Geometry eigenvalues and the scalar product from recoupling theory in loop quantum gravity

Physical Review D - Particles, Fields, Gravitation and Cosmology

Volumn 54, Issue 4, 1996, Pages 2664-2690

  De Pietri, Roberto ^a   Rovelli, Carlo ^b  

^a INFN SEZIONE DI MILANO   (Italy)

^b UNIVERSITY OF PITTSBURGH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2742537895     PISSN: 15507998     EISSN: 15502368     Source Type: Journal    
DOI: 10.1103/PhysRevD.54.2664     Document Type: Article

References (81)

1. R. Penrose, in Magic Without Magic: John Archibald Wheeler, edited by J. R. Klauder (W. H. Freeman and Company, San Francisco, 1972)
2. C. Rovelli and L. Smolin, Phys. Rev. Lett. 61, 1155 (1988)
3. C. Rovelli and L. Smolin, Nucl. Phys. B331, 80 (1990)
4. C. Isham, Structural Issues in Quantum Gravity, to appear in the proceedings of the GR14 (Florence 1995)
5. C. IshamDiffeomorphism Invariant Quantum Field Theory and Quantum Geometry [J. Math. Phys. 36 (1995)]
6. J. Baez, Knots and Quantum Gravity (Oxford University Press, Oxford, 1994)
7. J. Ehlers and H. Friedrich, Canonical Gravity: from Classical to Quantum (Springer-Verlag, Berlin, 1994)
8. C. Rovelli, Class. Quantum Grav. 8, 1613 (1991)
9. A. Asthekar, in Gravitation and Quantization, Les Houches, Proceedings of the Les Houches Summer School, Les Houches, France, 1992, edited by B. Julia and J. Zinn-Justin, Les Houches Summer School Proceedings Vol. 57 (Elsevier, Amsterdam, 1995)
10. L. Smolin, in Brill Feschrift Proceedings, edited by B. Hu and T. Jacobson (Cambridge University Press, Cambridge, England, 1993)
11. L. SmolinQuantum Gravity and Cosmology (World Scientific, Singapore, 1993)
12. R. Gambini and J. Pullin, Loops, Knots, Gauge Theories and Quantum Gravity (Cambridge University Press, Cambridge, England, 1996)
13. K. Ezawa, "Nonperturbative solutions for canonical quantum gravity: an overview," Report No. gr-qc/9601050 (unpublished)
14. A. Ashtekar and C. Isham, Class. Quantum Grav. 9, 1433 (1992)
15. A. Ashtekar, J. Lewandowsky, D. Marolf, J. Mourao, and T. Thiemann, J. Math. Phys. 36, 6456 (1995)
16. A. Ashtekar and J. Lewandowski, "Quantum Theory of Geometry: Area Operator," Report No. gr-qc/9602046 (unpublished)
17. J. Lewandoski, "Volume and Quantization," report, 1996 (unpublished)
18. R. De Pietri, "On the relation between the connection and the loop representation of quantum gravity," University of Parma Report No. UPRF-96-466, 1996 (unpublished)
19. C. Rovelli and L. Smolin, Phys. Rev. D 52, 5743 (1995)
20. J. C. Baez, Adv. Math. 117, 253 (1996) ADMTA4
21. J. C. BaezSpin Networks in Nonperturbative Quantum Gravity, in The Interface of Knots and Physics, edited by L.H. Kauffman (American Mathematical Society, Providence, RI, 1996)
22. T. Foxon, Class. Quantum Grav. 12, 951 (1995)
23. A. Ashtekar, C. Rovelli, and L. Smolin, Phys. Rev. Lett. 69, 237 (1992)
24. C. Rovelli and L. Smolin, Nucl. Phys. B442, 593 (1995)
25. R. Loll, Phys. Rev. Lett. 75, 3048 (1995)
26. A. Ashtekar and J. Lewandowski, J. Geom. Phys. 17, 191 (1995) JGPHE5
27. J. Lewandowski, lecture given at the Workshop on Canonical and Quantum Gravity, Warsaw, 1995 (unpublished)
28. C. Rovelli, Nucl. Phys. B405, 797 (1993)
29. C. Rovelli and L. Smolin, Phys. Rev. Lett. 72, 446 (1994)
30. R. Borissov, "Graphical Evolution of Spin Network States" (in preparation)
31. R. Gambini and J. Pullin, "A rigorous solution of the quantum Einstein equations," Report No. gr-qc/9511042 (unpublished)
32. R. GambiniJ. Pullin"The general solution of the quantum Einstein equations?" Report No. gr-qc/9603019 (unpublished)
33. C. Rovelli, J. Math. Phys. 36, 6529 (1995)
34. C. Rovelli and H. Morales-Tecotl, Phys. Rev. Lett. 72, 3642 (1994)
35. C. RovelliH. Morales-TecotlNucl. Phys. B451, 325 (1995)
36. K. Krasnov Phys. Rev. D 53, 1874 (1996)
37. R. Gambini and J. Pullim, Phys. Rev. D 47, R5214 (1993)
38. R. Penrose, in Combinatorial Mathematics and its Application, edited by D. Welsh (Academic Press, New York, 1971)
39. L. H. Kauffman and S. L. Lins, Temperley-Lieb Recoupling Theory and Invariant of 3-Manifolds (Princeton University Press, Princeton, NJ, 1994)
40. M. Reisenberger, "Spin-network states and operators" (in preparation)
41. S. Major and L. Smolin, "Quantum deformation of quantum gravity" Report No. CGPG-95/12-3 (1995)
42. S. MajorL. Smolingr-qc/9512020 (unpublished)
43. R. Borissov, S. Major, and L. Smolin, "The geometry of quantum spin networks," Report No. CGPG-95/12-4 (1995)
44. R. BorissovS. MajorL. Smolingr-qc/9512043 (unpublished)
45. R. Borissov, "Eigenvalue spectrum of the Volume Operator in Quantum Gravity" (in preparation)
46. R. Loll, Nucl. Phys. B460, 143 (1996)
47. J. F. Barbero, Phys. Rev. D 51, 5507 (1995)
48. A. Ashtekar, Phys. Rev. Lett. 57, 2244 (1986)
49. A. Ashtekar, Nonperturbative Canonical Gravity (World Scientific, Singapore, 1991)
50. T. Thiemann, Class. Quantum Grav. 13, 1383 (1996)
51. A. Ashtekar, Phys. Rev. D 53, 2865 (1996)
52. R. Loll, Phys. Rev. D (to be published)
53. G. Immirzi, "Quantizing Regge Calculus," Report No. gr-qc/9512040 (unpublished)
54. S. Major (private communication)
55. R. Penrose, in Quantum Theory and Beyond, edited by T. Bastin (Cambridge University Press, Cambridge, England, 1971)
56. C. Rovelli, T. Newman, and J. Lewandowski, J. Math. Phys. 34, 4646 (1993)
57. R. Gambini and A. Trias, Nucl. Phys. B278, 436 (1986)
58. R. GambiniA. TriasPhys. Rev. D 23, 553 (1981)
59. L. H. Kauffman, in Knots, Topology and Quantum Field Theories, edited by L. Lusanna (World Scientific, Singapore, 1991)
60. M. Behzad and G. Chartrand, Introduction to the Theory of Graphs (Allyn and Bacon, Boston, 1971)
61. K. Reidemeister, Knotentheorie (Springer, Berlin, 1932)
62. B. Brügmann, R. Gambini, and J. Pullin, Phys. Rev. Lett. 68, 431 (1992)
63. B. BrügmannR. GambiniJ. PullinNucl. Phys. B385, 587 (1992)
64. B. BrügmannR. GambiniJ. PullinGen. Relativ. Gravit. 25, 1 (1993) GRGVA8
65. B. Brügmann, in Canonical Gravity: From Classical to Quantum, edited by J. Ehlers and H. Friedrich (Springer-Verlag, Berlin, 1993)
66. J. Pullin, in Proceedings of the Vth Mexican School of Particles and Fields, edited by J. Lucio (World Scientific, Singapore, 1993)
67. C. Rovelli, Phys. Rev. D 47, 1703 (1993)
68. S. Frittelli, L. Lehner, and C. Rovelli, "The Complete Spectrum of the Area from Recoupling Theory in Loop Quantum Gravity," report (unpublished)
69. D. M. Brink and G. R. Satchler, Angular Momentum (Claredon Press, Oxford, 1968)
70. C. Rovelli, Class. Quantum Grav. 8, 297 (1991)
71. C. RovelliClass. Quantum Grav.8, 317 (1991)
72. J. Iwasaki and C. Rovelli, Int. J. Mod. Phys. D 1, 533 (1993)
73. J. IwasakiC. RovelliClass. Quantum Grav. 11, 1653 (1994)
74. S. Hawking, Commun. Math. Phys. 43, 199 (1975)
75. M. Barreira, M. Carfora, and C. Rovelli, "Physics with non-perturbative quantum gravity: radiation from a quantum black hole," Report No. gr-qc/9603064 (unpublished)
76. C. Rovelli, "Black Hole Entropy from Loop Quantum Gravity," Report No. gr-qc/9603063 (unpublished)
77. L. H. Kauffman, Trans. Am. Math. Soc. 318, 417 (1990)
78. Moussoris, in Advances in Twistor Theory, Research Notes in Mathematics, edited by Huston and Ward (Pitman, New York, 1979), pp. 308-312
79. L. H. Kauffman, Int. J. Mod. Phys. A 5, 93 (1990)
80. I. Levinson, Liet. TSR Mokslu Akad. Darb. B4, 3 (1956)
81. A. P. Yutsin, J. B. Levinson, and V. V. Vanagas, Mathematical Apparatus of the Theory of Angular Momentum (Israel program for Scientific Translation, Jerusalem, 1962)