The volume operator in spherically symmetric quantum geometry

Classical and Quantum Gravity

Volumn 21, Issue 21, 2004, Pages 4881-4900

  Bojowald, Martin ^a   Swiderski, Rafal ^a  

^a ALBERT EINSTEIN INSTITUT   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 8744228089     PISSN: 02649381     EISSN: None     Source Type: Journal    
DOI: 10.1088/0264-9381/21/21/009     Document Type: Article

References (47)

1. Rovelli C 1998 Loop quantum gravity Living Rev. Relativ. 1 1 online at http://www.livingreviews.org/Articles/Volume1/1998-1rovelli Thiemann T 2001 Introduction to modern canonical quantum general relativity Preprint gr-qc/0110034 Ashtekar A and Lewandowski J 2004 Background independent quantum gravity: a status report Class. Quantum Grav. 21 R53-152
2. Rovelli C 1998 Loop quantum gravity Living Rev. Relativ. 1 1 online at http://www.livingreviews.org/Articles/Volume1/1998-1rovelli Thiemann T 2001 Introduction to modern canonical quantum general relativity Preprint gr-qc/0110034 Ashtekar A and Lewandowski J 2004 Background independent quantum gravity: a status report Class. Quantum Grav. 21 R53-152
3. Rovelli C 1998 Loop quantum gravity Living Rev. Relativ. 1 1 online at http://www.livingreviews.org/Articles/Volume1/1998-1rovelli Thiemann T 2001 Introduction to modern canonical quantum general relativity Preprint gr-qc/0110034 Ashtekar A and Lewandowski J 2004 Background independent quantum gravity: a status report Class. Quantum Grav. 21 R53-152
4. Einstein A 1917 Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie Sitzber. Berlin 142
5. Friedmann A 1922 Über die Krümmung des Raumes Z. Phys. 10 377-86
6. Schwarzschild K 1916 Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie Sitzber. Deut. Akad. Wiss. Berlin, Phys. -Math. Klasse 189-96 Schwarzschild K 1990 Preprint physics/9905030 (Engl. Transi.)
7. Schwarzschild K 1916 Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie Sitzber. Deut. Akad. Wiss. Berlin, Phys. -Math. Klasse 189-96 Schwarzschild K 1990 Preprint physics/9905030 (Engl. Transi.)
8. DeWitt B S 1967 Quantum theory of gravity. I. the canonical theory Phys. Rev. 160 1113-48
9. Kuchař K V 1971 Canonical quantization of cylindrical gravitational waves Phys. Rev. D 4 955-86
10. Bojowald M and Kastrup H A 2000 Symmetry reduction for quantized diffeomorphism invariant theories of connections Class. Quantum Grav. 17 3009-43
11. Bojowald M 2000 Quantum geometry and symmetry PhD thesis RWTH Aachen (Aachen: Shaker)
12. Bojowald M 2001 Absence of a singularity in loop quantum cosmology Phys. Rev. Lett. 86 5227-30
13. Bojowald M 2002 Isotropic loop quantum cosmology Class. Quantum Grav. 19 2717-41
14. Bojowald M 2003 Homogeneous loop quantum cosmology Class. Quantum Grav. 20 2595-615
15. Bojowald M, Date G and Vandersloot K 2004 Homogeneous loop quantum cosmology: the role of the spin connection Class. Quantum Grav. 21 1253-78
16. Bojowald M 2001 Dynamical initial conditions in quantum cosmology Phys. Rev. Lett. 87 121301 Bojowald M 2003 Initial conditions for a universe Gen. Rel. Grav. 35 1877-83 Cartin D, Khanna G and Bojowald M 2004 Generating function techniques for loop quantum cosmology Class. Quantum Grav. 21 4495-509 (Preprint gr-qc/0405126)
17. Bojowald M 2001 Dynamical initial conditions in quantum cosmology Phys. Rev. Lett. 87 121301 Bojowald M 2003 Initial conditions for a universe Gen. Rel. Grav. 35 1877-83 Cartin D, Khanna G and Bojowald M 2004 Generating function techniques for loop quantum cosmology Class. Quantum Grav. 21 4495-509 (Preprint gr-qc/0405126)
18. Bojowald M 2001 Dynamical initial conditions in quantum cosmology Phys. Rev. Lett. 87 121301 Bojowald M 2003 Initial conditions for a universe Gen. Rel. Grav. 35 1877-83 Cartin D, Khanna G and Bojowald M 2004 Generating function techniques for loop quantum cosmology Class. Quantum Grav. 21 4495-509 (Preprint gr-qc/0405126)
19. Bojowald M 2002 Inflation from quantum geometry Phys. Rev. Lett. 89 261301 Bojowald M and Vandersloot K 2003 Loop quantum cosmology, boundary proposals and inflation Phys. Rev. D 67 124023 Tsujikawa S, Singh P and Maartens R 2003 Loop quantum gravity effects on inflation and the CMB Preprint astro-ph/0311015 Bojowald M, Lidsey J E, Mulryne D J, Singh P and Tavakol R 2004 Inflationary cosmology and quantization ambiguities in semi-classical loop quantum gravity Phys. Rev. D 70 043530 (Preprint gr-qc/0403106)
20. Bojowald M 2002 Inflation from quantum geometry Phys. Rev. Lett. 89 261301 Bojowald M and Vandersloot K 2003 Loop quantum cosmology, boundary proposals and inflation Phys. Rev. D 67 124023 Tsujikawa S, Singh P and Maartens R 2003 Loop quantum gravity effects on inflation and the CMB Preprint astro-ph/0311015 Bojowald M, Lidsey J E, Mulryne D J, Singh P and Tavakol R 2004 Inflationary cosmology and quantization ambiguities in semi-classical loop quantum gravity Phys. Rev. D 70 043530 (Preprint gr-qc/0403106)
21. Bojowald M 2002 Inflation from quantum geometry Phys. Rev. Lett. 89 261301 Bojowald M and Vandersloot K 2003 Loop quantum cosmology, boundary proposals and inflation Phys. Rev. D 67 124023 Tsujikawa S, Singh P and Maartens R 2003 Loop quantum gravity effects on inflation and the CMB Preprint astro-ph/0311015 Bojowald M, Lidsey J E, Mulryne D J, Singh P and Tavakol R 2004 Inflationary cosmology and quantization ambiguities in semi-classical loop quantum gravity Phys. Rev. D 70 043530 (Preprint gr-qc/0403106)
22. Bojowald M 2002 Inflation from quantum geometry Phys. Rev. Lett. 89 261301 Bojowald M and Vandersloot K 2003 Loop quantum cosmology, boundary proposals and inflation Phys. Rev. D 67 124023 Tsujikawa S, Singh P and Maartens R 2003 Loop quantum gravity effects on inflation and the CMB Preprint astro-ph/0311015 Bojowald M, Lidsey J E, Mulryne D J, Singh P and Tavakol R 2004 Inflationary cosmology and quantization ambiguities in semi-classical loop quantum gravity Phys. Rev. D 70 043530 (Preprint gr-qc/0403106)
23. Singh P and Toporensky A 2003 Big crunch avoidance in k = 1 loop quantum cosmology Phys. Rev. D 69 104008 Lidsey J E, Mulryne D J, Nunes N J and Tavakol R 2004 Oscillatory universes in loop quantum cosmology and initial conditions for inflation Phys. Rev. D 70 063521 (Preprint gr-qc/0406042) Vereshchagin G V 2004 Qualitative approach to semi-classical loop quantum cosmology J. Cosmol. Astropart. Phys. JCAP07(2004)013 (Preprint gr-qc/0406108)
24. Singh P and Toporensky A 2003 Big crunch avoidance in k = 1 loop quantum cosmology Phys. Rev. D 69 104008 Lidsey J E, Mulryne D J, Nunes N J and Tavakol R 2004 Oscillatory universes in loop quantum cosmology and initial conditions for inflation Phys. Rev. D 70 063521 (Preprint gr-qc/0406042) Vereshchagin G V 2004 Qualitative approach to semi-classical loop quantum cosmology J. Cosmol. Astropart. Phys. JCAP07(2004)013 (Preprint gr-qc/0406108)
25. Singh P and Toporensky A 2003 Big crunch avoidance in k = 1 loop quantum cosmology Phys. Rev. D 69 104008 Lidsey J E, Mulryne D J, Nunes N J and Tavakol R 2004 Oscillatory universes in loop quantum cosmology and initial conditions for inflation Phys. Rev. D 70 063521 (Preprint gr-qc/0406042) Vereshchagin G V 2004 Qualitative approach to semi-classical loop quantum cosmology J. Cosmol. Astropart. Phys. JCAP07(2004)013 (Preprint gr-qc/0406108)
26. Bojowald M and Date G 2004 Quantum suppression of the generic chaotic behaviour close to cosmological singularities Phys. Rev. Lett. 92 071302 Bojowald M, Date G and Hossain G M 2004 The Bianchi IX model in loop quantum cosmology Class. Quantum Grav. 21 3541-69
27. Bojowald M and Date G 2004 Quantum suppression of the generic chaotic behaviour close to cosmological singularities Phys. Rev. Lett. 92 071302 Bojowald M, Date G and Hossain G M 2004 The Bianchi IX model in loop quantum cosmology Class. Quantum Grav. 21 3541-69
28. Thiemann T and Kastrup H A 1993 Canonical quantization of spherically symmetric gravity in Ashtekar's self-dual representation Nucl. Phys. B 399 211-58 Kastrup H A and Thiemann T 1994 Spherically symmetric gravity as a completely integrable system Nucl. Phys. B 425 665-86
29. Thiemann T and Kastrup H A 1993 Canonical quantization of spherically symmetric gravity in Ashtekar's self-dual representation Nucl. Phys. B 399 211-58 Kastrup H A and Thiemann T 1994 Spherically symmetric gravity as a completely integrable system Nucl. Phys. B 425 665-86
30. Kuchař K V 1994 Geometrodynamics of Schwarzschild black holes Phys. Rev. D 50 3961-81
31. Bojowald M 2004 Spherically symmetric quantum geometry: states and basic operators Class. Quantum Grav. 21 3733-53
32. Bojowald M and Kastrup H A 2000 The area operator in the spherically symmetric sector of loop quantum gravity Class. Quantum Grav. 17 3009-43 (published as section 5 of [7])
33. Rovelli C and Smolin L 1995 Discreteness of area and volume in quantum gravity Nucl. Phys. B 442 593-619 Rovelli C and Smolin L 1995 Nucl. Phys. B 456 753 (erratum) Ashtekar A and Lewandowski J 1997 Quantum theory of geometry II. Volume operators Adv. Theor. Math. Phys. 1 388-429
34. Rovelli C and Smolin L 1995 Discreteness of area and volume in quantum gravity Nucl. Phys. B 442 593-619 Rovelli C and Smolin L 1995 Nucl. Phys. B 456 753 (erratum) Ashtekar A and Lewandowski J 1997 Quantum theory of geometry II. Volume operators Adv. Theor. Math. Phys. 1 388-429
35. Rovelli C and Smolin L 1995 Discreteness of area and volume in quantum gravity Nucl. Phys. B 442 593-619 Rovelli C and Smolin L 1995 Nucl. Phys. B 456 753 (erratum) Ashtekar A and Lewandowski J 1997 Quantum theory of geometry II. Volume operators Adv. Theor. Math. Phys. 1 388-429
36. Bojowald M 2000 Loop quantum cosmology: II. Volume operators Class. Quantum Grav. 17 1509-26
37. Ashtekar A, Bojowald M and Lewandowski J 2003 Mathematical structure of loop quantum cosmology Adv. Theor. Math. Phys. 7 233-68
38. Thiemann T 1998 Quantum spin dynamics (QSD) Class. Quantum Grav. 15 839-73
39. Morse P M and Feshbach H 1953 Methods of Theoretical Physics vol 1 (New York: McGraw-Hill)
40. Velhinho J M 2004 Comments on the kinematical structure of loop quantum cosmology Class. Quantum Grav. 21 L109-13
41. Bojowald M 2000 Loop quantum cosmology: I. Kinematics Class. Quantum Grav. 17 1489-508
42. Abramowitz M and Stegun I A 1984 Pocketbook of Mathematical Functions (Thun/Frankfurt/Main: Deutsch)
43. Bojowald M 2001 Inverse scale factor in isotropic quantum geometry Phys. Rev. D 64 084018
44. Bojowald M 2001 Loop quantum cosmology: IV. Discrete time evolution Class. Quantum Grav. 18 1071-88
45. Brunnemann J and Thiemann T 2004 Simplification of the spectral analysis of the volume operator in loop quantum gravity Preprint gr-qc/0405060
46. Bojowald M 2002 Quantization ambiguities in isotropic quantum geometry Class. Quantum Grav. 19 5113-30
47. Magnus W, Oberhettinger F and Soni R P 1966 Formulas and Theorems for the Special Functions of Theoretical Physics (Berlin: Springer)