Selected topics in Planck-scale physics

Modern Physics Letters A

Volumn 18, Issue 16, 2003, Pages 1073-1097

  Ng, Y Jack ^a  

^a UNIVERSITY OF NORTH CAROLINA   (United States)

Author keywords

Holographic principle; Planck scale physics; Quantum gravity phenomenology; Spacetime foam

Indexed keywords

CALCULATION; COMPUTER; COSMIC RADIATION; ENERGY; FOAM; GAMMA RADIATION; GRAVITY; HOLOGRAPHY; INTERFEROMETRY; LASER; PHENOMENOLOGY; PHOTON; PHYSICS; QUANTUM THEORY; RATING SCALE; REVIEW; SHORT SURVEY; SPACE; TIME;

EID: 0037663899     PISSN: 02177323     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0217732303010934     Document Type: Short Survey

References (139)

1. See, e.g., J. Polchinski, String Theory (Cambridge Univ. Press, 1998), and references therein.
2. See, e.g., R. Gambini and J. Pullin, Loops, Knots, Gauge Theories and Quantum Gravity (Cambridge Univ. Press, 1996), and references therein.
3. S. Hawking, Commun. Math. Phys. 87, 395 (1982); J. Ellis, J. S. Hagelin, D. V. Nanopoulos and M. Srednicki, Nucl. Phys. B241, 381 (1984).
4. S. Hawking, Commun. Math. Phys. 87, 395 (1982); J. Ellis, J. S. Hagelin, D. V. Nanopoulos and M. Srednicki, Nucl. Phys. B241, 381 (1984).
5. L. Smolin, hep-th/0303185.
6. Y. J. Ng and H. van Dam, Found. Phys. 30, 795 (2000).
7. Y. J. Ng, Int. J. Mod. Phys. D11, 1585 (2002).
8. E. P. Wigner, Rev. Mod. Phys. 29, 255 (1957); H. Salecker and E. P. Wigner, Phys. Rev. 109, 571 (1958).
9. E. P. Wigner, Rev. Mod. Phys. 29, 255 (1957); H. Salecker and E. P. Wigner, Phys. Rev. 109, 571 (1958).
10. Y. J. Ng and H. van Dam, Mod. Phys. Lett. A9, 335 (1994); A10, 2801 (1995). Also see F. Karolyhazy, Nuovo Cimento A42, 390 (1966); D. V. Ahluwalia, Phys. Lett. B339, 301 (1994); N. Sasakura, Prog. Theor. Phys. 102, 169 (1999).
11. Y. J. Ng and H. van Dam, Mod. Phys. Lett. A9, 335 (1994); A10, 2801 (1995). Also see F. Karolyhazy, Nuovo Cimento A42, 390 (1966); D. V. Ahluwalia, Phys. Lett. B339, 301 (1994); N. Sasakura, Prog. Theor. Phys. 102, 169 (1999).
12. Y. J. Ng and H. van Dam, Mod. Phys. Lett. A9, 335 (1994); A10, 2801 (1995). Also see F. Karolyhazy, Nuovo Cimento A42, 390 (1966); D. V. Ahluwalia, Phys. Lett. B339, 301 (1994); N. Sasakura, Prog. Theor. Phys. 102, 169 (1999).
13. Y. J. Ng and H. van Dam, Mod. Phys. Lett. A9, 335 (1994); A10, 2801 (1995). Also see F. Karolyhazy, Nuovo Cimento A42, 390 (1966); D. V. Ahluwalia, Phys. Lett. B339, 301 (1994); N. Sasakura, Prog. Theor. Phys. 102, 169 (1999).
14. Y. J. Ng and H. van Dam, Mod. Phys. Lett. A9, 335 (1994); A10, 2801 (1995). Also see F. Karolyhazy, Nuovo Cimento A42, 390 (1966); D. V. Ahluwalia, Phys. Lett. B339, 301 (1994); N. Sasakura, Prog. Theor. Phys. 102, 169 (1999).
15. B. L. Hu and E. Verdaguer, Class. Quantum Grav. 20, R1 (2003).
16. G. 't Hooft, in Salamfestschrift, edited by A. Ali et al. (World Scientific, 1993), p. 284; L. Susskind, J. Math. Phys. (N. Y.) 36, 6377 (1995). Also see J. A. Wheeler, Int. J. Theor. Phys. 21, 557 (1982); J. D. Bekenstein, Phys. Rev. D7, 2333 (1973); S. Hawking, Commun. Math. Phys. 43, 199 (1975).
17. G. 't Hooft, in Salamfestschrift, edited by A. Ali et al. (World Scientific, 1993), p. 284; L. Susskind, J. Math. Phys. (N. Y.) 36, 6377 (1995). Also see J. A. Wheeler, Int. J. Theor. Phys. 21, 557 (1982); J. D. Bekenstein, Phys. Rev. D7, 2333 (1973); S. Hawking, Commun. Math. Phys. 43, 199 (1975).
18. G. 't Hooft, in Salamfestschrift, edited by A. Ali et al. (World Scientific, 1993), p. 284; L. Susskind, J. Math. Phys. (N. Y.) 36, 6377 (1995). Also see J. A. Wheeler, Int. J. Theor. Phys. 21, 557 (1982); J. D. Bekenstein, Phys. Rev. D7, 2333 (1973); S. Hawking, Commun. Math. Phys. 43, 199 (1975).
19. G. 't Hooft, in Salamfestschrift, edited by A. Ali et al. (World Scientific, 1993), p. 284; L. Susskind, J. Math. Phys. (N. Y.) 36, 6377 (1995). Also see J. A. Wheeler, Int. J. Theor. Phys. 21, 557 (1982); J. D. Bekenstein, Phys. Rev. D7, 2333 (1973); S. Hawking, Commun. Math. Phys. 43, 199 (1975).
20. G. 't Hooft, in Salamfestschrift, edited by A. Ali et al. (World Scientific, 1993), p. 284; L. Susskind, J. Math. Phys. (N. Y.) 36, 6377 (1995). Also see J. A. Wheeler, Int. J. Theor. Phys. 21, 557 (1982); J. D. Bekenstein, Phys. Rev. D7, 2333 (1973); S. Hawking, Commun. Math. Phys. 43, 199 (1975).
21. Y. J. Ng, Phys. Rev. Lett. 86, 2946 (2001); 88, 139902-1(E) (2002); Y. J. Ng, in Proc. of OCPA 2000, eds. N. G. Chang et al. (World Scientific, 2002), p. 235.
22. Y. J. Ng, Phys. Rev. Lett. 86, 2946 (2001); 88, 139902-1(E) (2002); Y. J. Ng, in Proc. of OCPA 2000, eds. N. G. Chang et al. (World Scientific, 2002), p. 235.
23. Y. J. Ng, Phys. Rev. Lett. 86, 2946 (2001); 88, 139902-1(E) (2002); Y. J. Ng, in Proc. of OCPA 2000, eds. N. G. Chang et al. (World Scientific, 2002), p. 235.
24. J. D. Barrow, Phys. Rev. D54, 6563 (1996).
25. S. Lloyd, Nature (London) 406, 1047 (2000).
26. P. Zizzi, gr-qc/0304032.
27. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, 1973), p. 1190.
28. G. Amelino-Camelia, Nature 398, 216 (1999).
29. L. Diosi and B. Lukacs, Phys. Lett. A142, 331 (1989).
30. Y. J. Ng, W. Christiansen and H. van Dam, astro-ph/0302372.
31. Y. J. Ng and H. van Dam, in Proc. of Fundamental Problems in Quantum Theory, eds. D. M. Greenberger and A. Zeilinger, Ann. Phys. (N. Y.) 755, 579 (1995).
32. For lightcone fluctuations, see L. H. Ford, Phys. Rev. D51, 1692 (1995).
33. H. van Dam and M. Veltman, Nucl. Phys. B22, 397 (1970); Z. I. Zakharov, JETP Lett. 12, 312 (1970); A. Higuchi, Nucl. Phys. B282, 397 (1987).
34. H. van Dam and M. Veltman, Nucl. Phys. B22, 397 (1970); Z. I. Zakharov, JETP Lett. 12, 312 (1970); A. Higuchi, Nucl. Phys. B282, 397 (1987).
35. H. van Dam and M. Veltman, Nucl. Phys. B22, 397 (1970); Z. I. Zakharov, JETP Lett. 12, 312 (1970); A. Higuchi, Nucl. Phys. B282, 397 (1987).
36. Y. J. Ng, D. S. Lee, M. C. Oh and H. van Dam, Phys. Lett. B507, 236 (2001); hep-ph/0010152 and references therein. The arXiv preprint is a more informative version of the paper in Phys. Lett. B.
37. G. Amelino-Camelia, J. Ellis, N. E. Mavromatos, D. V. Nanopoulos and S. Sarkar, Nature 393, 763 (1998).
38. G. Amelino-Camelia and T. Piran, hep-th/0006210; astro-ph/0008107.
39. B. E. Schaefer, Phys. Rev. Lett. 82, 4964 (1999); S. D. Biller et al., ibid. 83, 2108 (1999).
40. B. E. Schaefer, Phys. Rev. Lett. 82, 4964 (1999); S. D. Biller et al., ibid. 83, 2108 (1999).
41. R. Lieu and L. W. Hillman, astro-ph/0211402.
42. R. Lieu and L. W. Hillman, astro-ph/0301184.
43. R. Ragazzoni, M. Turatto and W. Gaessler, astro-ph/0303043.
44. E. S. Perlman et al., Astro. J. 124, 2401 (2002).
45. G. T. van Belle, R. R. Thompson and M. J. Creech-Eakman, Astro. J. 124, 1706 (2002).
46. D. H. Coule, astro-ph/0302333.
47. V. Radeka, Annu. Rev. Nucl. Part. Sci. 38, 217 (1988).
48. A. Abramovici et al., Phys. Lett. A218, 157 (1996).
49. A. Abramovici et al., Science 256, 325 (1992).
50. K. Danzmann, Class. Quantum Grav. 13, A247 (1996).
51. G. Amelino-Camelia, gr-qc/0001100.
52. I. Percival, Phys. World, March 1997, p. 43.
53. F. Benatti and R. Floreanini, quant-ph/0204094, 0208164.
54. L. B. Crowell, Found. Phys., in press.
55. M. A. Lawrence et al., J. Phys. G17, 733 (1991); N. N. Efimov et al., in 22nd Int. Cosmic Ray Conf. (Dublin, 1991); D. J. Bird et al., Astrophys. J. 441, 144 (1995); M. Takeda et al., Phys. Rev. Lett. 81, 1163 (1998); A. Watson, in Proc. Snowmass Workshop, 126 (1996).
56. M. A. Lawrence et al., J. Phys. G17, 733 (1991); N. N. Efimov et al., in 22nd Int. Cosmic Ray Conf. (Dublin, 1991); D. J. Bird et al., Astrophys. J. 441, 144 (1995); M. Takeda et al., Phys. Rev. Lett. 81, 1163 (1998); A. Watson, in Proc. Snowmass Workshop, 126 (1996).
57. M. A. Lawrence et al., J. Phys. G17, 733 (1991); N. N. Efimov et al., in 22nd Int. Cosmic Ray Conf. (Dublin, 1991); D. J. Bird et al., Astrophys. J. 441, 144 (1995); M. Takeda et al., Phys. Rev. Lett. 81, 1163 (1998); A. Watson, in Proc. Snowmass Workshop, 126 (1996).
58. M. A. Lawrence et al., J. Phys. G17, 733 (1991); N. N. Efimov et al., in 22nd Int. Cosmic Ray Conf. (Dublin, 1991); D. J. Bird et al., Astrophys. J. 441, 144 (1995); M. Takeda et al., Phys. Rev. Lett. 81, 1163 (1998); A. Watson, in Proc. Snowmass Workshop, 126 (1996).
59. M. A. Lawrence et al., J. Phys. G17, 733 (1991); N. N. Efimov et al., in 22nd Int. Cosmic Ray Conf. (Dublin, 1991); D. J. Bird et al., Astrophys. J. 441, 144 (1995); M. Takeda et al., Phys. Rev. Lett. 81, 1163 (1998); A. Watson, in Proc. Snowmass Workshop, 126 (1996).
60. F. A. Aharonian et al., Astron. Astrophys. 349, 11A (1999); R. J. Protheroe and H. Meyer, astro-ph/0005349; F. W. Stecker, astro-ph/0010015; M. Harwit, P. J. Protheroe and P. L. Biermann, Ap. J. 524, L91 (1999).
61. F. A. Aharonian et al., Astron. Astrophys. 349, 11A (1999); R. J. Protheroe and H. Meyer, astro-ph/0005349; F. W. Stecker, astro-ph/0010015; M. Harwit, P. J. Protheroe and P. L. Biermann, Ap. J. 524, L91 (1999).
62. F. A. Aharonian et al., Astron. Astrophys. 349, 11A (1999); R. J. Protheroe and H. Meyer, astro-ph/0005349; F. W. Stecker, astro-ph/0010015; M. Harwit, P. J. Protheroe and P. L. Biermann, Ap. J. 524, L91 (1999).
63. F. A. Aharonian et al., Astron. Astrophys. 349, 11A (1999); R. J. Protheroe and H. Meyer, astro-ph/0005349; F. W. Stecker, astro-ph/0010015; M. Harwit, P. J. Protheroe and P. L. Biermann, Ap. J. 524, L91 (1999).
64. K. Greisen, Phys. Rev. Lett. 16, 748 (1966); G. T. Zatsepin and V. A. Kuz'min, JETP Lett. 41, 78 (1966).
65. K. Greisen, Phys. Rev. Lett. 16, 748 (1966); G. T. Zatsepin and V. A. Kuz'min, JETP Lett. 41, 78 (1966).
66. A. I. Nikishov, Sov. Phys. JETP 14, 393 (1962); J. Gould and G. Schreder, Phys. Rev. 155, 1404 (1967); W. Stecker, O. C. De Jager and M. H. Salomon, Ap. J. 390, L49 (1992).
67. A. I. Nikishov, Sov. Phys. JETP 14, 393 (1962); J. Gould and G. Schreder, Phys. Rev. 155, 1404 (1967); W. Stecker, O. C. De Jager and M. H. Salomon, Ap. J. 390, L49 (1992).
68. A. I. Nikishov, Sov. Phys. JETP 14, 393 (1962); J. Gould and G. Schreder, Phys. Rev. 155, 1404 (1967); W. Stecker, O. C. De Jager and M. H. Salomon, Ap. J. 390, L49 (1992).
69. C. J. Cesarsky, Nucl. Phys. (Proc. Suppl.) B28, 51 (1992); L. Gonzalez-Mestres, physics/9704017; R. Aloisio, P. Blasi, P. L. Ghia and A. F. Grillo, astro-ph/0001258; O. Bertolami and C. S. Carvalho, Phys. Rev. D61, 103002 (2000); H. Sato, astro-ph/0005218; T. Kifune, Astrophys. J. Lett. 518, L21 (1999); W. Kluzniak, astro-ph/9905308; S. Coleman and S. L. Glashow, Phys. Rev. D59, 116008 (1999); D. Colladay and A. Kostelecky, ibid. D55, 6760 (1997); R. Lieu, Ap. J. 568, L67 (2002). Also see F. W. Stecker, astro-ph/0304527.
70. C. J. Cesarsky, Nucl. Phys. (Proc. Suppl.) B28, 51 (1992); L. Gonzalez-Mestres, physics/9704017; R. Aloisio, P. Blasi, P. L. Ghia and A. F. Grillo, astro-ph/0001258; O. Bertolami and C. S. Carvalho, Phys. Rev. D61, 103002 (2000); H. Sato, astro-ph/0005218; T. Kifune, Astrophys. J. Lett. 518, L21 (1999); W. Kluzniak, astro-ph/9905308; S. Coleman and S. L. Glashow, Phys. Rev. D59, 116008 (1999); D. Colladay and A. Kostelecky, ibid. D55, 6760 (1997); R. Lieu, Ap. J. 568, L67 (2002). Also see F. W. Stecker, astro-ph/0304527.
71. C. J. Cesarsky, Nucl. Phys. (Proc. Suppl.) B28, 51 (1992); L. Gonzalez-Mestres, physics/9704017; R. Aloisio, P. Blasi, P. L. Ghia and A. F. Grillo, astro-ph/0001258; O. Bertolami and C. S. Carvalho, Phys. Rev. D61, 103002 (2000); H. Sato, astro-ph/0005218; T. Kifune, Astrophys. J. Lett. 518, L21 (1999); W. Kluzniak, astro-ph/9905308; S. Coleman and S. L. Glashow, Phys. Rev. D59, 116008 (1999); D. Colladay and A. Kostelecky, ibid. D55, 6760 (1997); R. Lieu, Ap. J. 568, L67 (2002). Also see F. W. Stecker, astro-ph/0304527.
72. C. J. Cesarsky, Nucl. Phys. (Proc. Suppl.) B28, 51 (1992); L. Gonzalez-Mestres, physics/9704017; R. Aloisio, P. Blasi, P. L. Ghia and A. F. Grillo, astro-ph/0001258; O. Bertolami and C. S. Carvalho, Phys. Rev. D61, 103002 (2000); H. Sato, astro-ph/0005218; T. Kifune, Astrophys. J. Lett. 518, L21 (1999); W. Kluzniak, astro-ph/9905308; S. Coleman and S. L. Glashow, Phys. Rev. D59, 116008 (1999); D. Colladay and A. Kostelecky, ibid. D55, 6760 (1997); R. Lieu, Ap. J. 568, L67 (2002). Also see F. W. Stecker, astro-ph/0304527.
73. C. J. Cesarsky, Nucl. Phys. (Proc. Suppl.) B28, 51 (1992); L. Gonzalez-Mestres, physics/9704017; R. Aloisio, P. Blasi, P. L. Ghia and A. F. Grillo, astro-ph/0001258; O. Bertolami and C. S. Carvalho, Phys. Rev. D61, 103002 (2000); H. Sato, astro-ph/0005218; T. Kifune, Astrophys. J. Lett. 518, L21 (1999); W. Kluzniak, astro-ph/9905308; S. Coleman and S. L. Glashow, Phys. Rev. D59, 116008 (1999); D. Colladay and A. Kostelecky, ibid. D55, 6760 (1997); R. Lieu, Ap. J. 568, L67 (2002). Also see F. W. Stecker, astro-ph/0304527.
74. C. J. Cesarsky, Nucl. Phys. (Proc. Suppl.) B28, 51 (1992); L. Gonzalez-Mestres, physics/9704017; R. Aloisio, P. Blasi, P. L. Ghia and A. F. Grillo, astro-ph/0001258; O. Bertolami and C. S. Carvalho, Phys. Rev. D61, 103002 (2000); H. Sato, astro-ph/0005218; T. Kifune, Astrophys. J. Lett. 518, L21 (1999); W. Kluzniak, astro-ph/9905308; S. Coleman and S. L. Glashow, Phys. Rev. D59, 116008 (1999); D. Colladay and A. Kostelecky, ibid. D55, 6760 (1997); R. Lieu, Ap. J. 568, L67 (2002). Also see F. W. Stecker, astro-ph/0304527.
75. C. J. Cesarsky, Nucl. Phys. (Proc. Suppl.) B28, 51 (1992); L. Gonzalez-Mestres, physics/9704017; R. Aloisio, P. Blasi, P. L. Ghia and A. F. Grillo, astro-ph/0001258; O. Bertolami and C. S. Carvalho, Phys. Rev. D61, 103002 (2000); H. Sato, astro-ph/0005218; T. Kifune, Astrophys. J. Lett. 518, L21 (1999); W. Kluzniak, astro-ph/9905308; S. Coleman and S. L. Glashow, Phys. Rev. D59, 116008 (1999); D. Colladay and A. Kostelecky, ibid. D55, 6760 (1997); R. Lieu, Ap. J. 568, L67 (2002). Also see F. W. Stecker, astro-ph/0304527.
76. C. J. Cesarsky, Nucl. Phys. (Proc. Suppl.) B28, 51 (1992); L. Gonzalez-Mestres, physics/9704017; R. Aloisio, P. Blasi, P. L. Ghia and A. F. Grillo, astro-ph/0001258; O. Bertolami and C. S. Carvalho, Phys. Rev. D61, 103002 (2000); H. Sato, astro-ph/0005218; T. Kifune, Astrophys. J. Lett. 518, L21 (1999); W. Kluzniak, astro-ph/9905308; S. Coleman and S. L. Glashow, Phys. Rev. D59, 116008 (1999); D. Colladay and A. Kostelecky, ibid. D55, 6760 (1997); R. Lieu, Ap. J. 568, L67 (2002). Also see F. W. Stecker, astro-ph/0304527.
77. C. J. Cesarsky, Nucl. Phys. (Proc. Suppl.) B28, 51 (1992); L. Gonzalez-Mestres, physics/9704017; R. Aloisio, P. Blasi, P. L. Ghia and A. F. Grillo, astro-ph/0001258; O. Bertolami and C. S. Carvalho, Phys. Rev. D61, 103002 (2000); H. Sato, astro-ph/0005218; T. Kifune, Astrophys. J. Lett. 518, L21 (1999); W. Kluzniak, astro-ph/9905308; S. Coleman and S. L. Glashow, Phys. Rev. D59, 116008 (1999); D. Colladay and A. Kostelecky, ibid. D55, 6760 (1997); R. Lieu, Ap. J. 568, L67 (2002). Also see F. W. Stecker, astro-ph/0304527.
78. C. J. Cesarsky, Nucl. Phys. (Proc. Suppl.) B28, 51 (1992); L. Gonzalez-Mestres, physics/9704017; R. Aloisio, P. Blasi, P. L. Ghia and A. F. Grillo, astro-ph/0001258; O. Bertolami and C. S. Carvalho, Phys. Rev. D61, 103002 (2000); H. Sato, astro-ph/0005218; T. Kifune, Astrophys. J. Lett. 518, L21 (1999); W. Kluzniak, astro-ph/9905308; S. Coleman and S. L. Glashow, Phys. Rev. D59, 116008 (1999); D. Colladay and A. Kostelecky, ibid. D55, 6760 (1997); R. Lieu, Ap. J. 568, L67 (2002). Also see F. W. Stecker, astro-ph/0304527.
79. C. J. Cesarsky, Nucl. Phys. (Proc. Suppl.) B28, 51 (1992); L. Gonzalez-Mestres, physics/9704017; R. Aloisio, P. Blasi, P. L. Ghia and A. F. Grillo, astro-ph/0001258; O. Bertolami and C. S. Carvalho, Phys. Rev. D61, 103002 (2000); H. Sato, astro-ph/0005218; T. Kifune, Astrophys. J. Lett. 518, L21 (1999); W. Kluzniak, astro-ph/9905308; S. Coleman and S. L. Glashow, Phys. Rev. D59, 116008 (1999); D. Colladay and A. Kostelecky, ibid. D55, 6760 (1997); R. Lieu, Ap. J. 568, L67 (2002). Also see F. W. Stecker, astro-ph/0304527.
80. G. Amelino-Camelia, J. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Int. J. Mod. Phys. A12, 607 (1997).
81. G. Amelino-Camelia, Y. J. Ng and H. van Dam, gr-qc/0204077, to appear in Astropart. Phys.
82. R. Aloisio, P. Blasi, A. Galante, P. L. Ghia and A. F. Grillo, Astropart. Phys. 19, 127 (2003).
83. Y. J. Ng, talk given in the Huntsville Workshop 2002 (unpublished).
84. See, e.g., G. Amelino-Camelia, Int. J. Mod. Phys. D11, 35 (2002); J. Magueijo and L. Smolin, Phys. Rev. Lett. 88, 190403 (2002); S. Judes and M. Visser, gr-qc/0205067; J. Kowalski-Glikman, hep-th/0207279. Also see D. V. Ahluwalia, gr-qc/0212128.
85. See, e.g., G. Amelino-Camelia, Int. J. Mod. Phys. D11, 35 (2002); J. Magueijo and L. Smolin, Phys. Rev. Lett. 88, 190403 (2002); S. Judes and M. Visser, gr-qc/0205067; J. Kowalski-Glikman, hep-th/0207279. Also see D. V. Ahluwalia, gr-qc/0212128.
86. See, e.g., G. Amelino-Camelia, Int. J. Mod. Phys. D11, 35 (2002); J. Magueijo and L. Smolin, Phys. Rev. Lett. 88, 190403 (2002); S. Judes and M. Visser, gr-qc/0205067; J. Kowalski-Glikman, hep-th/0207279. Also see D. V. Ahluwalia, gr-qc/0212128.
87. See, e.g., G. Amelino-Camelia, Int. J. Mod. Phys. D11, 35 (2002); J. Magueijo and L. Smolin, Phys. Rev. Lett. 88, 190403 (2002); S. Judes and M. Visser, gr-qc/0205067; J. Kowalski-Glikman, hep-th/0207279. Also see D. V. Ahluwalia, gr-qc/0212128.
88. See, e.g., G. Amelino-Camelia, Int. J. Mod. Phys. D11, 35 (2002); J. Magueijo and L. Smolin, Phys. Rev. Lett. 88, 190403 (2002); S. Judes and M. Visser, gr-qc/0205067; J. Kowalski-Glikman, hep-th/0207279. Also see D. V. Ahluwalia, gr-qc/0212128.
89. R. Aloisio, P. Blasi, A. Galante and A. F. Grillo, astro-ph/0304050; R. Le Gallou, astro-ph/0304560.
90. R. Aloisio, P. Blasi, A. Galante and A. F. Grillo, astro-ph/0304050; R. Le Gallou, astro-ph/0304560.
91. P. Chen and T. Tajima, Phys. Rev. Lett. 83, 256 (1999).
92. J. Ellis, J. S. Hagelin, D. V. Nanopoulos and M. Srednicki, Nucl. Phys. B241, 381 (1984); V. A. Kostelecky and R. Potting, Phys. Rev. D51, 3923 (1995); P. Huet and M. E. Peskin, Nucl. Phys. B434, 3 (1995).
93. J. Ellis, J. S. Hagelin, D. V. Nanopoulos and M. Srednicki, Nucl. Phys. B241, 381 (1984); V. A. Kostelecky and R. Potting, Phys. Rev. D51, 3923 (1995); P. Huet and M. E. Peskin, Nucl. Phys. B434, 3 (1995).
94. J. Ellis, J. S. Hagelin, D. V. Nanopoulos and M. Srednicki, Nucl. Phys. B241, 381 (1984); V. A. Kostelecky and R. Potting, Phys. Rev. D51, 3923 (1995); P. Huet and M. E. Peskin, Nucl. Phys. B434, 3 (1995).
95. R. C. Myers and M. Pospelov, hep-ph/0301124; T. E. Chupp et al., Phys. Rev. Lett. 72, 2363 (1994); E. G. Adelberger et al., in Physics Beyond the Standard Model (World Scientific, 1999); R. Gambini and J. Pullin, Phys. Rev. D59, 124021 (1999); J. Alfaro, H. A. Morales-Tecotl and L. F. Urrutia, ibid. D65, 103509 (2002).
96. R. C. Myers and M. Pospelov, hep-ph/0301124; T. E. Chupp et al., Phys. Rev. Lett. 72, 2363 (1994); E. G. Adelberger et al., in Physics Beyond the Standard Model (World Scientific, 1999); R. Gambini and J. Pullin, Phys. Rev. D59, 124021 (1999); J. Alfaro, H. A. Morales-Tecotl and L. F. Urrutia, ibid. D65, 103509 (2002).
97. R. C. Myers and M. Pospelov, hep-ph/0301124; T. E. Chupp et al., Phys. Rev. Lett. 72, 2363 (1994); E. G. Adelberger et al., in Physics Beyond the Standard Model (World Scientific, 1999); R. Gambini and J. Pullin, Phys. Rev. D59, 124021 (1999); J. Alfaro, H. A. Morales-Tecotl and L. F. Urrutia, ibid. D65, 103509 (2002).
98. R. C. Myers and M. Pospelov, hep-ph/0301124; T. E. Chupp et al., Phys. Rev. Lett. 72, 2363 (1994); E. G. Adelberger et al., in Physics Beyond the Standard Model (World Scientific, 1999); R. Gambini and J. Pullin, Phys. Rev. D59, 124021 (1999); J. Alfaro, H. A. Morales-Tecotl and L. F. Urrutia, ibid. D65, 103509 (2002).
99. R. C. Myers and M. Pospelov, hep-ph/0301124; T. E. Chupp et al., Phys. Rev. Lett. 72, 2363 (1994); E. G. Adelberger et al., in Physics Beyond the Standard Model (World Scientific, 1999); R. Gambini and J. Pullin, Phys. Rev. D59, 124021 (1999); J. Alfaro, H. A. Morales-Tecotl and L. F. Urrutia, ibid. D65, 103509 (2002).
100. T. J. Konopka and S. A. Major, New J. Phys. 4, 57 (2002); T. Jacobson, S. Liberati and D. Mattingly, hep-ph/0112207.
101. T. J. Konopka and S. A. Major, New J. Phys. 4, 57 (2002); T. Jacobson, S. Liberati and D. Mattingly, hep-ph/0112207.
102. G. Amelino-Camelia, gr-qc/0107086; E. E. Antonov et al., Pis'ma ZhETF 73, 506 (2001).
103. G. Amelino-Camelia, gr-qc/0107086; E. E. Antonov et al., Pis'ma ZhETF 73, 506 (2001).
104. L. Gonzalez-Mestres, astro-ph/0011182; T. Jacobson, S. Liberati and D. Mattingly, astro-ph/0212190; gr-qc/0303001; G. Amelino-Camelia, gr-qc/0212002.
105. L. Gonzalez-Mestres, astro-ph/0011182; T. Jacobson, S. Liberati and D. Mattingly, astro-ph/0212190; gr-qc/0303001; G. Amelino-Camelia, gr-qc/0212002.
106. L. Gonzalez-Mestres, astro-ph/0011182; T. Jacobson, S. Liberati and D. Mattingly, astro-ph/0212190; gr-qc/0303001; G. Amelino-Camelia, gr-qc/0212002.
107. R. J. Gleiser and C. N. Kozameh, gr-qc/0102093; R. Gambini and J. Pullin, Phys. Rev. D59, 124021 (1999).
108. R. J. Gleiser and C. N. Kozameh, gr-qc/0102093; R. Gambini and J. Pullin, Phys. Rev. D59, 124021 (1999).
109. B. E. Schaefer, Phys. Rev. Lett. 82, 4964 (1999); S. D. Biller et al., ibid. 83, 2108 (1999).
110. B. E. Schaefer, Phys. Rev. Lett. 82, 4964 (1999); S. D. Biller et al., ibid. 83, 2108 (1999).
111. J. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Phys. Rev. D63, 124025 (2001).
112. See, e.g., J. Lukierski et al., Phys. Lett. B264, 331 (1991); S. Majid and H. Ruegg, ibid. B334, 348 (1994).
113. See, e.g., J. Lukierski et al., Phys. Lett. B264, 331 (1991); S. Majid and H. Ruegg, ibid. B334, 348 (1994).
114. R. V. Mendes, J. Phys. A27, 8091 (1994).
115. R. Brandenberger, hep-th/0210186; A. Kempf, Phys. Rev. D63, 083514 (2001); R. Easther, B. Greene, W. Kinney and G. Shiu, ibid. D67, 063508 (2003); S. Shankaranarayanan, Class. Quantum Grav. 20, 75 (2003).
116. R. Brandenberger, hep-th/0210186; A. Kempf, Phys. Rev. D63, 083514 (2001); R. Easther, B. Greene, W. Kinney and G. Shiu, ibid. D67, 063508 (2003); S. Shankaranarayanan, Class. Quantum Grav. 20, 75 (2003).
117. R. Brandenberger, hep-th/0210186; A. Kempf, Phys. Rev. D63, 083514 (2001); R. Easther, B. Greene, W. Kinney and G. Shiu, ibid. D67, 063508 (2003); S. Shankaranarayanan, Class. Quantum Grav. 20, 75 (2003).
118. R. Brandenberger, hep-th/0210186; A. Kempf, Phys. Rev. D63, 083514 (2001); R. Easther, B. Greene, W. Kinney and G. Shiu, ibid. D67, 063508 (2003); S. Shankaranarayanan, Class. Quantum Grav. 20, 75 (2003).
119. L. Mersini, M. Bastero-Gil and P. Kanti, Phys. Rev. D64, 043508 (2001); M. Bastero-Gil, P. H. Frampton and L. Mersini, ibid. D65, 106002 (2002); P. H. Frampton, Phys. Lett. B555, 139 (2003).
120. L. Mersini, M. Bastero-Gil and P. Kanti, Phys. Rev. D64, 043508 (2001); M. Bastero-Gil, P. H. Frampton and L. Mersini, ibid. D65, 106002 (2002); P. H. Frampton, Phys. Lett. B555, 139 (2003).
121. L. Mersini, M. Bastero-Gil and P. Kanti, Phys. Rev. D64, 043508 (2001); M. Bastero-Gil, P. H. Frampton and L. Mersini, ibid. D65, 106002 (2002); P. H. Frampton, Phys. Lett. B555, 139 (2003).
122. Y. J. Ng and H. van Dam, Phys. Lett. A309, 335 (2003).
123. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965); also see T. Jacobson, J. Phys. A17, 2433 (1984); I. Bialynicki-Birula, Phys. Rev. D49, 6920 (1994); L. H. Kauffman and H. P. Noyes, Phys. Lett. A218, 139 (1996).
124. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965); also see T. Jacobson, J. Phys. A17, 2433 (1984); I. Bialynicki-Birula, Phys. Rev. D49, 6920 (1994); L. H. Kauffman and H. P. Noyes, Phys. Lett. A218, 139 (1996).
125. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965); also see T. Jacobson, J. Phys. A17, 2433 (1984); I. Bialynicki-Birula, Phys. Rev. D49, 6920 (1994); L. H. Kauffman and H. P. Noyes, Phys. Lett. A218, 139 (1996).
126. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965); also see T. Jacobson, J. Phys. A17, 2433 (1984); I. Bialynicki-Birula, Phys. Rev. D49, 6920 (1994); L. H. Kauffman and H. P. Noyes, Phys. Lett. A218, 139 (1996).
127. P. A. M. Dirac, Quantum Mechanics (Oxford Univ. Press, 1958), 4th edition.
128. D. V. Ahluwalia and M. Kirchbach, Int. J. Mod. Phys. D10, 811 (2001).
129. T. Regge, Nuovo Cimento 19, 558 (1961).
130. J. J. van der Bij, H. van Dam and Y. J. Ng, Physica A116, 307 (1982).
131. Y. J. Ng and H. van Dam, Phys. Rev. Lett. 65, 1972 (1990); Int. J. Mod. Phys. D10, 49 (2001), and the many references on unimodular gravity contained therein.
132. Y. J. Ng and H. van Dam, Phys. Rev. Lett. 65, 1972 (1990); Int. J. Mod. Phys. D10, 49 (2001), and the many references on unimodular gravity contained therein.
133. M. Henneaux and C. Teitelboim, Phys. Lett. B222, 195 (1989).
134. R. D. Sorkin, in Relativity and Gravitation: Classical and Quantum, eds. J. C. D'Olivo et al. (World Scientific, 1991); Int. J. Theor. Phys. 36, 2759 (1997); T. Padmanabhan, gr-qc/0112068.
135. R. D. Sorkin, in Relativity and Gravitation: Classical and Quantum, eds. J. C. D'Olivo et al. (World Scientific, 1991); Int. J. Theor. Phys. 36, 2759 (1997); T. Padmanabhan, gr-qc/0112068.
136. R. D. Sorkin, in Relativity and Gravitation: Classical and Quantum, eds. J. C. D'Olivo et al. (World Scientific, 1991); Int. J. Theor. Phys. 36, 2759 (1997); T. Padmanabhan, gr-qc/0112068.
137. E. Baum, Phys. Lett. B133, 185 (1984).
138. S. W. Hawking, Phys. Lett. B134, 403 (1984).
139. A of this form was proposed by W. Chen and Y. S. Wu, Phys. Rev. D41, 695 (1990), based on other reasons.