Sum over surfaces” form of loop quantum gravity

Physical Review D - Particles, Fields, Gravitation and Cosmology

Volumn 56, Issue 6, 1997, Pages 3490-3508

  Reisenberger, Michael P ^a   Rovelli, Carlo ^b  

^a UNIVERSIDAD DE LA REPÚBLICA   (Uruguay)

^b UNIVERSITY OF PITTSBURGH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0009249782     PISSN: 15507998     EISSN: 15502368     Source Type: Journal    
DOI: 10.1103/PhysRevD.56.3490     Document Type: Article

References (106)

1. For an overview, see C. Isham, gr-qc/9510063.
2. J. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983);
3. S. W. Hawking, in Relativity, Groups and Topology, Les Houches Session XL, edited by B. DeWitt and R. Stora (North Holland, Amsterdam, 1984).
4. R. Feynman, Rev. Mod. Phys. 20, 367 (1948).
5. C. Teitelboim, Phys. Lett. 96B, 77 (1980).
6. R. Feynman, Phys. Rev. 80, 440 (1950);
7. Y. Nambu, Prog. Theor. Phys. 5, 82 (1950).
8. L. Smolin, gr-qc/9609034.
9. J. Baez, in Knots and Quantum Gravity, edited by J. Baez (Oxford University Press, Oxford, 1994).
10. M. Reisenberger, gr-qc/9412035.
11. M. Reisenberger, gr-qc/9609002, (in preparation).
12. H. Ooguri, Mod. Phys. Lett. A 7, 2799 (1992).
13. L. Crane and D. N. Yetter, in Quantum Topology, edited by R. A. Baadhio and L. H. Kauffman (World Scientific, Singapore, 1993);L. Crane, L. H. Kauffman, and D. N. Yetter, in, pp. 131–138. See also 58.
14. G. Ponzano and T. Regge, in Spectroscopy and Group Theoretical Methods in Physics, edited by F. Bloch (North Holland, Amsterdam, 1968);
15. H. Ooguri and N. Sasakura, Mod. Phys. Lett. A 6, 3591 (1991);
16. H. Ooguri, Nucl. Phys. B382, 276 (1992).
17. V. G. Turaev and O. V. Viro, Topology 21, 865 (1992).
18. C. Rovelli, Phys. Rev. D 48, 2702 (1993).
19. L. Crane, J. Math. Phys. (N.Y.) 36, 6180 (1995);
20. J. Baez, 36, 6073 (1995);
21. M. Atiyah, 36, 6069 (1995);
22. A. S. Cattaneo, P. Cotta-Ramusino, J. Frölich, and M. Martellini, 36, 6137 (1995);
23. J. Barret, 36, 6161 (1995);
24. L. Smolin, 36, 6417 (1995);L. Crane, in Knots and Quantum Gravity 7.
25. J. B. Hartle, in Gravitation and Quantization, Les Houches, Session LVII, edited by B. Julia and J. Zinn-Justin (Elsevier Science, New York, 1995).
26. C. Rovelli and L. Smolin, Phys. Rev. Lett. 61, 1155 (1988);
27. Nucl. Phys. B331, 80 (1990).
28. For a recent introductive review, see C. Rovelli, Helv. Phys. Acta 69, 582 (1996), and complete references therein. For a detailed introduction to the formalism needed to perform the calculations in the scheme developed here, see 21.
29. General introductions to the loop formalism can be found in C. Rovelli, Class. Quantum Grav. 8, 1613 (1991);
30. R. Gambini and J. Pullin, Loops, Knots and Quantum Gravity (Cambridge University Press, Cambridge, England, 1997);A. Ashtekar, in Gravitation and Quantization, Les Houches, Session LVII 16.
31. C. Rovelli and L. Smolin, Nucl. Phys. B442, 593 (1995).
32. A. Ashtekar and J. Lewandowski, J. Geom. Phys. 17, 191 (1995); JGPHE5J. Lewandowski, talk given at the Workshop on Canonical and Quantum Gravity, Warsaw, 1995 (unpublished).
33. R. Loll, Phys. Rev. Lett. 75, 3048 (1995);
34. Nucl. Phys. B444, 619 (1995);
35. B460, 143 (1996);
36. Phys. Rev. D 54, 5381 (1996).
37. R. DePietri and C. Rovelli, Phys. Rev. D 54, 2664 (1996).
38. S. Frittelli, L. Lehner, and C. Rovelli, Class. Quantum Grav. 13, 2921 (1996).
39. J. Lewandowski, Class. Quantum Grav. 14, 71 (1997).
40. T. Jacobson and L. Smolin, Nucl. Phys. B229, 295 (1988);
41. V. Husain, B313, 711 (1988);
42. W. P. Blencowe, B341, 213 (1990);
43. B. Brügman and J. Pullin, B341, 221 (1990);
44. B. Brügman, R. Gambini, and J. Pullin, Phys. Rev. Lett. 68, 431 (1992);
45. Nucl. Phys. B385, 587 (1992);
46. Gen. Relativ. Gravit. 25, 1 (1993); GRGVA8
47. C. Rovelli and L. Smolin, Phys. Rev. Lett. 72, 446 (1994);
48. R. Gambini and J. Pullin, Phys. Rev. D 54, 5935 (1996);
49. Class. Quantum Grav. 13, L125,446 (1996); gr-qc/9703042;
50. R. Borissov, Phys. Rev. D 55, 2059 (1997);
51. 55, 6099 (1997);
52. K. Ezawa, gr-qc/9601050.
53. T. Thiemann, Phys. Lett. B 380, 257 (1996); gr-qc/9606089;gr-qc/9606090.
54. R. Borrisov, R. DePietri, and C. Rovelli, Class. Quantum Grav. (to be published).
55. J. Lewandowski, Int. J. Mod. Phys. D 3, 207 (1994);
56. C. Rovelli and L. Smolin, Phys. Rev. D 52, 5743 (1995).
57. L. Smolin, gr-qc/9702030.
58. J. Baez, Adv. Math. 117, 253 (1996); ADMTA4in The Interface of Knots and Physics, edited by L. H. Kauffman (American Mathematical Society, Providence, RI, 1966).
59. T. Thiemann, hep-th/9601105.
60. A. Ashtekar and J. Lewandowski, in Knots and Quantum Gravity (Clarendon, Oxford, 1994);
61. A. Ashtekar and C. Isham, Class. Quantum Grav. 9, 1433 (1992);
62. A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao, and T. Thiemann, J. Math. Phys. (N.Y.) 36, 6456 (1995).
63. R. DePietri, Class. Quantum Grav. 14, 53 (1997).
64. D. Maroff, Ph.D. dissertation, 1992;
65. Class. Quantum Grav. 12, 1199 (1995);
66. D. Marolf, The Spectral Analysis Inner Product, Proceedings of the VIIth Marcel Grossman Conference, edited by R. Ruffini and M. Keiser (World Scientific, Singapore, 1995).
67. A. Higuchi, Class. Quantum Grav. 8, 1983 (1991).
68. J. F. Barbero, Phys. Rev. D 51, 5507 (1995);
69. T. Thiemann, Class. Quantum Grav. 13, 1383 (1996);
70. A. Ashtekar, Phys. Rev. D 53, 2865 (1996).
71. C. Rovelli, J. Math. Phys. (N.Y.) 36, 5629 (1995).
72. B. Brügmann, Nucl. Phys. B474, 249 (1996).
73. J. Lewandowski (private communication).
74. C. Rovelli, Class. Quantum Grav. 8, 297 (1991).
75. J. D. Brown and K. Kuchar, Phys. Rev. D 51, 5600 (1995).
76. J. Lewandowski (private communication).
77. N. Grot and C. Rovelli, J. Math. Phys. (N.Y.) 37, 3014 (1996).
78. J. A. Zapata, gr-qc/9703037;gr-qc/9703038.
79. E. P. T. Liang, Phys. Rev. D 5, 2458 (1972);
80. C. J. Isham, Proc. R. Soc. London A351, 209 (1976);
81. C. Teitelboim, in An Einstein Centenary Volume, edited by A. Held (Plenum, New York, 1980).
82. J.S. Carter and M. Saito, in Knots and Quantum Gravity 7.
83. R. M. Williams, J. Math. Phys. (N.Y.) 36, 6276 (1995).
84. M. Reisenberger, Nucl. Phys. B457, 643 (1995).
85. B. Brügmann (private communication).
86. L. H. Kauffman and S. L. Lins, Temperley-Lieb Recoupling Theory and Invariant of 3-Manifolds (Princeton University Press, Princeton, 1994).
87. H. Fort, R. Gambini, and J. Pullin, Phys. Rev. D 56, 2127 (1997);
88. G. Immirzi, Class. Quantum Grav. 13, 2385 (1996).
89. J. Iwasaki, J. Math. Phys. (N.Y.) 36, 6288 (1995).
90. J. A. Zapata, Class. Quantum Grav. 13, 2617 (1986).
91. S. Frittelli, E. T. Newman, and C. N. Kozameh, J. Math. Phys. (N.Y.) 36, 6397 (1995).
92. L. Smolin, gr-qc/9609031.
93. A. S. Schwarz, Lett. Math. Phys. 2, 247 (1978);
94. Commun. Math. Phys. 67, 1 (1979);
95. G. Horowitz, 125, 417 (1989);
96. M. Blau and G. Thompson, Ann. Phys. (N.Y.) 209, 129 (1991).APNYA6
97. J. F. Plebanski, J. Math. Phys. (N.Y.) 18, 2511 (1977);
98. T. Jacobson and L. Smolin, Phys. Lett. B 196, 39 (1987);
99. Class. Quantum Grav. 5, 583 (1988);
100. R. Capovilla, J. Dell, and T. Jacobson, 8, 41 (1991);
101. 8, 59 (1991);
102. R. Capovilla and T. Jacobson, Mod. Phys. Lett. A 7, 1871 (1992).
103. R. Borissov, S. Major, and L. Smolin, Class. Quantum Grav. 13, 3183 (1996).
104. J. C. Baez, Adv. Math. 117, 253 (1996); ADMTA4gr-qc/9411007;in The Interface of Knots and Physics 28;gr-qc/9504036.
105. L. Crane and I. Frenkel, J. Math. Phys. (N.Y.) 35, 5136 (1994); L. Crane and D. N. Yetter (unpublished);J. Knot Theory Ramif. (to be published);
106. L. Crane, L. H. Kauffman, and D. N. Yetter, J. Math. Phys. (N.Y.) 6, 177 (1997);P. Cotta-Ramusino and M. Martellini, in Knots and Quantum Gravity 7.