Candidate for a background independent formulation of M theory

Physical Review D - Particles, Fields, Gravitation and Cosmology

Volumn 62, Issue 8, 2000, Pages 1-24

  Smolin, Lee ^a  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 17144390255     PISSN: 05562821     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevD.62.086001     Document Type: Article

References (146)

1. C. M. Hull and P. K. Townsend, Nucl. Phys. B348, 109 (1995); E. Witten, ibid. B443, 85 (1995).
2. C. M. Hull and P. K. Townsend, Nucl. Phys. B348, 109 (1995); E. Witten, ibid. B443, 85 (1995).
3. F. Markopoulou and L. Smolin, Phys. Rev. D 58, 084032 (1998).
4. T. Jacobson and L. Smolin, Nucl. Phys. B299, 295 (1988).
5. C. Rovelli and L. Smolin, Phys. Rev. Lett. 61, 1155 (1988); Nucl. Phys. B133, 80 (1990); Phys. Rev. Lett. 72, 446 (1994).
6. C. Rovelli and L. Smolin, Phys. Rev. Lett. 61, 1155 (1988); Nucl. Phys. B133, 80 (1990); Phys. Rev. Lett. 72, 446 (1994).
7. C. Rovelli and L. Smolin, Phys. Rev. Lett. 61, 1155 (1988); Nucl. Phys. B133, 80 (1990); Phys. Rev. Lett. 72, 446 (1994).
8. C. Rovelli and L. Smolin, Nucl. Phys. B442, 593 (1995); B456, 734(E) (1995).
9. C. Rovelli and L. Smolin, Nucl. Phys. B442, 593 (1995); B456, 734(E) (1995).
10. C. Rovelli and L. Smolin, Phys. Rev. D 52, 5743 (1995).
11. L. Smolin, in Quantum Gravity and Cosmology, edited by J. Pérez-Mercader et al. (World Scientific, Singapore, 1992); "The future of spin networks," gr-qc/9702030.
12. L. Smolin, in Quantum Gravity and Cosmology, edited by J. Pérez-Mercader et al. (World Scientific, Singapore, 1992); "The future of spin networks," gr-qc/9702030.
13. C. Rovelli, gr-qc/9710008.
14. T. Jaocobson, Class. Quantum Grav. 5, 923 (1988); D. Armand-Ugon, R. Gambini, O. Obregon, and J. Pullin, Nucl. Phys. B460, 615 (1996); H. Kunitomo and T. Sano, Int. J. Mod. Phys. D 1, 559 (1993); Prog. Thebr. Phys. Suppl. 114, 31 (1993); Takashi Sano and J. Shiraishi, Nucl. Phys. B410, 423 (1993); "The Ashtekar Formalism and WKB Wave Functions of N=1,2 Supergravities," hep-th/9211103; K. Ezawa, Prog. Theor. Phys. 95, 863 (1996); T. Kadoyoshi and S. Nojiri, Mod. Phys. Lett. A 12, 1165 (1997); L. F. Urrutia "Towards a loop representation of connection theories defined over a super-Lie algebra," hep-th/9609010.
15. T. Jaocobson, Class. Quantum Grav. 5, 923 (1988); D. Armand-Ugon, R. Gambini, O. Obregon, and J. Pullin, Nucl. Phys. B460, 615 (1996); H. Kunitomo and T. Sano, Int. J. Mod. Phys. D 1, 559 (1993); Prog. Thebr. Phys. Suppl. 114, 31 (1993); Takashi Sano and J. Shiraishi, Nucl. Phys. B410, 423 (1993); "The Ashtekar Formalism and WKB Wave Functions of N=1,2 Supergravities," hep-th/9211103; K. Ezawa, Prog. Theor. Phys. 95, 863 (1996); T. Kadoyoshi and S. Nojiri, Mod. Phys. Lett. A 12, 1165 (1997); L. F. Urrutia "Towards a loop representation of connection theories defined over a super-Lie algebra," hep-th/9609010.
16. T. Jaocobson, Class. Quantum Grav. 5, 923 (1988); D. Armand-Ugon, R. Gambini, O. Obregon, and J. Pullin, Nucl. Phys. B460, 615 (1996); H. Kunitomo and T. Sano, Int. J. Mod. Phys. D 1, 559 (1993); Prog. Thebr. Phys. Suppl. 114, 31 (1993); Takashi Sano and J. Shiraishi, Nucl. Phys. B410, 423 (1993); "The Ashtekar Formalism and WKB Wave Functions of N=1,2 Supergravities," hep-th/9211103; K. Ezawa, Prog. Theor. Phys. 95, 863 (1996); T. Kadoyoshi and S. Nojiri, Mod. Phys. Lett. A 12, 1165 (1997); L. F. Urrutia "Towards a loop representation of connection theories defined over a super-Lie algebra," hep-th/9609010.
17. T. Jaocobson, Class. Quantum Grav. 5, 923 (1988); D. Armand-Ugon, R. Gambini, O. Obregon, and J. Pullin, Nucl. Phys. B460, 615 (1996); H. Kunitomo and T. Sano, Int. J. Mod. Phys. D 1, 559 (1993); Prog. Thebr. Phys. Suppl. 114, 31 (1993); Takashi Sano and J. Shiraishi, Nucl. Phys. B410, 423 (1993); "The Ashtekar Formalism and WKB Wave Functions of N=1,2 Supergravities," hep-th/9211103; K. Ezawa, Prog. Theor. Phys. 95, 863 (1996); T. Kadoyoshi and S. Nojiri, Mod. Phys. Lett. A 12, 1165 (1997); L. F. Urrutia "Towards a loop representation of connection theories defined over a super-Lie algebra," hep-th/9609010.
18. T. Jaocobson, Class. Quantum Grav. 5, 923 (1988); D. Armand-Ugon, R. Gambini, O. Obregon, and J. Pullin, Nucl. Phys. B460, 615 (1996); H. Kunitomo and T. Sano, Int. J. Mod. Phys. D 1, 559 (1993); Prog. Thebr. Phys. Suppl. 114, 31 (1993); Takashi Sano and J. Shiraishi, Nucl. Phys. B410, 423 (1993); "The Ashtekar Formalism and WKB Wave Functions of N=1,2 Supergravities," hep-th/9211103; K. Ezawa, Prog. Theor. Phys. 95, 863 (1996); T. Kadoyoshi and S. Nojiri, Mod. Phys. Lett. A 12, 1165 (1997); L. F. Urrutia "Towards a loop representation of connection theories defined over a super-Lie algebra," hep-th/9609010.
19. T. Jaocobson, Class. Quantum Grav. 5, 923 (1988); D. Armand-Ugon, R. Gambini, O. Obregon, and J. Pullin, Nucl. Phys. B460, 615 (1996); H. Kunitomo and T. Sano, Int. J. Mod. Phys. D 1, 559 (1993); Prog. Thebr. Phys. Suppl. 114, 31 (1993); Takashi Sano and J. Shiraishi, Nucl. Phys. B410, 423 (1993); "The Ashtekar Formalism and WKB Wave Functions of N=1,2 Supergravities," hep-th/9211103; K. Ezawa, Prog. Theor. Phys. 95, 863 (1996); T. Kadoyoshi and S. Nojiri, Mod. Phys. Lett. A 12, 1165 (1997); L. F. Urrutia "Towards a loop representation of connection theories defined over a super-Lie algebra," hep-th/9609010.
20. T. Jaocobson, Class. Quantum Grav. 5, 923 (1988); D. Armand-Ugon, R. Gambini, O. Obregon, and J. Pullin, Nucl. Phys. B460, 615 (1996); H. Kunitomo and T. Sano, Int. J. Mod. Phys. D 1, 559 (1993); Prog. Thebr. Phys. Suppl. 114, 31 (1993); Takashi Sano and J. Shiraishi, Nucl. Phys. B410, 423 (1993); "The Ashtekar Formalism and WKB Wave Functions of N=1,2 Supergravities," hep-th/9211103; K. Ezawa, Prog. Theor. Phys. 95, 863 (1996); T. Kadoyoshi and S. Nojiri, Mod. Phys. Lett. A 12, 1165 (1997); L. F. Urrutia "Towards a loop representation of connection theories defined over a super-Lie algebra," hep-th/9609010.
21. T. Jaocobson, Class. Quantum Grav. 5, 923 (1988); D. Armand-Ugon, R. Gambini, O. Obregon, and J. Pullin, Nucl. Phys. B460, 615 (1996); H. Kunitomo and T. Sano, Int. J. Mod. Phys. D 1, 559 (1993); Prog. Thebr. Phys. Suppl. 114, 31 (1993); Takashi Sano and J. Shiraishi, Nucl. Phys. B410, 423 (1993); "The Ashtekar Formalism and WKB Wave Functions of N=1,2 Supergravities," hep-th/9211103; K. Ezawa, Prog. Theor. Phys. 95, 863 (1996); T. Kadoyoshi and S. Nojiri, Mod. Phys. Lett. A 12, 1165 (1997); L. F. Urrutia "Towards a loop representation of connection theories defined over a super-Lie algebra," hep-th/9609010.
22. T. Jaocobson, Class. Quantum Grav. 5, 923 (1988); D. Armand-Ugon, R. Gambini, O. Obregon, and J. Pullin, Nucl. Phys. B460, 615 (1996); H. Kunitomo and T. Sano, Int. J. Mod. Phys. D 1, 559 (1993); Prog. Thebr. Phys. Suppl. 114, 31 (1993); Takashi Sano and J. Shiraishi, Nucl. Phys. B410, 423 (1993); "The Ashtekar Formalism and WKB Wave Functions of N=1,2 Supergravities," hep-th/9211103; K. Ezawa, Prog. Theor. Phys. 95, 863 (1996); T. Kadoyoshi and S. Nojiri, Mod. Phys. Lett. A 12, 1165 (1997); L. F. Urrutia "Towards a loop representation of connection theories defined over a super-Lie algebra," hep-th/9609010.
23. Yi Li and L. Smolin, Phys. Rev. D 61, 044008 (2000).
24. See, for example, P. S. Aspinwall, B. R. Greene, and D. R. Morrison, Phys. Lett. B 303, 249 (1993); Nucl. Phys. B420, 184 (1994); Brian R. Greene, David R. Morrison, and Andrew Strominger, ibid. B451, 109 (1995).
25. See, for example, P. S. Aspinwall, B. R. Greene, and D. R. Morrison, Phys. Lett. B 303, 249 (1993); Nucl. Phys. B420, 184 (1994); Brian R. Greene, David R. Morrison, and Andrew Strominger, ibid. B451, 109 (1995).
26. See, for example, P. S. Aspinwall, B. R. Greene, and D. R. Morrison, Phys. Lett. B 303, 249 (1993); Nucl. Phys. B420, 184 (1994); Brian R. Greene, David R. Morrison, and Andrew Strominger, ibid. B451, 109 (1995).
27. L. Smolin and C. Soo, Nucl. Phys. B449, 289 (1995).
28. L. Smolin, J. Math. Phys. 36, 6417 (1995).
29. S. Kauffman and L. Smolin, "Combinatorial dynamics in quantum gravity," report.
30. F. Markopoulou and L. Smolin, Phys. Rev. D 58, 084033 (1998).
31. L. Smolin, "Towards a background independent approach to M theory," hep-th/9808192.
32. B. DeWitt, J. Hoppe, and H. Nicolai, Nucl. Phys. B305, 545 (1988).
33. T. Banks, W. Fishler, S. H. Shenker, and L. Susskind, Phys. Rev. D 55, 5112 (1997); Lubos Motl, "Proposals on nonperturbative superstring interactions," hep-th/9701025; Tom Banks and Nathan Seiberg, Nucl. Phys. B497, 41 (1997); R. Dijkgraaf, E. Verlinde, and H. Verlinde, ibid. B500, 43 (1997).
34. T. Banks, W. Fishler, S. H. Shenker, and L. Susskind, Phys. Rev. D 55, 5112 (1997); Lubos Motl, "Proposals on nonperturbative superstring interactions," hep-th/9701025; Tom Banks and Nathan Seiberg, Nucl. Phys. B497, 41 (1997); R. Dijkgraaf, E. Verlinde, and H. Verlinde, ibid. B500, 43 (1997).
35. T. Banks, W. Fishler, S. H. Shenker, and L. Susskind, Phys. Rev. D 55, 5112 (1997); Lubos Motl, "Proposals on nonperturbative superstring interactions," hep-th/9701025; Tom Banks and Nathan Seiberg, Nucl. Phys. B497, 41 (1997); R. Dijkgraaf, E. Verlinde, and H. Verlinde, ibid. B500, 43 (1997).
36. T. Banks, W. Fishler, S. H. Shenker, and L. Susskind, Phys. Rev. D 55, 5112 (1997); Lubos Motl, "Proposals on nonperturbative superstring interactions," hep-th/9701025; Tom Banks and Nathan Seiberg, Nucl. Phys. B497, 41 (1997); R. Dijkgraaf, E. Verlinde, and H. Verlinde, ibid. B500, 43 (1997).
37. L. Smolin, "Holography, inflation and causality" (in preparation).
38. F. Markopoulou and L. Smolin, "Holography in a quantum spacetime," hep-th/9910146.
39. L. Smolin, "The Bekenstein bound, topological quantum field theory and pluralistic quantum cosmology," gr-qc/9508064.
40. J. D. Bekenstein, Lett. Nuovo Cimento 4, 737 (1972).
41. L. Smolin, Phys. Rev. D 61, 084007 (2000).
42. Yi Ling and L. Smolin, "Holographic formulation of N=2 supergravity" (in preparation).
43. F. Markopoulou, "Dual formulation of spin network evolution," gr-qc/9704013.
44. F. Markopoulou and L. Smolin, Nucl. Phys. B508, 409 (1997).
45. J. Ambjorn and R. Loll, Nucl. Phys. B536, 407 (1998); J. Ambjorn, J. L. Nielsen, J. Rolf, and R. Loll, Chaos, Solitons and Fractals 10, 177 (1999).
46. J. Ambjorn and R. Loll, Nucl. Phys. B536, 407 (1998); J. Ambjorn, J. L. Nielsen, J. Rolf, and R. Loll, Chaos, Solitons and Fractals 10, 177 (1999).
47. R. Borissov and S. Gupta, Phys. Rev. D 60, 024002 (2000).
48. L. Smolin, "Strings as perturbations of evolving spin-networks," hep-th/9801022.
49. L. Crane, J. Math. Phys. 36, 6180 (1995); in Knot Theory and Quantum Gravity, edited by J. Baez (Oxford University Press, New York, 1994); "Categorical Physics," hep-th/9301061.
50. L. Crane, J. Math. Phys. 36, 6180 (1995); in Knot Theory and Quantum Gravity, edited by J. Baez (Oxford University Press, New York, 1994); "Categorical Physics," hep-th/9301061.
51. L. Crane, J. Math. Phys. 36, 6180 (1995); in Knot Theory and Quantum Gravity, edited by J. Baez (Oxford University Press, New York, 1994); "Categorical Physics," hep-th/9301061.
52. R. Penrose, "Theory of quantized directions" (unpublished); in Quantum Theory and Beyond, edited by T. Bastin (Cambridge University Press, Cambridge, England, 1971); in Advances in Twistor Theory, edited by L. P. Hughston and R. S. Ward (Pitman, New York, 1979), p. 301; in Combinatorial Mathematics and its Application, edited by D. J. A. Welsh (Academic Press, New York, 1971); "Theory of quantized directions" (unpublished).
53. R. Penrose, "Theory of quantized directions" (unpublished); in Quantum Theory and Beyond, edited by T. Bastin (Cambridge University Press, Cambridge, England, 1971); in Advances in Twistor Theory, edited by L. P. Hughston and R. S. Ward (Pitman, New York, 1979), p. 301; in Combinatorial Mathematics and its Application, edited by D. J. A. Welsh (Academic Press, New York, 1971); "Theory of quantized directions" (unpublished).
54. R. Penrose, "Theory of quantized directions" (unpublished); in Quantum Theory and Beyond, edited by T. Bastin (Cambridge University Press, Cambridge, England, 1971); in Advances in Twistor Theory, edited by L. P. Hughston and R. S. Ward (Pitman, New York, 1979), p. 301; in Combinatorial Mathematics and its Application, edited by D. J. A. Welsh (Academic Press, New York, 1971); "Theory of quantized directions" (unpublished).
55. R. Penrose, "Theory of quantized directions" (unpublished); in Quantum Theory and Beyond, edited by T. Bastin (Cambridge University Press, Cambridge, England, 1971); in Advances in Twistor Theory, edited by L. P. Hughston and R. S. Ward (Pitman, New York, 1979), p. 301; in Combinatorial Mathematics and its Application, edited by D. J. A. Welsh (Academic Press, New York, 1971); "Theory of quantized directions" (unpublished).
56. R. Penrose, "Theory of quantized directions" (unpublished); in Quantum Theory and Beyond, edited by T. Bastin (Cambridge University Press, Cambridge, England, 1971); in Advances in Twistor Theory, edited by L. P. Hughston and R. S. Ward (Pitman, New York, 1979), p. 301; in Combinatorial Mathematics and its Application, edited by D. J. A. Welsh (Academic Press, New York, 1971); "Theory of quantized directions" (unpublished).
57. Juan M. Maldacena, Adv. Theor. Math. Phys. 2, 23 (1998); Phys. Rev. Lett. 80, 4859 (1998).
58. Juan M. Maldacena, Adv. Theor. Math. Phys. 2, 23 (1998); Phys. Rev. Lett. 80, 4859 (1998).
59. E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998); 2, 505 (1998).
60. E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998); 2, 505 (1998).
61. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B 428, 105 (1998).
62. G. Moore and N. Seiberg, Commun. Math. Phys. 123, 177 (1988).
63. E. Witten, Commun. Math. Phys. 121, 351 (1989).
64. L. Crane, Commun. Math. Phys. 135, 615 (1991); Phys. Lett. B 259, 243 (1991).
65. L. Crane, Commun. Math. Phys. 135, 615 (1991); Phys. Lett. B 259, 243 (1991).
66. E. Verlinde, Nucl. Phys. B300, 360 (1988); G. Moore and N. Yu. Reshetikhin, ibid. B328, 557 (1989); N. Yu. Reshetikhin and V. G. Turaev, Commun. Math. Phys. 127, 1 (1990); Invent. Math. 103, 547 (1991); V. Turaev and O. Viro, Topology 31, 865 (1992).
67. E. Verlinde, Nucl. Phys. B300, 360 (1988); G. Moore and N. Yu. Reshetikhin, ibid. B328, 557 (1989); N. Yu. Reshetikhin and V. G. Turaev, Commun. Math. Phys. 127, 1 (1990); Invent. Math. 103, 547 (1991); V. Turaev and O. Viro, Topology 31, 865 (1992).
68. E. Verlinde, Nucl. Phys. B300, 360 (1988); G. Moore and N. Yu. Reshetikhin, ibid. B328, 557 (1989); N. Yu. Reshetikhin and V. G. Turaev, Commun. Math. Phys. 127, 1 (1990); Invent. Math. 103, 547 (1991); V. Turaev and O. Viro, Topology 31, 865 (1992).
69. E. Verlinde, Nucl. Phys. B300, 360 (1988); G. Moore and N. Yu. Reshetikhin, ibid. B328, 557 (1989); N. Yu. Reshetikhin and V. G. Turaev, Commun. Math. Phys. 127, 1 (1990); Invent. Math. 103, 547 (1991); V. Turaev and O. Viro, Topology 31, 865 (1992).
70. E. Verlinde, Nucl. Phys. B300, 360 (1988); G. Moore and N. Yu. Reshetikhin, ibid. B328, 557 (1989); N. Yu. Reshetikhin and V. G. Turaev, Commun. Math. Phys. 127, 1 (1990); Invent. Math. 103, 547 (1991); V. Turaev and O. Viro, Topology 31, 865 (1992).
71. L. Kauffman and S. L. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (Princeton University Press, Princeton, 1994).
72. R. Loll, Nucl. Phys. B444, 619 (1995); B460, 143 (1996); R. DePietri and C. Rovelli, Phys. Rev. D 54, 2664 (1996); Simonetta Frittelli, Luis Lehner, and Carlo Rovelli, Class. Quantum Grav. 13, 2921 (1996); R. Borissov, Ph.D. thesis, Temple University, 1996.
73. R. Loll, Nucl. Phys. B444, 619 (1995); B460, 143 (1996); R. DePietri and C. Rovelli, Phys. Rev. D 54, 2664 (1996); Simonetta Frittelli, Luis Lehner, and Carlo Rovelli, Class. Quantum Grav. 13, 2921 (1996); R. Borissov, Ph.D. thesis, Temple University, 1996.
74. R. Loll, Nucl. Phys. B444, 619 (1995); B460, 143 (1996); R. DePietri and C. Rovelli, Phys. Rev. D 54, 2664 (1996); Simonetta Frittelli, Luis Lehner, and Carlo Rovelli, Class. Quantum Grav. 13, 2921 (1996); R. Borissov, Ph.D. thesis, Temple University, 1996.
75. R. Loll, Nucl. Phys. B444, 619 (1995); B460, 143 (1996); R. DePietri and C. Rovelli, Phys. Rev. D 54, 2664 (1996); Simonetta Frittelli, Luis Lehner, and Carlo Rovelli, Class. Quantum Grav. 13, 2921 (1996); R. Borissov, Ph.D. thesis, Temple University, 1996.
76. R. Loll, Nucl. Phys. B444, 619 (1995); B460, 143 (1996); R. DePietri and C. Rovelli, Phys. Rev. D 54, 2664 (1996); Simonetta Frittelli, Luis Lehner, and Carlo Rovelli, Class. Quantum Grav. 13, 2921 (1996); R. Borissov, Ph.D. thesis, Temple University, 1996.
77. H. Kodama, Prog. Theor. Phys. 80, 1024 (1988).
78. A. Sen, Phys. Lett. 119B, 89 (1982); Int. J. Theor. Phys. 21, 1 (1982).
79. A. Sen, Phys. Lett. 119B, 89 (1982); Int. J. Theor. Phys. 21, 1 (1982).
80. A. Ashtekar, Phys. Rev. Lett. 57, 2244 (1986); Phys. Rev. D 36, 1587 (1987).
81. A. Ashtekar, Phys. Rev. Lett. 57, 2244 (1986); Phys. Rev. D 36, 1587 (1987).
82. B. Bruegmann, R. Gambini, and J. Pullin, Phys. Rev. Lett. 68, 431 (1992); Gen. Relativ. Gravit. 25, 1 (1993).
83. B. Bruegmann, R. Gambini, and J. Pullin, Phys. Rev. Lett. 68, 431 (1992); Gen. Relativ. Gravit. 25, 1 (1993).
84. R. Gambini and J. Pullin, Loops, Knots, Gauge Theories and Quantum Gravity (Cambridge University Press, Cambridge, England, 1996).
85. S. Major and L. Smolin, Nucl. Phys. B473, 267 (1996); R. Borissov, S. Major, and L. Smolin, Class. Quantum Grav. 12, 3183 (1996).
86. S. Major and L. Smolin, Nucl. Phys. B473, 267 (1996); R. Borissov, S. Major, and L. Smolin, Class. Quantum Grav. 12, 3183 (1996).
87. J. Barrett and L. Crane, J. Math. Phys. 39, 3296 (1998); L. Crane, "On the interpretation of relativistic spin networks and the balanced state sum," gr-qc/9710108; J. Baez, Class. Quantum Grav. 15, 1827 (1998); A. Barbieri, Nucl. Phys. B518, 714 (1998).
88. J. Barrett and L. Crane, J. Math. Phys. 39, 3296 (1998); L. Crane, "On the interpretation of relativistic spin networks and the balanced state sum," gr-qc/9710108; J. Baez, Class. Quantum Grav. 15, 1827 (1998); A. Barbieri, Nucl. Phys. B518, 714 (1998).
89. J. Barrett and L. Crane, J. Math. Phys. 39, 3296 (1998); L. Crane, "On the interpretation of relativistic spin networks and the balanced state sum," gr-qc/9710108; J. Baez, Class. Quantum Grav. 15, 1827 (1998); A. Barbieri, Nucl. Phys. B518, 714 (1998).
90. J. Barrett and L. Crane, J. Math. Phys. 39, 3296 (1998); L. Crane, "On the interpretation of relativistic spin networks and the balanced state sum," gr-qc/9710108; J. Baez, Class. Quantum Grav. 15, 1827 (1998); A. Barbieri, Nucl. Phys. B518, 714 (1998).
91. D. Rayner, Class. Quantum Grav. 7, 111 (1990); 7, 651 (1990).
92. D. Rayner, Class. Quantum Grav. 7, 111 (1990); 7, 651 (1990).
93. A. Ashtekar and C. J. Isham, Class. Quantum Grav. 9, 1069 (1992).
94. A. Ashtekar, J. Lewandowski, D. Marlof, J. Mourãu, and T. Thiemann, J. Math. Phys. 36, 6456 (1995); A. Ashtekar and J. Lewandowski, Class. Quantum Grav. 14, A55 (1997); J. Lewandowski, ibid. 14, 71 (1997).
95. A. Ashtekar, J. Lewandowski, D. Marlof, J. Mourãu, and T. Thiemann, J. Math. Phys. 36, 6456 (1995); A. Ashtekar and J. Lewandowski, Class. Quantum Grav. 14, A55 (1997); J. Lewandowski, ibid. 14, 71 (1997).
96. A. Ashtekar, J. Lewandowski, D. Marlof, J. Mourãu, and T. Thiemann, J. Math. Phys. 36, 6456 (1995); A. Ashtekar and J. Lewandowski, Class. Quantum Grav. 14, A55 (1997); J. Lewandowski, ibid. 14, 71 (1997).
97. T. Thiemann, Class. Quantum Grav. 15, 839 (1998); 15, 1207 (1998); J. Math. Phys. 39, 3372 (1998); Class. Quantum Grav. 15, 575 (1998); 15, 1463 (1998); 15, 1457 (1998); 15, 1281 (1998); 15, 1249 (1998).
98. T. Thiemann, Class. Quantum Grav. 15, 839 (1998); 15, 1207 (1998); J. Math. Phys. 39, 3372 (1998); Class. Quantum Grav. 15, 575 (1998); 15, 1463 (1998); 15, 1457 (1998); 15, 1281 (1998); 15, 1249 (1998).
99. T. Thiemann, Class. Quantum Grav. 15, 839 (1998); 15, 1207 (1998); J. Math. Phys. 39, 3372 (1998); Class. Quantum Grav. 15, 575 (1998); 15, 1463 (1998); 15, 1457 (1998); 15, 1281 (1998); 15, 1249 (1998).
100. T. Thiemann, Class. Quantum Grav. 15, 839 (1998); 15, 1207 (1998); J. Math. Phys. 39, 3372 (1998); Class. Quantum Grav. 15, 575 (1998); 15, 1463 (1998); 15, 1457 (1998); 15, 1281 (1998); 15, 1249 (1998).
101. T. Thiemann, Class. Quantum Grav. 15, 839 (1998); 15, 1207 (1998); J. Math. Phys. 39, 3372 (1998); Class. Quantum Grav. 15, 575 (1998); 15, 1463 (1998); 15, 1457 (1998); 15, 1281 (1998); 15, 1249 (1998).
102. T. Thiemann, Class. Quantum Grav. 15, 839 (1998); 15, 1207 (1998); J. Math. Phys. 39, 3372 (1998); Class. Quantum Grav. 15, 575 (1998); 15, 1463 (1998); 15, 1457 (1998); 15, 1281 (1998); 15, 1249 (1998).
103. T. Thiemann, Class. Quantum Grav. 15, 839 (1998); 15, 1207 (1998); J. Math. Phys. 39, 3372 (1998); Class. Quantum Grav. 15, 575 (1998); 15, 1463 (1998); 15, 1457 (1998); 15, 1281 (1998); 15, 1249 (1998).
104. T. Thiemann, Class. Quantum Grav. 15, 839 (1998); 15, 1207 (1998); J. Math. Phys. 39, 3372 (1998); Class. Quantum Grav. 15, 575 (1998); 15, 1463 (1998); 15, 1457 (1998); 15, 1281 (1998); 15, 1249 (1998).
105. G. 't Hooft, "Dimensional reduction in quantum gravity," gr-qc/9310006.
106. L. Susskind, J. Math. Phys. 36, 6377 (1995); Phys. Rev. Lett. 71, 2367 (1993); Phys. Rev. D 49, 6606 (1994); 50, 2700 (1994); L. Susskind and P. Griffin, "Partons and black holes," hep-ph/9410306.
107. L. Susskind, J. Math. Phys. 36, 6377 (1995); Phys. Rev. Lett. 71, 2367 (1993); Phys. Rev. D 49, 6606 (1994); 50, 2700 (1994); L. Susskind and P. Griffin, "Partons and black holes," hep-ph/9410306.
108. L. Susskind, J. Math. Phys. 36, 6377 (1995); Phys. Rev. Lett. 71, 2367 (1993); Phys. Rev. D 49, 6606 (1994); 50, 2700 (1994); L. Susskind and P. Griffin, "Partons and black holes," hep-ph/9410306.
109. L. Susskind, J. Math. Phys. 36, 6377 (1995); Phys. Rev. Lett. 71, 2367 (1993); Phys. Rev. D 49, 6606 (1994); 50, 2700 (1994); L. Susskind and P. Griffin, "Partons and black holes," hep-ph/9410306.
110. L. Susskind, J. Math. Phys. 36, 6377 (1995); Phys. Rev. Lett. 71, 2367 (1993); Phys. Rev. D 49, 6606 (1994); 50, 2700 (1994); L. Susskind and P. Griffin, "Partons and black holes," hep-ph/9410306.
111. W. Fischler and L. Susskind, "Holography and Cosmology," hep-th/9806039.
112. Gary T. Horowitz and Donald Marolf, J. High Energy Phys. 07, 014 (1998).
113. R. Bousso, J. High. Energy Phys. 07, 004 (1999); 06, 028 (1999).
114. R. Bousso, J. High. Energy Phys. 07, 004 (1999); 06, 028 (1999).
115. C. Rovelli, J. Theor. Phys. 35, 1637 (1996).
116. L. Smolin, The Life of the Cosmos (Oxford University Press, New York, 1997).
117. F. Markopoulou, Commun. Math. Phys. 211, 559 (2000).
118. F. Markopoulou, Class. Quantum Grav. 17, 2059 (2000).
119. F. Markopoulou and L. Smolin, "Causal quantum cosmology," (in preparation).
120. L. Bombelli, J. Lee, D. Meyer, and R. D. Sorkin, Phys. Rev. Lett. 59, 521 (1987).
121. G. 't Hooft, Cargèse Summer School Lectures 1978, edited by M. Lévy and S. Deser (Plenum, New York, 1978), p. 323; lectures given at the NATO Advanced Study Institute on "Quantum Fields and Quantum Space Time," Cargèse, 1996, gr-qc/9608037; Int. J. Mod. Phys. A 11, 4623 (1996).
122. G. 't Hooft, Cargèse Summer School Lectures 1978, edited by M. Lévy and S. Deser (Plenum, New York, 1978), p. 323; lectures given at the NATO Advanced Study Institute on "Quantum Fields and Quantum Space Time," Cargèse, 1996, gr-qc/9608037; Int. J. Mod. Phys. A 11, 4623 (1996).
123. G. 't Hooft, Cargèse Summer School Lectures 1978, edited by M. Lévy and S. Deser (Plenum, New York, 1978), p. 323; lectures given at the NATO Advanced Study Institute on "Quantum Fields and Quantum Space Time," Cargèse, 1996, gr-qc/9608037; Int. J. Mod. Phys. A 11, 4623 (1996).
124. M. Reisenberger, "A lattice worldsheet sum for 4-d Euclidean general relativity," gr-qc/9711052; M. Reisenberger and C. Rovelli, Phys. Rev. D 56, 3490 (1997).
125. M. Reisenberger, "A lattice worldsheet sum for 4-d Euclidean general relativity," gr-qc/9711052; M. Reisenberger and C. Rovelli, Phys. Rev. D 56, 3490 (1997).
126. Simonetta Frittelli, Carlos N. Kozameh, Ezra T. Newman, Carlo Rovelli, and Ranjeet S. Tate, Class. Quantum Grav. 14, A143 (1997); Phys. Rev. D 56, 889 (1997); Savitri V. Iyer, Carlos N. Kozameh, and Ezra T. Newman, "The Vacuum Einstein Equations via Holonomy around Closed Loops on Characteristic Surfaces," gr-qc/9502020; Simonetta Frittelli, Carlos Kozameh, and Ted Newman, J. Math. Phys. 36, 4975 (1995).
127. Simonetta Frittelli, Carlos N. Kozameh, Ezra T. Newman, Carlo Rovelli, and Ranjeet S. Tate, Class. Quantum Grav. 14, A143 (1997); Phys. Rev. D 56, 889 (1997); Savitri V. Iyer, Carlos N. Kozameh, and Ezra T. Newman, "The Vacuum Einstein Equations via Holonomy around Closed Loops on Characteristic Surfaces," gr-qc/9502020; Simonetta Frittelli, Carlos Kozameh, and Ted Newman, J. Math. Phys. 36, 4975 (1995).
128. Simonetta Frittelli, Carlos N. Kozameh, Ezra T. Newman, Carlo Rovelli, and Ranjeet S. Tate, Class. Quantum Grav. 14, A143 (1997); Phys. Rev. D 56, 889 (1997); Savitri V. Iyer, Carlos N. Kozameh, and Ezra T. Newman, "The Vacuum Einstein Equations via Holonomy around Closed Loops on Characteristic Surfaces," gr-qc/9502020; Simonetta Frittelli, Carlos Kozameh, and Ted Newman, J. Math. Phys. 36, 4975 (1995).
129. Simonetta Frittelli, Carlos N. Kozameh, Ezra T. Newman, Carlo Rovelli, and Ranjeet S. Tate, Class. Quantum Grav. 14, A143 (1997); Phys. Rev. D 56, 889 (1997); Savitri V. Iyer, Carlos N. Kozameh, and Ezra T. Newman, "The Vacuum Einstein Equations via Holonomy around Closed Loops on Characteristic Surfaces," gr-qc/9502020; Simonetta Frittelli, Carlos Kozameh, and Ted Newman, J. Math. Phys. 36, 4975 (1995).
130. Y. Ling and L. Smolin, Phys. Rev. D 61, 044008 (2000).
131. L. Smolin, "M theory as a matrix extension of Chern-Simons theory," hep-th/0002009; "The cubic matrix model and the duality between strings and loops," hep-th/0006137.
132. L. Smolin, "M theory as a matrix extension of Chern-Simons theory," hep-th/0002009; "The cubic matrix model and the duality between strings and loops," hep-th/0006137.
133. P. Townsend, Nucl. Phys. B (Proc. Suppl.) 68, 11 (1998); in Particles, Strings, and Cosmology, edited by J. Bagger et al. (World Scientific, Singapore, 1996); hep/9612121; I. Bars, hep-th/9608061; L. Buffon et al., Mod. Phys. Lett. A 11, 2339 (1996); Petr Horava, Phys. Rev. D 59, 046004 (1999).
134. P. Townsend, Nucl. Phys. B (Proc. Suppl.) 68, 11 (1998); in Particles, Strings, and Cosmology, edited by J. Bagger et al. (World Scientific, Singapore, 1996); hep/9612121; I. Bars, hep-th/9608061; L. Buffon et al., Mod. Phys. Lett. A 11, 2339 (1996); Petr Horava, Phys. Rev. D 59, 046004 (1999).
135. P. Townsend, Nucl. Phys. B (Proc. Suppl.) 68, 11 (1998); in Particles, Strings, and Cosmology, edited by J. Bagger et al. (World Scientific, Singapore, 1996); hep/9612121; I. Bars, hep-th/9608061; L. Buffon et al., Mod. Phys. Lett. A 11, 2339 (1996); Petr Horava, Phys. Rev. D 59, 046004 (1999).
136. P. Townsend, Nucl. Phys. B (Proc. Suppl.) 68, 11 (1998); in Particles, Strings, and Cosmology, edited by J. Bagger et al. (World Scientific, Singapore, 1996); hep/9612121; I. Bars, hep-th/9608061; L. Buffon et al., Mod. Phys. Lett. A 11, 2339 (1996); Petr Horava, Phys. Rev. D 59, 046004 (1999).
137. S. Gupta (in preparation).
138. P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988); Per Bak, How Nature Works (Oxford University Press, Oxford, 1997).
139. P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988); Per Bak, How Nature Works (Oxford University Press, Oxford, 1997).
140. P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988); Per Bak, How Nature Works (Oxford University Press, Oxford, 1997).
141. Y. Ling and L. Smolin (in preparation).
142. K. Krasnov, Phys. Rev. D 55, 3505 (1997); Gen. Relativ. Gravit. 30, 53 (1998); A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Phys. Rev. Lett. 80, 904 (1998); K. Krasnov, Class. Quantum Grav. 16, L15 (1999).
143. K. Krasnov, Phys. Rev. D 55, 3505 (1997); Gen. Relativ. Gravit. 30, 53 (1998); A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Phys. Rev. Lett. 80, 904 (1998); K. Krasnov, Class. Quantum Grav. 16, L15 (1999).
144. K. Krasnov, Phys. Rev. D 55, 3505 (1997); Gen. Relativ. Gravit. 30, 53 (1998); A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Phys. Rev. Lett. 80, 904 (1998); K. Krasnov, Class. Quantum Grav. 16, L15 (1999).
145. K. Krasnov, Phys. Rev. D 55, 3505 (1997); Gen. Relativ. Gravit. 30, 53 (1998); A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Phys. Rev. Lett. 80, 904 (1998); K. Krasnov, Class. Quantum Grav. 16, L15 (1999).
146. F. Markopoulou, to appear in the proceedings of Quantum Gravity 99 meeting, held in Sardinia, Italy, 1999, hep-th/9912137.