Ward identity in noncommutative QED

International Journal of Modern Physics A

Volumn 18, Issue 29, 2003, Pages 5433-5444

  Mariz, T ^a   Pires, C A De S ^a   Ribeiro, R F ^a  

^a FEDERAL UNIVERSITY OF PARAÍBA   (Brazil)

Author keywords

Noncommutative QED; Pair annihilation process; Ward identity

Indexed keywords

ARTICLE; CALCULATION; DYNAMICS; ELEMENTARY PARTICLE; FORCE; MATHEMATICAL ANALYSIS; PHOTON; POLARIZATION; QUANTUM ELECTRODYNAMICS; QUANTUM MECHANICS; SPACE; TIME; VELOCITY;

EID: 0346895096     PISSN: 0217751X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0217751X03015945     Document Type: Article

References (56)

1. H. S. Snyder, Phys. Rev. D71, 38 (1947).
2. A. Connes, M. R. Douglas and A. Schwarz, J. High Energy Phys. 9802, 003 (1998).
3. N. Seiberg and E. Witten, J. High Energy Phys. 9909, 032 (1999).
4. M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001).
5. S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. Lane and T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001); Z. Guralnik, R. Jackiw, S. Y. Pi and A. P. Polychronakos, Phys. Lett. B517, 450 (2001).
6. S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. Lane and T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001); Z. Guralnik, R. Jackiw, S. Y. Pi and A. P. Polychronakos, Phys. Lett. B517, 450 (2001).
7. M. V. Raamsdonk and N. Seiberg, J. High Energy Phys. 03, 035 (2000); H. Grosse, C. Klimcik and P. Presnajder, Commun. Math. Phys. 180, 429 (1996); I. Ya, Aref'eva, D. M. Belov and A. S. Koshelev, Phys. Lett. B476, 431 (2000).
8. M. V. Raamsdonk and N. Seiberg, J. High Energy Phys. 03, 035 (2000); H. Grosse, C. Klimcik and P. Presnajder, Commun. Math. Phys. 180, 429 (1996); I. Ya, Aref'eva, D. M. Belov and A. S. Koshelev, Phys. Lett. B476, 431 (2000).
9. M. V. Raamsdonk and N. Seiberg, J. High Energy Phys. 03, 035 (2000); H. Grosse, C. Klimcik and P. Presnajder, Commun. Math. Phys. 180, 429 (1996); I. Ya, Aref'eva, D. M. Belov and A. S. Koshelev, Phys. Lett. B476, 431 (2000).
10. S. Minwalla, M. V. Raamsdonk and N. Seiberg, J. High Energy Phys. 02, 020 (2000).
11. Ihab. F. Riad and M. M. Sheikh-Jabbari, J. High Energy Phys. 08, 045 (2000).
12. M. Hayakama, Phys. Lett. B478, 394 (2000).
13. A. Matusis, L. Susskind and N. Toumbas, J. High Energy Phys. 0012, 002 (2000).
14. Gomis and T. Mehen, Nucl. Phys. B591, 265 (2000); C.-S. Chu, J. Lukierski and W. J. Zakrzewski, Nucl. Phys. B632, 219 (2002); Y. Liao and K. Sibold, Eur. Phys. J. C25, 479 (2002).
15. Gomis and T. Mehen, Nucl. Phys. B591, 265 (2000); C.-S. Chu, J. Lukierski and W. J. Zakrzewski, Nucl. Phys. B632, 219 (2002); Y. Liao and K. Sibold, Eur. Phys. J. C25, 479 (2002).
16. Gomis and T. Mehen, Nucl. Phys. B591, 265 (2000); C.-S. Chu, J. Lukierski and W. J. Zakrzewski, Nucl. Phys. B632, 219 (2002); Y. Liao and K. Sibold, Eur. Phys. J. C25, 479 (2002).
17. G. Arcioni and M. A. Vazquez-Mozo, J. High Energy Phys. 01, 028 (2000); F. T. Brandt, J. Frenkel and D. G. C. McKeon, Phys. Rev. D65, 125029 (2002); F. T. Brandt, A. Das and J. Frenkel, Phys. Rev. D65, 085017 (2002).
18. G. Arcioni and M. A. Vazquez-Mozo, J. High Energy Phys. 01, 028 (2000); F. T. Brandt, J. Frenkel and D. G. C. McKeon, Phys. Rev. D65, 125029 (2002); F. T. Brandt, A. Das and J. Frenkel, Phys. Rev. D65, 085017 (2002).
19. G. Arcioni and M. A. Vazquez-Mozo, J. High Energy Phys. 01, 028 (2000); F. T. Brandt, J. Frenkel and D. G. C. McKeon, Phys. Rev. D65, 125029 (2002); F. T. Brandt, A. Das and J. Frenkel, Phys. Rev. D65, 085017 (2002).
20. T. Krajewski and R. Wulkenhaar, Int. J. Mod. Phys. A15, 1011 (2000); A. Armoni, Nucl. Phys. B593, 229 (2001).
21. T. Krajewski and R. Wulkenhaar, Int. J. Mod. Phys. A15, 1011 (2000); A. Armoni, Nucl. Phys. B593, 229 (2001).
22. M. M. Sheikh-Jabbari, J. High Energy Phys. 9906, 015 (1999); R. J. Szabo, hep-th/0109162.
23. M. M. Sheikh-Jabbari, J. High Energy Phys. 9906, 015 (1999); R. J. Szabo, hep-th/0109162.
24. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
25. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
26. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
27. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
28. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
29. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
30. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
31. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
32. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
33. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
34. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
35. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
36. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
37. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
38. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
39. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
40. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
41. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
42. For an incomplete list of works on NCQFT see: T. Filk, Phys. Lett. B376, 53 (1996); I. Mociou, M. Pospelov and R. Roiban, Phys. Lett. B489, 390 (2000); H. O. Girotti, M. Gomes, V. O. Rivelles and A. J. da Silva, Nucl. Phys. B587, 299 (2000); M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000); L. Alvarez-Gaumé and S. Wadia, Phys. Lett. B501, 319 (2001); C. P. Martin and F. Ruiz, Nucl. Phys. B597, 197 (2001); C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. B518, 201 (2001); A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar J. High Energy Phys. 0106, 013 (2001); J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D64, 067901 (2001); C. E. Carlson, C. D. Carone and N. Zobin, Phys. Rev. D66, 075001-1 (2002); R. Banerjee and S. Ghosh, Phys. Lett. B533, 162 (2002); P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88, 151602-1 (2002); H. Falomir, J. Gamboa, M. Loewe, F. Mendez and J. C. Rojas, Phys. Rev. D66, 045018-1 (2002); M. Lubo, Phys. Rev. D65, 066003 (2002); H. O. Girotti, M. Gomes, A. Y. Petrov, V. O. Rivelles and A. J. da Silva, hep-th/0207220; K. Morita, hep-th/0209234; C. E. Carlson and C. D. Carone, hep-ph/0209077; R. Banerjee, hep-th/0210259; C. P. Martin, hep-th/0211164.
43. For an excelente review on the subject see: I. Hinchliffe and N. Kersting, hep-ph/0205040.
44. P. Mathews, Phys. Rev. D63, 075007-1 (2001); T. Rizzo, hep-ph/0203240; S. Godfrey and M. A. Doricheski, hep-ph/0111147; A. Anisimov, T. Banks, M. Dine, M. Graesser, H. Grosse and Y. Liao Phys. Rev. D64, 115007 (2001).
45. P. Mathews, Phys. Rev. D63, 075007-1 (2001); T. Rizzo, hep-ph/0203240; S. Godfrey and M. A. Doricheski, hep-ph/0111147; A. Anisimov, T. Banks, M. Dine, M. Graesser, H. Grosse and Y. Liao Phys. Rev. D64, 115007 (2001).
46. P. Mathews, Phys. Rev. D63, 075007-1 (2001); T. Rizzo, hep-ph/0203240; S. Godfrey and M. A. Doricheski, hep-ph/0111147; A. Anisimov, T. Banks, M. Dine, M. Graesser, H. Grosse and Y. Liao Phys. Rev. D64, 115007 (2001).
47. P. Mathews, Phys. Rev. D63, 075007-1 (2001); T. Rizzo, hep-ph/0203240; S. Godfrey and M. A. Doricheski, hep-ph/0111147; A. Anisimov, T. Banks, M. Dine, M. Graesser, H. Grosse and Y. Liao Phys. Rev. D64, 115007 (2001).
48. N. Chair and M. M. Sheikh-Jabbari, Phys. Lett. B504, 146 (2001).
49. J. L. Hewett, F. J. Petriello and T. G. Rizzo, Phys. Rev. D64, 075012 (2001); ibid. Phys. Rev. D64, 075012 (2001); S. Godfrey and M. A. Doncheski, Phys. Rev. D65, 015005 (2002); S.-W. Baek, D. K. Ghosh, X.-G. He and W. Y. P. Hwang, Phys. Rev. D64, 056001 (2001); N. Mahajan, hep-ph/0110148.
50. J. L. Hewett, F. J. Petriello and T. G. Rizzo, Phys. Rev. D64, 075012 (2001); ibid. Phys. Rev. D64, 075012 (2001); S. Godfrey and M. A. Doncheski, Phys. Rev. D65, 015005 (2002); S.-W. Baek, D. K. Ghosh, X.-G. He and W. Y. P. Hwang, Phys. Rev. D64, 056001 (2001); N. Mahajan, hep-ph/0110148.
51. J. L. Hewett, F. J. Petriello and T. G. Rizzo, Phys. Rev. D64, 075012 (2001); ibid. Phys. Rev. D64, 075012 (2001); S. Godfrey and M. A. Doncheski, Phys. Rev. D65, 015005 (2002); S.-W. Baek, D. K. Ghosh, X.-G. He and W. Y. P. Hwang, Phys. Rev. D64, 056001 (2001); N. Mahajan, hep-ph/0110148.
52. J. L. Hewett, F. J. Petriello and T. G. Rizzo, Phys. Rev. D64, 075012 (2001); ibid. Phys. Rev. D64, 075012 (2001); S. Godfrey and M. A. Doncheski, Phys. Rev. D65, 015005 (2002); S.-W. Baek, D. K. Ghosh, X.-G. He and W. Y. P. Hwang, Phys. Rev. D64, 056001 (2001); N. Mahajan, hep-ph/0110148.
53. J. L. Hewett, F. J. Petriello and T. G. Rizzo, Phys. Rev. D64, 075012 (2001); ibid. Phys. Rev. D64, 075012 (2001); S. Godfrey and M. A. Doncheski, Phys. Rev. D65, 015005 (2002); S.-W. Baek, D. K. Ghosh, X.-G. He and W. Y. P. Hwang, Phys. Rev. D64, 056001 (2001); N. Mahajan, hep-ph/0110148.
54. The works Refs. 8 and 12 have checked the Ward identity at loop level for two point functions.
55. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, New York, 1995).
56. This mapping can be drawn straightforward from a reading of the action or of the Feynman rules of the NCQED. This was first perceived by Jabbari in Ref. 14. However from such reading we are notable to infer the role played by mapping in the NCQED.