Phys. Lett. A 162, 35 (1992); Y. V. Shtanov, Phys. Rev. D 54, 2564 (1996); R. Colistete, Jr., J. C. Fabris, and N. Pinto-Neto, Phys. Rev. D 62, 083507 (2000); F. Shojai and A. Shojai, J. High Energy Phys. 05 (2001) 037; Classical Quantum Gravity 21, 1 (2004); W.-H. Huang and I.-C. Wang, gr-qc/0309042.
48. J. Kowalski-Glikman and J. C. Vink, Classical Quantum Gravity 7, 901 (1990); Phys. Lett. A 162, 35 (1992); Y. V. Shtanov, Phys. Rev. D 54, 2564 (1996); R. Colistete, Jr., J. C. Fabris, and N. Pinto-Neto, Phys. Rev. D 62, 083507 (2000); F. Shojai and A. Shojai, J. High Energy Phys. 05 (2001) 037; Classical Quantum Gravity 21, 1 (2004); W.-H. Huang and I.-C. Wang, gr-qc/0309042.
49. J. Kowalski-Glikman and J. C. Vink, Classical Quantum Gravity 7, 901 (1990); Phys. Lett. A 162, 35 (1992); Y. V. Shtanov, Phys. Rev. D 54, 2564 (1996); R. Colistete, Jr., J. C. Fabris, and N. Pinto-Neto, Phys. Rev. D 62, 083507 (2000); F. Shojai and A. Shojai, J. High Energy Phys. 05 (2001) 037; Classical Quantum Gravity 21, 1 (2004); W.-H. Huang and I.-C. Wang, gr-qc/0309042.
50. J. Kowalski-Glikman and J. C. Vink, Classical Quantum Gravity 7, 901 (1990); Phys. Lett. A 162, 35 (1992); Y. V. Shtanov, Phys. Rev. D 54, 2564 (1996); R. Colistete, Jr., J. C. Fabris, and N. Pinto-Neto, Phys. Rev. D 62, 083507 (2000); F. Shojai and A. Shojai, J. High Energy Phys. 05 (2001) 037; Classical Quantum Gravity 21, 1 (2004); W.-H. Huang and I.-C. Wang, gr-qc/0309042.
51. J. Kowalski-Glikman and J. C. Vink, Classical Quantum Gravity 7, 901 (1990); Phys. Lett. A 162, 35 (1992); Y. V. Shtanov, Phys. Rev. D 54, 2564 (1996); R. Colistete, Jr., J. C. Fabris, and N. Pinto-Neto, Phys. Rev. D 62, 083507 (2000); F. Shojai and A. Shojai, J. High Energy Phys. 05 (2001) 037; Classical Quantum Gravity 21, 1 (2004); W.-H. Huang and I.-C. Wang, gr-qc/0309042.
52. N. Pinto-Neto and E. S. Santini, Phys. Rev. D 59, 123517 (1999).
53. J. Acacio de Barros, N. Pinto-Neto, and M. A. Sagioro-Leal, Phys. Lett. A 241, 229 (1998).
54. D. Durr, S. Goldstein, R. Tumulka, and N. Zanghi, J. Phys. A 36, 4143 (2003); C. Colijn and E. R. Vrscay, Phys. Lett. A 300, 334 (2002); A. S. Sans, F. Borondo, and S. Miret-Artés, J. Phys. Condens. Matter 14, 6109 (2002); D. Home and A. S. Majumdar, Found. Phys. 29, 721 (1999); L. Delle Site, Europhys. Lett. 57, 20 (2002); G. Gruebl, R. Moser, and K. Rheinberger, J. Phys. A 34, 2753 (2001).
55. D. Durr, S. Goldstein, R. Tumulka, and N. Zanghi, J. Phys. A 36, 4143 (2003); C. Colijn and E. R. Vrscay, Phys. Lett. A 300, 334 (2002); A. S. Sans, F. Borondo, and S. Miret-Artés, J. Phys. Condens. Matter 14, 6109 (2002); D. Home and A. S. Majumdar, Found. Phys. 29, 721 (1999); L. Delle Site, Europhys. Lett. 57, 20 (2002); G. Gruebl, R. Moser, and K. Rheinberger, J. Phys. A 34, 2753 (2001).
56. D. Durr, S. Goldstein, R. Tumulka, and N. Zanghi, J. Phys. A 36, 4143 (2003); C. Colijn and E. R. Vrscay, Phys. Lett. A 300, 334 (2002); A. S. Sans, F. Borondo, and S. Miret-Artés, J. Phys. Condens. Matter 14, 6109 (2002); D. Home and A. S. Majumdar, Found. Phys. 29, 721 (1999); L. Delle Site, Europhys. Lett. 57, 20 (2002); G. Gruebl, R. Moser, and K. Rheinberger, J. Phys. A 34, 2753 (2001).
57. D. Durr, S. Goldstein, R. Tumulka, and N. Zanghi, J. Phys. A 36, 4143 (2003); C. Colijn and E. R. Vrscay, Phys. Lett. A 300, 334 (2002); A. S. Sans, F. Borondo, and S. Miret-Artés, J. Phys. Condens. Matter 14, 6109 (2002); D. Home and A. S. Majumdar, Found. Phys. 29, 721 (1999); L. Delle Site, Europhys. Lett. 57, 20 (2002); G. Gruebl, R. Moser, and K. Rheinberger, J. Phys. A 34, 2753 (2001).
58. D. Durr, S. Goldstein, R. Tumulka, and N. Zanghi, J. Phys. A 36, 4143 (2003); C. Colijn and E. R. Vrscay, Phys. Lett. A 300, 334 (2002); A. S. Sans, F. Borondo, and S. Miret-Artés, J. Phys. Condens. Matter 14, 6109 (2002); D. Home and A. S. Majumdar, Found. Phys. 29, 721 (1999); L. Delle Site, Europhys. Lett. 57, 20 (2002); G. Gruebl, R. Moser, and K. Rheinberger, J. Phys. A 34, 2753 (2001).
59. D. Durr, S. Goldstein, R. Tumulka, and N. Zanghi, J. Phys. A 36, 4143 (2003); C. Colijn and E. R. Vrscay, Phys. Lett. A 300, 334 (2002); A. S. Sans, F. Borondo, and S. Miret-Artés, J. Phys. Condens. Matter 14, 6109 (2002); D. Home and A. S. Majumdar, Found. Phys. 29, 721 (1999); L. Delle Site, Europhys. Lett. 57, 20 (2002); G. Gruebl, R. Moser, and K. Rheinberger, J. Phys. A 34, 2753 (2001).
60. R. Arnowitt, S. Deser, and C. Misner, in Gravitation: An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962).
61. C. Kiefer, Lect. Notes Phys. 541, 158 (2000); D. L. Wiltshire, in Proceedings of the 8th Physics Summer School, Australian National University, Australia, 1995 (World Scientific, Singapore, 1996); N. Pinto Neto, Quantum Cosmology, VIII Brazilian School of Cosmology and Gravitation (Editions Frontieres, Gifsur-Yvette, 1996).
62. C. Kiefer, Lect. Notes Phys. 541, 158 (2000); D. L. Wiltshire, in Proceedings of the 8th Physics Summer School, Australian National University, Australia, 1995 (World Scientific, Singapore, 1996); N. Pinto Neto, Quantum Cosmology, VIII Brazilian School of Cosmology and Gravitation (Editions Frontieres, Gifsur-Yvette, 1996).
63. C. Kiefer, Lect. Notes Phys. 541, 158 (2000); D. L. Wiltshire, in Proceedings of the 8th Physics Summer School, Australian National University, Australia, 1995 (World Scientific, Singapore, 1996); N. Pinto Neto, Quantum Cosmology, VIII Brazilian School of Cosmology and Gravitation (Editions Frontieres, Gifsur-Yvette, 1996).
64. N. Pinto Neto and E. S. Santini, Lett. A 315, 36 (2003).
65. S. W. Hawking and G. F. Ellis, The Large Scale Structure of Spacetime (Cambridge University Press, Cambridge, 1973).
66. M. J. Gotay and J. Demaret, Phys. Rev. D 28, 2402 (1983).
67. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, San Diego, 1980), 4th ed., p. 958 (8.432).
68. G. D. Barbosa, J. High Energy Phys. 05 (2003) 024.
69. G. D. Barbosa and N. Pinto-Neto, Phys. Rev. D 69, 065014 (2004).
70. M. Chaichian, M. M. Sheikh-Jabbari, and A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001); Eur. Phys. J. C 36, 251 (2004); V. P. Nair and A. P. Polychronakos, Phys. Lett. B 505, 267 (2001); M. Demetrian and D. Kochan, Acta Physica Slovaca 52, 1 (2002); J. Gamboa, M. Loewe, and J. C. Rojas, Phys. Rev. D 64, 067901 (2001); B. Muthukumar and P. Mitra, Phys. Rev. D 66, 027701 (2002);
71. M. Chaichian, M. M. Sheikh-Jabbari, and A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001); Eur. Phys. J. C 36, 251 (2004); V. P. Nair and A. P. Polychronakos, Phys. Lett. B 505, 267 (2001); M. Demetrian and D. Kochan, Acta Physica Slovaca 52, 1 (2002); J. Gamboa, M. Loewe, and J. C. Rojas, Phys. Rev. D 64, 067901 (2001); B. Muthukumar and P. Mitra, Phys. Rev. D 66, 027701 (2002);
72. M. Chaichian, M. M. Sheikh-Jabbari, and A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001); Eur. Phys. J. C 36, 251 (2004); V. P. Nair and A. P. Polychronakos, Phys. Lett. B 505, 267 (2001); M. Demetrian and D. Kochan, Acta Physica Slovaca 52, 1 (2002); J. Gamboa, M. Loewe, and J. C. Rojas, Phys. Rev. D 64, 067901 (2001); B. Muthukumar and P. Mitra, Phys. Rev. D 66, 027701 (2002);
73. M. Chaichian, M. M. Sheikh-Jabbari, and A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001); Eur. Phys. J. C 36, 251 (2004); V. P. Nair and A. P. Polychronakos, Phys. Lett. B 505, 267 (2001); M. Demetrian and D. Kochan, Acta Physica Slovaca 52, 1 (2002); J. Gamboa, M. Loewe, and J. C. Rojas, Phys. Rev. D 64, 067901 (2001); B. Muthukumar and P. Mitra, Phys. Rev. D 66, 027701 (2002);
74. M. Chaichian, M. M. Sheikh-Jabbari, and A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001); Eur. Phys. J. C 36, 251 (2004); V. P. Nair and A. P. Polychronakos, Phys. Lett. B 505, 267 (2001); M. Demetrian and D. Kochan, Acta Physica Slovaca 52, 1 (2002); J. Gamboa, M. Loewe, and J. C. Rojas, Phys. Rev. D 64, 067901 (2001); B. Muthukumar and P. Mitra, Phys. Rev. D 66, 027701 (2002);
75. M. Chaichian, M. M. Sheikh-Jabbari, and A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001); Eur. Phys. J. C 36, 251 (2004); V. P. Nair and A. P. Polychronakos, Phys. Lett. B 505, 267 (2001); M. Demetrian and D. Kochan, Acta Physica Slovaca 52, 1 (2002); J. Gamboa, M. Loewe, and J. C. Rojas, Phys. Rev. D 64, 067901 (2001); B. Muthukumar and P. Mitra, Phys. Rev. D 66, 027701 (2002);
76. S. Bellucci and A. Nersessian, Phys. Lett. B 542, 295 (2002); A. Yelnikov, hep-th/0112134.