Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 082, 43 pages      arXiv:1206.6088      https://doi.org/10.3842/SIGMA.2013.082

Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models

Martin Bojowald
Institute for Gravitation and the Cosmos, The Pennsylvania State University, 104 Davey Lab, University Park, PA 16802, USA

Received August 08, 2013, in final form December 22, 2013; Published online December 30, 2013

Abstract
The mathematical structure of homogeneous loop quantum cosmology is analyzed, starting with and taking into account the general classification of homogeneous connections not restricted to be Abelian. As a first consequence, it is seen that the usual approach of quantizing Abelian models using spaces of functions on the Bohr compactification of the real line does not capture all properties of homogeneous connections. A new, more general quantization is introduced which applies to non-Abelian models and, in the Abelian case, can be mapped by an isometric, but not unitary, algebra morphism onto common representations making use of the Bohr compactification. Physically, the Bohr compactification of spaces of Abelian connections leads to a degeneracy of edge lengths and representations of holonomies. Lifting this degeneracy, the new quantization gives rise to several dynamical properties, including lattice refinement seen as a direct consequence of state-dependent regularizations of the Hamiltonian constraint of loop quantum gravity. The representation of basic operators - holonomies and fluxes - can be derived from the full theory specialized to lattices. With the new methods of this article, loop quantum cosmology comes closer to the full theory and is in a better position to produce reliable predictions when all quantum effects of the theory are taken into account.

Key words: loop quantum cosmology; symmetry reduction.

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References

  1. Anderson I.M., Fels M.E., Torre C.G., Group invariant solutions without transversality, Comm. Math. Phys. 212 (2000), 653-686, math-ph/9910014.
  2. Ashtekar A., Bojowald M., Lewandowski J., Mathematical structure of loop quantum cosmology, Adv. Theor. Math. Phys. 7 (2003), 233-268, gr-qc/0304074.
  3. Ashtekar A., Lewandowski J., Projective techniques and functional integration for gauge theories, J. Math. Phys. 36 (1995), 2170-2191, gr-qc/9411046.
  4. Ashtekar A., Lewandowski J., Background independent quantum gravity: a status report, Classical Quantum Gravity 21 (2004), R53-R152, gr-qc/0404018.
  5. Ashtekar A., Pawlowski T., Singh P., Quantum nature of the big bang: an analytical and numerical investigation, Phys. Rev. D 73 (2006), 124038, 33 pages, gr-qc/0604013.
  6. Ashtekar A., Pawlowski T., Singh P., Quantum nature of the big bang: improved dynamics, Phys. Rev. D 74 (2006), 084003, 23 pages, gr-qc/0607039.
  7. Ashtekar A., Pawlowski T., Singh P., Vandersloot K., Loop quantum cosmology of k=1 FRW models, Phys. Rev. D 75 (2007), 024035, 26 pages, gr-qc/0612104.
  8. Ashtekar A., Wilson-Ewing E., Loop quantum cosmology of Bianchi type I models, Phys. Rev. D 79 (2009), 083535, 21 pages, arXiv:0903.3397.
  9. Ashtekar A., Wilson-Ewing E., Loop quantum cosmology of Bianchi type II models, Phys. Rev. D 80 (2009), 123532, 16 pages, arXiv:0910.1278.
  10. Bianchi E., Rovelli C., Vidotto F., Towards spinfoam cosmology, Phys. Rev. D 82 (2010), 084035, 8 pages, arXiv:1003.3483.
  11. Bojowald M., Loop quantum cosmology. I. Kinematics, Classical Quantum Gravity 17 (2000), 1489-1508, gr-qc/9910103.
  12. Bojowald M., Loop quantum cosmology. II. Volume operators, Classical Quantum Gravity 17 (2000), 1509-1526, gr-qc/9910104.
  13. Bojowald M., Loop quantum cosmology. III. Wheeler-DeWitt operators, Classical Quantum Gravity 18 (2001), 1055-1069, gr-qc/0008052.
  14. Bojowald M., Absence of a singularity in loop quantum cosmology, Phys. Rev. Lett. 86 (2001), 5227-5230, gr-qc/0102069.
  15. Bojowald M., The semiclassical limit of loop quantum cosmology, Classical Quantum Gravity 18 (2001), L109-L116, gr-qc/0105113.
  16. Bojowald M., Isotropic loop quantum cosmology, Classical Quantum Gravity 19 (2002), 2717-2741, gr-qc/0202077.
  17. Bojowald M., Homogeneous loop quantum cosmology, Classical Quantum Gravity 20 (2003), 2595-2615, gr-qc/0303073.
  18. Bojowald M., Degenerate configurations, singularities and the non-abelian nature of loop quantum gravity, Classical Quantum Gravity 23 (2006), 987-1008, gr-qc/0508118.
  19. Bojowald M., Loop quantum cosmology and inhomogeneities, Gen. Relativity Gravitation 38 (2006), 1771-1795, gr-qc/0609034.
  20. Bojowald M., Dynamical coherent states and physical solutions of quantum cosmological bounces, Phys. Rev. D 75 (2007), 123512, 16 pages, gr-qc/0703144.
  21. Bojowald M., Singularities and quantum gravity, in Cosmology and Gravitation, AIP Conf. Proc., Vol. 910, Amer. Inst. Phys., Melville, NY, 2007, 294-333, gr-qc/0702144.
  22. Bojowald M., Loop quantum cosmology, Living Rev. Relativity 11 (2008), 4, 131 pages, gr-qc/0601085.
  23. Bojowald M., The dark side of a patchwork universe, Gen. Relativity Gravitation 40 (2008), 639-660, arXiv:0705.4398.
  24. Bojowald M., Consistent loop quantum cosmology, Classical Quantum Gravity 26 (2009), 075020, 10 pages, arXiv:0811.4129.
  25. Bojowald M., Canonical gravity and applications: cosmology, black holes, and quantum gravity, Cambridge University Press, Cambridge, 2010.
  26. Bojowald M., Quantum cosmology: a fundamental theory of the Universe, Lecture Notes in Physics, Vol. 835, Springer, New York, 2011.
  27. Bojowald M., Calcagni G., Inflationary observables in loop quantum cosmology, J. Cosmol. Astropart. Phys. 2011 (2011), no. 3, 032, 35 pages, arXiv:1011.2779.
  28. Bojowald M., Calcagni G., Tsujikawa S., Observational constraints on loop quantum cosmology, Phys. Rev. Lett. 107 (2011), 211302, 5 pages, arXiv:1101.5391.
  29. Bojowald M., Calcagni G., Tsujikawa S., Observational test of inflation in loop quantum cosmology, J. Cosmol. Astropart. Phys. 2011 (2011), no. 11, 046, 32 pages, arXiv:1107.1540.
  30. Bojowald M., Cartin D., Khanna G., Lattice refining loop quantum cosmology, anisotropic models, and stability, Phys. Rev. D 76 (2007), 064018, 13 pages, arXiv:0704.1137.
  31. Bojowald M., Das R., Canonical gravity with fermions, Phys. Rev. D 78 (2008), 064009, 16 pages, arXiv:0710.5722.
  32. Bojowald M., Das R., Fermions in loop quantum cosmology and the role of parity, Classical Quantum Gravity 25 (2008), 195006, 23 pages, arXiv:0806.2821.
  33. Bojowald M., Date G., Vandersloot K., Homogeneous loop quantum cosmology: the role of the spin connection, Classical Quantum Gravity 21 (2004), 1253-1278, gr-qc/0311004.
  34. Bojowald M., Hossain G.M., Kagan M., Shankaranarayanan S., Anomaly freedom in perturbative loop quantum gravity, Phys. Rev. D 78 (2008), 063547, 31 pages, arXiv:0806.3929.
  35. Bojowald M., Kagan M., Hernández H.H., Skirzewski A., Effective constraints of loop quantum gravity, Phys. Rev. D 75 (2007), 064022, 25 pages, gr-qc/0611112.
  36. Bojowald M., Kastrup H.A., Symmetry reduction for quantized diffeomorphism-invariant theories of connections, Classical Quantum Gravity 17 (2000), 3009-3043, hep-th/9907042.
  37. Bojowald M., Paily G.M., Deformed general relativity and effective actions from loop quantum gravity, Phys. Rev. D 86 (2012), 104018, 24 pages, arXiv:1112.1899.
  38. Bojowald M., Paily G.M., Deformed general relativity, Phys. Rev. D 87 (2013), 044044, 7 pages, arXiv:1212.4773.
  39. Bojowald M., Paily G.M., Reyes J.D., Tibrewala R., Black-hole horizons in modified spacetime structures arising from canonical quantum gravity, Classical Quantum Gravity 28 (2011), 185006, 34 pages, arXiv:1105.1340.
  40. Bojowald M., Perez A., Spin foam quantization and anomalies, Gen. Relativity Gravitation 42 (2010), 877-907, gr-qc/0303026.
  41. Bojowald M., Reyes J.D., Tibrewala R., Nonmarginal Lemaitre-Tolman-Bondi-like models with inverse triad corrections from loop quantum gravity, Phys. Rev. D 80 (2009), 084002, 19 pages, arXiv:0906.4767.
  42. Bojowald M., Sandhöfer B., Skirzewski A., Tsobanjan A., Effective constraints for quantum systems, Rev. Math. Phys. 21 (2009), 111-154, arXiv:0804.3365.
  43. Bojowald M., Skirzewski A., Effective equations of motion for quantum systems, Rev. Math. Phys. 18 (2006), 713-745, math-ph/0511043.
  44. Brodbeck O., On symmetric gauge fields for arbitrary gauge and symmetry groups, Helv. Phys. Acta 69 (1996), 321-324, gr-qc/9610024.
  45. Brunnemann J., Fleischhack C., On the configuration spaces of homogeneous loop quantum cosmology and loop quantum gravity, arXiv:0709.1621.
  46. Brunnemann J., Rideout D., Properties of the volume operator in loop quantum gravity. I. Results, Classical Quantum Gravity 25 (2008), 065001, 32 pages, arXiv:0706.0469.
  47. Brunnemann J., Rideout D., Properties of the volume operator in loop quantum gravity. II. Detailed presentation, Classical Quantum Gravity 25 (2008), 065002, 104 pages, arXiv:0706.0382.
  48. Brunnemann J., Thiemann T., Unboundedness of triad-like operators in loop quantum gravity, Classical Quantum Gravity 23 (2006), 1429-1483, gr-qc/0505033.
  49. Buchert T., Toward physical cosmology: focus on inhomogeneous geometry and its non-perturbative effects, Classical Quantum Gravity 28 (2011), 164007, 29 pages, arXiv:1103.2016.
  50. Cailleteau T., Mielczarek J., Barrau A., Grain J., Anomaly-free scalar perturbations with holonomy corrections in loop quantum cosmology, Classical Quantum Gravity 29 (2012), 095010, 17 pages, arXiv:1111.3535.
  51. Chiou D.W., Effective dynamics for the cosmological bounces in Bianchi type I loop quantum cosmology, gr-qc/0703010.
  52. Chiou D.W., Vandersloot K., Behavior of nonlinear anisotropies in bouncing Bianchi I models of loop quantum cosmology, Phys. Rev. D 76 (2007), 084015, 15 pages, arXiv:0707.2548.
  53. Cianfrani F., Marchini A., Montani G., The picture of the Bianchi I model via gauge fixing in loop quantum gravity, Europhys. Lett. 99 (2011), 10003, 6 pages, arXiv:1201.2588.
  54. Cianfrani F., Montani G., Implications of the gauge-fixing in loop quantum cosmology, Phys. Rev. D 85 (2012), 024027, 7 pages, arXiv:1104.4546.
  55. Dittrich B., Guedes C., Oriti D., On the space of generalized fluxes for loop quantum gravity, Classical Quantum Gravity 30 (2013), 055008, 24 pages, arXiv:1205.6166.
  56. Dittrich B., Höhn P.A., Canonical simplicial gravity, Classical Quantum Gravity 29 (2012), 115009, 52 pages, arXiv:1108.1974.
  57. Doldán R., Gambini R., Mora P., Quantum mechanics for totally constrained dynamical systems and evolving Hilbert spaces, Internat. J. Theoret. Phys. 35 (1996), 2057-2074, hep-th/9404169.
  58. Ellis G.F.R., Buchert T., The universe seen at different scales, Phys. Lett. A 347 (2005), 38-46, gr-qc/0506106.
  59. Engle J., Quantum field theory and its symmetry reduction, Classical Quantum Gravity 23 (2006), 2861-2893, gr-qc/0511107.
  60. Engle J., Relating loop quantum cosmology to loop quantum gravity: symmetric sectors and embeddings, Classical Quantum Gravity 24 (2007), 5777-5802, gr-qc/0701132.
  61. Fewster C.J., Sahlmann H., Phase space quantization and loop quantum cosmology: a Wigner function for the Bohr-compactified real line, Classical Quantum Gravity 25 (2008), 225015, 20 pages, arXiv:0804.2541.
  62. Fleischhack C., Representations of the Weyl algebra in quantum geometry, Comm. Math. Phys. 285 (2009), 67-140, math-ph/0407006.
  63. Gaul M., Rovelli C., A generalized Hamiltonian constraint operator in loop quantum gravity and its simplest Euclidean matrix elements, Classical Quantum Gravity 18 (2001), 1593-1624, gr-qc/0011106.
  64. Giesel K., Thiemann T., Consistency check on volume and triad operator quantization in loop quantum gravity. I, Classical Quantum Gravity 23 (2006), 5667-5691, gr-qc/0507036.
  65. Henderson A., Laddha A., Tomlin C., Constraint algebra in loop quantum gravity reloaded. I. Toy model of a U(1)3 gauge theory, Phys. Rev. D 88 (2013), 044028, 38 pages, arXiv:1204.0211.
  66. Höhn P.A., From classical to quantum: new canonical tools for the dynamics of gravity, Ph.D. Thesis, Utrecht University, 2012.
  67. Jacobson T., Trans-Planckian redshifts and the substance of the space-time river, Progr. Theoret. Phys. Suppl. (1999), 1-17, hep-th/0001085.
  68. Kamiński W., Lewandowski J., The flat FRW model in LQC: self-adjointness, Classical Quantum Gravity 25 (2008), 035001, 11 pages, arXiv:0709.3120.
  69. Kamiński W., Pawlowski T., Loop quantum cosmology evolution operator of an FRW universe with a positive cosmological constant, Phys. Rev. D 81 (2010), 024014, 9 pages, arXiv:0912.0162.
  70. Klauder J.R., Affine quantum gravity, Internat. J. Modern Phys. D 12 (2003), 1769-1773, gr-qc/0305067.
  71. Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. I, Interscience Publishers, New York - London, 1963.
  72. Kreienbuehl A., Husain V., Seahra S.S., Modified general relativity as a model for quantum gravitational collapse, Classical Quantum Gravity 29 (2012), 095008, 19 pages, arXiv:1011.2381.
  73. Kuchar K.V., Ryan Jr. M.P., Is minisuperspace quantization valid?: Taub in mixmaster, Phys. Rev. D 40 (1989), 3982-3996.
  74. Lewandowski J., Okołów A., Sahlmann H., Thiemann T., Uniqueness of diffeomorphism invariant states on holonomy-flux algebras, Comm. Math. Phys. 267 (2006), 703-733, gr-qc/0504147.
  75. Livine E.R., Martín-Benito M., Classical setting and effective dynamics for spinfoam cosmology, Classical Quantum Gravity 30 (2013), 035006, 50 pages, arXiv:1111.2867.
  76. MacCallum M.A.H., Taub A.H., Variational principles and spatially-homogeneous universes, including rotation, Comm. Math. Phys. 25 (1972), 173-189.
  77. Małkiewicz P., Reduced phase space approach to Kasner universe and the problem of time in quantum theory, Classical Quantum Gravity 29 (2012), 075008, 21 pages, arXiv:1105.6030.
  78. Martín-Benito M., Mena Marugán G.A., Pawlowski T., Physical evolution in loop quantum cosmology: the example of the vacuum Bianchi I model, Phys. Rev. D 80 (2009), 084038, 23 pages, arXiv:0906.3751.
  79. Misner C.W., Mixmaster Universe, Phys. Rev. Lett. 22 (1969), 1071-1074.
  80. Murugan J., Weltman A., Ellis G.F.R. (Editors), Foundations of space and time. Reflections on quantum gravity, Cambridge University Press, Cambridge, 2012.
  81. Perez A., Spin foam models for quantum gravity, Classical Quantum Gravity 20 (2003), R43-R104, gr-qc/0301113.
  82. Perez A., Regularization ambiguities in loop quantum gravity, Phys. Rev. D 73 (2006), 044007, 18 pages, gr-qc/0509118.
  83. Perez A., Pranzetti D., On the regularization of the constraint algebra of quantum gravity in 2+1 dimensions with a nonvanishing cosmological constant, Classical Quantum Gravity 27 (2010), 145009, 20 pages, arXiv:1001.3292.
  84. Reisenberger M.P., Rovelli C., "Sum over surfaces" form of loop quantum gravity, Phys. Rev. D 56 (1997), 3490-3508, gr-qc/9612035.
  85. Reyes J.D., Spherically symmetric loop quantum gravity: connections to 2-dimensional models and applications to gravitational collapse, Ph.D. Thesis, The Pennsylvania State University, 2009.
  86. Rovelli C., Loop quantum gravity, Living Rev. Relativ. 1 (1998), 1, 75 pages, gr-qc/9710008.
  87. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  88. Rovelli C., Smolin L., The physical Hamiltonian in nonperturbative quantum gravity, Phys. Rev. Lett. 72 (1994), 446-449, gr-qc/9308002.
  89. Rovelli C., Smolin L., Spin networks and quantum gravity, Phys. Rev. D 52 (1995), 5743-5759.
  90. Rovelli C., Vidotto F., Stepping out of homogeneity in loop quantum cosmology, Classical Quantum Gravity 25 (2008), 225024, 16 pages, arXiv:0805.4585.
  91. Thiemann T., Quantum spin dynamics (QSD), Classical Quantum Gravity 15 (1998), 839-873, gr-qc/9606089.
  92. Thiemann T., Quantum spin dynamics (QSD). V. Quantum gravity as the natural regulator of the Hamiltonian constraint of matter quantum field theories, Classical Quantum Gravity 15 (1998), 1281-1314, gr-qc/9705019.
  93. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007, gr-qc/0110034.
  94. Torre C.G., Midisuperspace models of canonical quantum gravity, Internat. J. Theoret. Phys. 38 (1999), 1081-1102, gr-qc/9806122.
  95. Unruh W., Time, gravity, and quantum mechanics, in Time's Arrows Today (Vancouver, BC, 1992), Editor S.F. Savitt, Cambridge University Press, Cambridge, 1995, 23-65, gr-qc/9312027.
  96. Vandersloot K., Hamiltonian constraint of loop quantum cosmology, Phys. Rev. D 71 (2005), 103506, 13 pages, gr-qc/0502082.
  97. Velhinho J.M., Comments on the kinematical structure of loop quantum cosmology, Classical Quantum Gravity 21 (2004), L109-L113, gr-qc/0406008.
  98. Velhinho J.M., The quantum configuration space of loop quantum cosmology, Classical Quantum Gravity 24 (2007), 3745-3758, arXiv:0704.2397.
  99. Weiss N., Constraints on Hamiltonian lattice formulations of field theories in an expanding universe, Phys. Rev. D 32 (1985), 3228-3232.
  100. Wilson-Ewing E., Loop quantum cosmology of Bianchi type IX models, Phys. Rev. D 82 (2010), 043508, 13 pages, arXiv:1005.5565.
  101. Wilson-Ewing E., Holonomy corrections in the effective equations for scalar mode perturbations in loop quantum cosmology, Classical Quantum Gravity 29 (2012), 085005, 19 pages, arXiv:1108.6265.

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