Properties of the volume operator in loop quantum gravity: I. Results

Classical and Quantum Gravity

Volumn 25, Issue 6, 2008, Pages

  Brunnemann, Johannes ^a   Rideout, David ^b  

^a UNIVERSITY OF HAMBURG   (Germany)

^b PERIMETER INSTITUTE FOR THEORETICAL PHYSICS   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 41149088272     PISSN: 02649381     EISSN: 13616382     Source Type: Journal    
DOI: 10.1088/0264-9381/25/6/065001     Document Type: Article

References (65)

1. Thiemann T 2006 Introduction to Modern Canonical Quantum General Relativity (Cambridge: Cambridge University Press) (Preprint gr-qc/0110034)
2. Rovelli C 2004 Quantum Gravity (Cambridge: Cambridge University Press)
3. Ashtekar A and Lewandowski J 2004 Background independent quantum gravity: a status report Class. Quantum Grav. 21 R53 (Preprint gr-qc/0404018)
4. Arnowitt R, Deser S and Misner C W 1962 Gravitation: An Introduction to Current Research ed L Witten (New York: Wiley)
5. Ashtekar A 1986 New variables for classical and quantum gravity Phys. Rev. Lett. 57 2244
6. Ashtekar A 1987 New variables for classical and quantum gravity Phys. Rev. D 36 1587
7. Immirzi G 1997 Quantum gravity and Regge calculus Nucl. Phys. Proc. Suppl. 57 65 (Preprint gr-qc/9701052)
8. Thiemann T 1998 Quantum spin dynamics (QSD) Class. Quantum Grav. 15 839-73 (Preprint gr-qc/9606089)
9. Thiemann T 1998 Quantum spin dynamics (QSD): V. Quantum gravity as the natural regulator of the Hamiltonian constraint of matter quantum field theories Class. Quantum Grav. 15 1281-314 (Preprint gr-qc/9705019)
10. Thiemann T 2006 Quantum spin dynamics: VIII. The master constraint Class. Quantum Grav. 23 2249 (Preprint gr-qc/0510011)
11. Thiemann T 1998 A length operator for canonical quantum gravity J. Math. Phys. 39 3372-92 (Preprint gr-qc/9606092)
12. Thiemann T 2001 Gauge field theory coherent states (GCS): I. General properties Class. Quantum Grav. 18 2025 (Preprint hep-th/0005233)
13. Sahlmann H and Thiemann T 2006 Towards the QFT on curved spacetime limit of QGR: I. A general scheme Class. Quantum Grav. 23 867 (Preprint gr-qc/0207030)
14. Sahlmann H and Thiemann T 2006 Towards the QFT on curved spacetime limit of QGR: II. A concrete implementation Class. Quantum Grav. 23 909 (Preprint gr-qc/0207031)
15. Sahlmann H, Thiemann T and Winkler O 2001 Coherent states for canonical quantum general relativity and the infinite tensor product extension Nucl. Phys. B 606 401 (Preprint gr-qc/0102038)
16. Rovelli C 1996 Black hole entropy from loop quantum gravity Phys. Rev. Lett. 77 3288 (Preprint gr-qc/9603063)
17. Ashtekar A, Baez J, Corichi A and Krasnov K 1998 Quantum geometry and black hole entropy Phys. Rev. Lett. 80 904 (Preprint gr-qc/9710007)
18. Meissner K A 2004 Black hole entropy in loop quantum gravity Class. Quantum Grav. 21 5245 (Preprint gr-qc/0407052)
19. Giesel K and Thiemann T 2006 Algebraic quantum gravity (AQG). I: Conceptual setup Class. Quantum Grav. 24 2465 (Preprint gr-qc/0607099)
20. Giesel K and Thiemann T 2006 Algebraic quantum gravity (AQG): II. Semiclassical analysis Class. Quantum Grav. 24 2499 (Preprint gr-qc/0607100)
21. Giesel K and Thiemann T 2006 Algebraic quantum gravity (AQG): III. Semiclassical perturbation theory Class. Quantum Grav. 24 2565 (Preprint gr-qc/0607101)
22. Bojowald M 2006 Quantum Riemannian geometry and black holes Preprint gr-qc/0602100
23. Bojowald M 2005 Loop quantum cosmology Living Rev. Rel. 8 11 (Preprint gr-qc/0601085)
24. Ashtekar A, Pawlowski T, Singh P and Vandersloot K 2007 Loop quantum cosmology of k = 1 FRW models Phys. Rev. D 75 024035 (Preprint gr-qc/0612104)
25. Husain V and Winkler O 2004 On singularity resolution in quantum gravity Phys. Rev. D 69 084016 (Preprint gr-qc/0312094)
26. Husain V and Winkler O 2005 Quantum resolution of black hole singularities Class. Quantum Grav. 22 L127 (Preprint gr-qc/0410125)
27. Bombelli L, Corichi A and Winkler O 2005 Semiclassical quantum gravity: statistics of combinatorial Riemannian geometries Ann. Phys. 14 499 (Preprint gr-qc/0409006)
28. Brunnemann J and Thiemann T 2006 On (cosmological) singularity avoidance in loop quantum gravity Class. Quantum Grav. 23 1395 (Preprint gr-qc/0505032)
29. Brunnemann J and Thiemann T 2006 Unboundedness of triad-like operators in loop quantum gravity Class. Quantum Grav. 23 1429 (Preprint gr-qc/0505033)
30. Grot N and Rovelli C 1996 Moduli-space structure of knots with intersections J. Math. Phys. 37 3014 (Preprint gr-qc/9604010)
31. Zapata J A 1997 A combinatorial approach to diffeomorphism invariant quantum gauge theories J. Math. Phys. 38 5663 (Preprint gr-qc/9703037)
32. Baez J C and Sawin S 1998 Diffeomorphism-invariant spin network states J. Funct. Anal. 158 253-66 (Preprint q-alg/9708005)
33. Fairbairn W and Rovelli C 2004 Separable Hilbert space in loop quantum gravity J. Math. Phys. 45 2802 (Preprint gr-qc/0403047)
34. Fleischhack C 2004 Representations of the Weyl algebra in quantum geometry Preprint math-ph/0407006
35. Lewandowski J, Okolow A, Sahlmann H and Thiemann T 2006 Uniqueness of diffeomorphism invariant states on holonomy-flux algebras Commun. Math. Phys. 267 703 (Preprint gr-qc/0504147)
36. Koslowski T A 2006 Physical diffeomorphisms in loop quantum gravity Preprint gr-qc/0610017
37. Rovelli C and Smolin L 1995 Discreteness of area and volume in quantum gravity Nucl. Phys. B 442 593
38. Rovelli C and Smolin L 1995 Discreteness of area and volume in quantum gravity Nucl. Phys. B 456 753 (Preprint gr-qc/9411005) (erratum)
39. De Pietri R and Rovelli C 1996 Geometry eigenvalues and scalar product from recoupling theory in loop quantum gravity Phys. Rev. D 54 2664 (Preprint gr-qc/9602023)
40. De Pietri R 1997 Spin networks and recoupling in loop quantum gravity Nucl. Phys. Proc. Suppl. 57 251
41. Loll R 1995 The volume operator in discretized quantum gravity Phys. Rev. Lett. 75 3048 (Preprint gr-qc/9506014)
42. Loll R 1996 Spectrum of the volume operator in quantum gravity Nucl. Phys. B 460 143 (Preprint gr-qc/9511030)
43. Lewandowski J 1997 Volume and quantizations Class. Quantum Grav. 14 71 (Preprint gr-qc/9602035)
44. Thiemann T 1998 Closed formula for the matrix elements of the volume operator in canonical quantum gravity J. Math. Phys. 39 3347 (Preprint gr-qc/9606091)
45. Ashtekar A and Lewandowski J 1997 Quantum theory of geometry: I. Area operators Class. Quantum Grav. 14 A55 (Preprint gr-qc/9602046)
46. Ashtekar A and Lewandowski J 1995 Differential geometry on the space of connections via graphs and projective limits J. Geom. Phys. 17 191 (Preprint hep-th/9412073)
47. Ashtekar A and Lewandowski J 1998 Quantum theory of geometry: II. Volume operators Adv. Theor. Math. Phys. 1 388 (Preprint gr-qc/9711031)
48. Seifert M D 2001 Angle and volume studies in quantized space Preprint gr-qc/0108047
49. Major S A and Seifert M D 2002 Modeling space with an atom of quantum geometry Class. Quantum Grav. 19 2211 (Preprint gr-qc/0109056)
50. Carbone G, Carfora M and Marzouli A 2002 Quantum states of elementary three-geometry Class. Quantum Grav. 19 3761 (Preprint gr-qc/0112043)
51. Helesfai G and Bene G 2003 A numerical study of spectral properties of the area operator in loop quantum gravity Preprint gr-qc/0306124
52. Meissner K A 2006 Eigenvalues of the volume operator in loop quantum gravity Class. Quantum Grav. 23 617 (Preprint gr-qc/0509049)
53. Giesel K and Thiemann T 2006 Consistency check on volume and triad operator quantisation in loop quantum gravity. I Class. Quantum Grav. 23 5667 (Preprint gr-qc/0507036)
54. Giesel K and Thiemann T 2006 Consistency check on volume and triad operator quantisation in loop quantum gravity: II Class. Quantum Grav. 23 5693 (Preprint gr-qc/0507037)
55. Brunnemann J and Thiemann T 2006 Simplification of the spectral analysis of the volume operator in loop quantum gravity Class. Quantum Grav. 23 1289 (Preprint gr-qc/0405060)
56. Brunnemann J and Rideout D 2008 Properties of the volume operator in loop quantum gravity: II. Detailed analysis Class. Quantum Grav. 25 065002 (Preprint 0706.0382)
57. Brunnemann J and Rideout D Classification of embeddable vertex geometries in three dimensional Riemannian space (in preparation)
58. Deift P 1999 Orthogonal polynomials and random matrices: a Riemann-Hilbert approach Courant Lecture Notes 3 (Courant Institute of Mathematical Sciences, New York University)
59. Mehta M L 1990 Random Matrices 2nd edn (New York: Academic)
60. Sexl R U and Urbantke H K 2001 Relativity, Groups, Particles: Special Relativity and Relativistic Symmetry in Field and Particle Physics revised edn (Wien: Springer)
61. Dirac P A M 1967 Lectures on Quantum Mechanics (New York: Academic) (second printing)
62. Marcus M and Minc H 1992 A Survey of Matrix Theory and Matrix Inequalities (New York: Dover)
63. Dreyer O 2004 Private communication
64. LAPACK Linear Algebra Package http://www.netlib.org/lapack
65. Goodale T, Allen G, Lanfermann G, Massó J, Radke T, Seidel E and Shalf J 2003 The Cactus framework and Toolkit: design and applications Vector and Parallel Processing-VECPAR 2002 5th Int. Conf. (Berlin: Springer) pp 197-227