A generalized Hamiltonian constraint operator in loop quantum gravity and its simplest Euclidean matrix elements

Classical and Quantum Gravity

Volumn 18, Issue 9, 2001, Pages 1593-1624

  Gaul, Marcus ^a,b,d   Rovelli, Carlo ^a,c  

^a CENTRE DE PHYSIQUE THÉORIQUE   (France)

^b MAX PLANCK INSTITUT FÜR PHYSIK   (Germany)

^c UNIVERSITY OF PITTSBURGH   (United States)

^d UNIVERSITY OF MUNICH   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035532013     PISSN: 02649381     EISSN: None     Source Type: Journal    
DOI: 10.1088/0264-9381/18/9/301     Document Type: Article

References (63)

1. Rovelli C 1998 Loop quantum gravity Living. Rev. Relat. http://www.livingreviews.org/Articles/Volume1/1998-lrovelli/ (Rovelli C 1997 Preprint gr-qc/9710008)
2. Rovelli C 1998 Loop quantum gravity Living. Rev. Relat. http://www.livingreviews.org/Articles/Volume1/1998-lrovelli/ (Rovelli C 1997 Preprint gr-qc/9710008)
3. Rovelli C and Smolin L 1990 Loop space representation of quantum general relativity Nucl. Phys. B 331 80 Rovelli C and Smolin L 1988 Knot theory and quantum gravity Phys. Rev. Lett. 61 1155
4. Rovelli C and Smolin L 1990 Loop space representation of quantum general relativity Nucl. Phys. B 331 80 Rovelli C and Smolin L 1988 Knot theory and quantum gravity Phys. Rev. Lett. 61 1155
5. Rovelli C 1991 Ashtekar formulation of general relativity and loop space nonperturbative quantum gravity: a report Class. Quantum Grav. 8 1613 Husain V 1989 Intersecting Loop solutions of the Hamiltonian constraint Nucl. Phys. B 313 711 Brügmann B 1991 Intersecting N loop solutions of the Hamiltonian constraint Nucl. Phys. B 363 221 Gambini R 1989 Loop space representation of quantum general relativity Phys. Lett. B 255 180 Brügmann B, Gambini R and Pullin J 1992 Jones polynomials for intersecting knots as physical states of quantum gravity Nucl. Phys. B 385 587 (Brügmann B, Gambini R and Pullin J 1992 Preprint hep-th/9202018)
6. Rovelli C 1991 Ashtekar formulation of general relativity and loop space nonperturbative quantum gravity: a report Class. Quantum Grav. 8 1613 Husain V 1989 Intersecting Loop solutions of the Hamiltonian constraint Nucl. Phys. B 313 711 Brügmann B 1991 Intersecting N loop solutions of the Hamiltonian constraint Nucl. Phys. B 363 221 Gambini R 1989 Loop space representation of quantum general relativity Phys. Lett. B 255 180 Brügmann B, Gambini R and Pullin J 1992 Jones polynomials for intersecting knots as physical states of quantum gravity Nucl. Phys. B 385 587 (Brügmann B, Gambini R and Pullin J 1992 Preprint hep-th/9202018)
7. Rovelli C 1991 Ashtekar formulation of general relativity and loop space nonperturbative quantum gravity: a report Class. Quantum Grav. 8 1613 Husain V 1989 Intersecting Loop solutions of the Hamiltonian constraint Nucl. Phys. B 313 711 Brügmann B 1991 Intersecting N loop solutions of the Hamiltonian constraint Nucl. Phys. B 363 221 Gambini R 1989 Loop space representation of quantum general relativity Phys. Lett. B 255 180 Brügmann B, Gambini R and Pullin J 1992 Jones polynomials for intersecting knots as physical states of quantum gravity Nucl. Phys. B 385 587 (Brügmann B, Gambini R and Pullin J 1992 Preprint hep-th/9202018)
8. Rovelli C 1991 Ashtekar formulation of general relativity and loop space nonperturbative quantum gravity: a report Class. Quantum Grav. 8 1613 Husain V 1989 Intersecting Loop solutions of the Hamiltonian constraint Nucl. Phys. B 313 711 Brügmann B 1991 Intersecting N loop solutions of the Hamiltonian constraint Nucl. Phys. B 363 221 Gambini R 1989 Loop space representation of quantum general relativity Phys. Lett. B 255 180 Brügmann B, Gambini R and Pullin J 1992 Jones polynomials for intersecting knots as physical states of quantum gravity Nucl. Phys. B 385 587 (Brügmann B, Gambini R and Pullin J 1992 Preprint hep-th/9202018)
9. Rovelli C 1991 Ashtekar formulation of general relativity and loop space nonperturbative quantum gravity: a report Class. Quantum Grav. 8 1613 Husain V 1989 Intersecting Loop solutions of the Hamiltonian constraint Nucl. Phys. B 313 711 Brügmann B 1991 Intersecting N loop solutions of the Hamiltonian constraint Nucl. Phys. B 363 221 Gambini R 1989 Loop space representation of quantum general relativity Phys. Lett. B 255 180 Brügmann B, Gambini R and Pullin J 1992 Jones polynomials for intersecting knots as physical states of quantum gravity Nucl. Phys. B 385 587 (Brügmann B, Gambini R and Pullin J 1992 Preprint hep-th/9202018)
10. Rovelli C 1991 Ashtekar formulation of general relativity and loop space nonperturbative quantum gravity: a report Class. Quantum Grav. 8 1613 Husain V 1989 Intersecting Loop solutions of the Hamiltonian constraint Nucl. Phys. B 313 711 Brügmann B 1991 Intersecting N loop solutions of the Hamiltonian constraint Nucl. Phys. B 363 221 Gambini R 1989 Loop space representation of quantum general relativity Phys. Lett. B 255 180 Brügmann B, Gambini R and Pullin J 1992 Jones polynomials for intersecting knots as physical states of quantum gravity Nucl. Phys. B 385 587 (Brügmann B, Gambini R and Pullin J 1992 Preprint hep-th/9202018)
11. Di Bartolo C, Gambini R, Griego R and Pullin J 2000 Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants Phys. Rev. Lett. 84 2314 (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9909063) (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911009) (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911010) Gambini R and Pullin J 2000 Making classical and quantum canonical general relativity computable through a power series expansion in the inverse cosmological constant Phys. Rev. Lett. 85 5272 (Gambini R and Pullin J 2000 Preprint gr-qc/0008031)
12. Di Bartolo C, Gambini R, Griego R and Pullin J 2000 Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants Phys. Rev. Lett. 84 2314 (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9909063) (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911009) (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911010) Gambini R and Pullin J 2000 Making classical and quantum canonical general relativity computable through a power series expansion in the inverse cosmological constant Phys. Rev. Lett. 85 5272 (Gambini R and Pullin J 2000 Preprint gr-qc/0008031)
13. Di Bartolo C, Gambini R, Griego R and Pullin J 2000 Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants Phys. Rev. Lett. 84 2314 (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9909063) (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911009) (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911010) Gambini R and Pullin J 2000 Making classical and quantum canonical general relativity computable through a power series expansion in the inverse cosmological constant Phys. Rev. Lett. 85 5272 (Gambini R and Pullin J 2000 Preprint gr-qc/0008031)
14. Di Bartolo C, Gambini R, Griego R and Pullin J 2000 Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants Phys. Rev. Lett. 84 2314 (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9909063) (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911009) (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911010) Gambini R and Pullin J 2000 Making classical and quantum canonical general relativity computable through a power series expansion in the inverse cosmological constant Phys. Rev. Lett. 85 5272 (Gambini R and Pullin J 2000 Preprint gr-qc/0008031)
15. Di Bartolo C, Gambini R, Griego R and Pullin J 2000 Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants Phys. Rev. Lett. 84 2314 (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9909063) (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911009) (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911010) Gambini R and Pullin J 2000 Making classical and quantum canonical general relativity computable through a power series expansion in the inverse cosmological constant Phys. Rev. Lett. 85 5272 (Gambini R and Pullin J 2000 Preprint gr-qc/0008031)
16. Di Bartolo C, Gambini R, Griego R and Pullin J 2000 Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants Phys. Rev. Lett. 84 2314 (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9909063) (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911009) (Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911010) Gambini R and Pullin J 2000 Making classical and quantum canonical general relativity computable through a power series expansion in the inverse cosmological constant Phys. Rev. Lett. 85 5272 (Gambini R and Pullin J 2000 Preprint gr-qc/0008031)
17. Thiemann T 1998 Quantum spin dynamics (QSD) Class. Quantum Grav. 15 839 (Thiemann T 1996 Preprint gr-qc/9606089)
18. Thiemann T 1998 Quantum spin dynamics (QSD) Class. Quantum Grav. 15 839 (Thiemann T 1996 Preprint gr-qc/9606089)
19. Thiemann T 1998 Quantum spin dynamics (QSD) II Class. Quantum Grav. 15 875 (Thiemann T 1996 Preprint gr-qc/9606090)
20. Thiemann T 1998 Quantum spin dynamics (QSD) II Class. Quantum Grav. 15 875 (Thiemann T 1996 Preprint gr-qc/9606090)
21. Rovelli C and Smolin L 1994 The physical Hamiltonian in nonperturbative quantum gravity Phys. Rev. Lett. 72 446 (Rovelli C and Smolin L 1993 Preprint gr-qc/9308002)
22. Rovelli C and Smolin L 1994 The physical Hamiltonian in nonperturbative quantum gravity Phys. Rev. Lett. 72 446 (Rovelli C and Smolin L 1993 Preprint gr-qc/9308002)
23. Rovelli C 1995 Outline of a generally covariant quantum field theory and a quantum theory of gravity J. Math. Phys. 36 6529 (Rovelli C 1995 Preprint gr-qc/9503067)
24. Rovelli C 1995 Outline of a generally covariant quantum field theory and a quantum theory of gravity J. Math. Phys. 36 6529 (Rovelli C 1995 Preprint gr-qc/9503067)
25. Gambini R, Lewandowski J, Marolf D and Pullin J 1998 On the consistency of the constraint algebra in spin network quantum gravity Int. J. Mod. Phys. D 7 97 (Gambini R, Lewandowski J, Marolf D and Pullin J 1997 Preprint gr-qc/9710018)
26. Gambini R, Lewandowski J, Marolf D and Pullin J 1998 On the consistency of the constraint algebra in spin network quantum gravity Int. J. Mod. Phys. D 7 97 (Gambini R, Lewandowski J, Marolf D and Pullin J 1997 Preprint gr-qc/9710018)
27. Lewandowski J and Marolf D 1998 Loop constraints: a habitat and their algebra Int. J. Mod. Phys. D 7 299 (Lewandowski J and Marolf D 1997 Preprint gr-qc/9710016)
28. Lewandowski J and Marolf D 1998 Loop constraints: a habitat and their algebra Int. J. Mod. Phys. D 7 299 (Lewandowski J and Marolf D 1997 Preprint gr-qc/9710016)
29. Smolin L 1996 The classical limit and the form of the Hamiltonian constraint in nonperturbative quantum gravity Preprint gr-qc/9609034
30. Thiemann T 2000 Gauge field theory coherent states (GCS): I. General properties Preprint hep-th/0005233
31. Thiemann T and Winkler O 2000 Gauge field theory coherent states (GCS) : II. Peakedness properties Preprint hep-th/0005237
32. Thiemann T and Winkler O 2000 Gauge field theory coherent states (GCS) : III. Ehrenfest Theorems Preprint hep-th/0005234
33. de Pietri R and Freidel L Private communication
34. Reisenberger M and Rovelli C 1997 Sum over surfaces form of loop quantum gravity Phys. Rev. D 56 3490 (Reisenberger M and Rovelli C 1996 Preprint gr-qc/9612035)
35. Reisenberger M and Rovelli C 1997 Sum over surfaces form of loop quantum gravity Phys. Rev. D 56 3490 (Reisenberger M and Rovelli C 1996 Preprint gr-qc/9612035)
36. Baez J C 1998 Spin foam models Class. Quantum Grav. 15 1827 (Baez J C 1997 Preprint gr-qc/9709052)
37. Baez J C 1998 Spin foam models Class. Quantum Grav. 15 1827 (Baez J C 1997 Preprint gr-qc/9709052)
38. Baez J C 2000 An introduction to spin foam models of quantum gravity and BF theory Geometry and Quantum Physics (Lecture Notes in Physics 543) ed H Gausterer and H Grosse (Berlin: Springer) p 25 (Baez J C 1999 Preprint gr-qc/9905087)
39. Baez J C 2000 An introduction to spin foam models of quantum gravity and BF theory Geometry and Quantum Physics (Lecture Notes in Physics 543) ed H Gausterer and H Grosse (Berlin: Springer) p 25 (Baez J C 1999 Preprint gr-qc/9905087)
40. Barrett J D and Crane L 1998 Relativistic spin networks and quantum gravity J. Math. Phys. 39 3296 (Barrett J D and Crane L 1997 Preprint gr-qc/9709028)
41. Barrett J D and Crane L 1998 Relativistic spin networks and quantum gravity J. Math. Phys. 39 3296 (Barrett J D and Crane L 1997 Preprint gr-qc/9709028)
42. Borissov R, de Pietri R and Rovelli C 1997 Matrix elements of Thiemann's Hamiltonian constraint in loop quantum gravity Class. Quantum Grav. 14 2793 (Borissov R, de Pietri R and Rovelli C 1997 Preprint gr-qc/9703090)
43. Borissov R, de Pietri R and Rovelli C 1997 Matrix elements of Thiemann's Hamiltonian constraint in loop quantum gravity Class. Quantum Grav. 14 2793 (Borissov R, de Pietri R and Rovelli C 1997 Preprint gr-qc/9703090)
44. Gaul M and Rovelli C 2000 Loop Quantum Gravity and the meaning of diffeomorphism invariance Towards Quantum Gravity: Proc. 35th Karpacz Int. Winter School on Theoretical Physics (Lecture Notes in Physics 541) ed J Kowalski-Glikman (Berlin: Springer) p 277 (Gaul M and Rovelli C 1999 Preprint gr-qc/9910079)
45. Gaul M and Rovelli C 2000 Loop Quantum Gravity and the meaning of diffeomorphism invariance Towards Quantum Gravity: Proc. 35th Karpacz Int. Winter School on Theoretical Physics (Lecture Notes in Physics 541) ed J Kowalski-Glikman (Berlin: Springer) p 277 (Gaul M and Rovelli C 1999 Preprint gr-qc/9910079)
46. de Pietri R and Rovelli C 1996 Geometry eigenvalues and scalar product from recoupling theory in loop quantum gravity Phys. Rev. D 54 2664 (de Pietri R and Rovelli C 1996 Preprint gr-qc/9602023)
47. de Pietri R and Rovelli C 1996 Geometry eigenvalues and scalar product from recoupling theory in loop quantum gravity Phys. Rev. D 54 2664 (de Pietri R and Rovelli C 1996 Preprint gr-qc/9602023)
48. de Pietri R 1997 On the relation between the connection and the loop representation of quantum gravity Class. Quantum Grav. 14 53 (de Pietri R 1996 Preprint gr-qc/9605064)
49. de Pietri R 1997 On the relation between the connection and the loop representation of quantum gravity Class. Quantum Grav. 14 53 (de Pietri R 1996 Preprint gr-qc/9605064)
50. Barbero F 1995 Real Ashtekar variables for Lorentzian signature space-times Phys. Rev. D 51 5507 (Barbero F 1994 Preprint gr-qc/9410014)
51. Barbero F 1995 Real Ashtekar variables for Lorentzian signature space-times Phys. Rev. D 51 5507 (Barbero F 1994 Preprint gr-qc/9410014)
52. Lewandowski J 1997 Volume and quantizations Class. Quantum Grav. 14 71 (Lewandowski J 1996 Preprint gr-qc/9602035)
53. Lewandowski J 1997 Volume and quantizations Class. Quantum Grav. 14 71 (Lewandowski J 1996 Preprint gr-qc/9602035)
54. Rovelli C and Smolin L 1995 Discreteness of area and volume in quantum gravity Nucl. Phys. B 442 593 (Rovelli C and Smolin L 1994 Preprint gr-qc/9411005) Rovelli C and Smolin L 1995 Nucl. Phys. B 456 734 (erratum)
55. Rovelli C and Smolin L 1995 Discreteness of area and volume in quantum gravity Nucl. Phys. B 442 593 (Rovelli C and Smolin L 1994 Preprint gr-qc/9411005) Rovelli C and Smolin L 1995 Nucl. Phys. B 456 734 (erratum)
56. Ashtekar A and Lewandowski J 1998 Quantum theory of geometry II: volume operators Adv. Theor. Math. Phys. 1 388 (Ashtekar A and Lewandowski J 1997 Preprint gr-qc/9711031 )
57. Ashtekar A and Lewandowski J 1998 Quantum theory of geometry II: volume operators Adv. Theor. Math. Phys. 1 388 (Ashtekar A and Lewandowski J 1997 Preprint gr-qc/9711031 )
58. Kauffman L H and Lins S L 1994 Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (Annals of Mathematics Studies 134) (Princeton, NJ: Princeton University Press)
59. Ashtekar A and Lewandowski J 1997 Quantum theory of geometry I: area operators Class. Quantum Grav. 14 A55 (Ashtekar A and Lewandowski J 1996 Preprint gr-qc/9602046)
60. Ashtekar A and Lewandowski J 1997 Quantum theory of geometry I: area operators Class. Quantum Grav. 14 A55 (Ashtekar A and Lewandowski J 1996 Preprint gr-qc/9602046)
61. Gaul M 2001 Doctoral Thesis
62. Gaul M and Rovelli C 2001 On a crossing symmetric extension of loop quantum gravity (in preparation)
63. Brink D M and Satchler G R 1968 Angular Momentum 2nd edn (Oxford: Clarendon)