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

SIGMA 12 (2016), 082, 49 pages      arXiv:1602.08104      https://doi.org/10.3842/SIGMA.2016.082
Contribution to the Special Issue on Tensor Models, Formalism and Applications

Quantum Cosmology from Group Field Theory Condensates: a Review

Steffen Gielen a and Lorenzo Sindoni b
a) Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK
b) Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, 14476 Golm, Germany

Received February 29, 2016, in final form August 12, 2016; Published online August 18, 2016

Abstract
We give, in some detail, a critical overview over recent work towards deriving a cosmological phenomenology from the fundamental quantum dynamics of group field theory (GFT), based on the picture of a macroscopic universe as a ''condensate'' of a large number of quanta of geometry which are given by excitations of the GFT field over a ''no-space'' vacuum. We emphasise conceptual foundations, relations to other research programmes in GFT and the wider context of loop quantum gravity (LQG), and connections to the quantum physics of real Bose-Einstein condensates. We show how to extract an effective dynamics for GFT condensates from the microscopic GFT physics, and how to compare it with predictions of more conventional quantum cosmology models, in particular loop quantum cosmology (LQC). No detailed familiarity with the GFT formalism is assumed.

Key words: group field theory; quantum cosmology; loop quantum gravity.

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References

  1. Alesci E., Cianfrani F., Loop quantum cosmology from quantum reduced loop gravity, Europhys. Lett. 111 (2015), 40002, 5 pages, arXiv:1410.4788.
  2. Alexander S., Jyoti D., Magueijo J., Inflation and the quantum measurement problem, Phys. Rev. D 94 (2016), 043502, 5 pages, arXiv:1602.01216.
  3. Alexandrov S., Geiller M., Noui K., Spin foams and canonical quantization, SIGMA 8 (2012), 055, 79 pages, arXiv:1112.1961.
  4. Ambjørn J., Görlich A., Jurkiewicz J., Loll R., Nonperturbative quantum gravity, Phys. Rep. 519 (2012), 127-210, arXiv:1203.3591.
  5. Ashtekar A., New variables for classical and quantum gravity, Phys. Rev. Lett. 57 (1986), 2244-2247.
  6. Ashtekar A., Lewandowski J., Representation theory of analytic holonomy $C^*$-algebras, in Knots and Quantum Gravity (Riverside, CA, 1993), Oxford Lecture Ser. Math. Appl., Vol. 1, Oxford University Press, New York, 1994, 21-61, gr-qc/9311010.
  7. Ashtekar A., Lewandowski J., Quantum theory of geometry. II. Volume operators, Adv. Theor. Math. Phys. 1 (1997), 388-429, gr-qc/9711031.
  8. Ashtekar A., Lewandowski J., Background independent quantum gravity: a status report, Classical Quantum Gravity 21 (2004), R53-R152, gr-qc/0404018.
  9. 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.
  10. Ashtekar A., Rovelli C., Smolin L., Weaving a classical metric with quantum threads, Phys. Rev. Lett. 69 (1992), 237-240, hep-th/9203079.
  11. Ashtekar A., Singh P., Loop quantum cosmology: a status report, Classical Quantum Gravity 28 (2011), 213001, 122 pages, arXiv:1108.0893.
  12. Baez J.C., An introduction to spin foam models of BF theory and quantum gravity, in Geometry and Quantum Physics (Schladming, 1999), Lecture Notes in Phys., Vol. 543, Springer, Berlin, 2000, 25-93, gr-qc/9905087.
  13. Bahr B., Dittrich B., Geiller M., A new realization of quantum geometry, arXiv:1506.08571.
  14. Balian R., Incomplete descriptions and relevant entropies, Amer. J. Phys. 67 (1999), 1078-1090, cond-mat/9907015.
  15. Baratin A., Dittrich B., Oriti D., Tambornino J., Non-commutative flux representation for loop quantum gravity, Classical Quantum Gravity 28 (2011), 175011, 19 pages, arXiv:1004.3450.
  16. Baratin A., Oriti D., Group field theory with noncommutative metric variables, Phys. Rev. Lett. 105 (2010), 221302, 4 pages, arXiv:1002.4723.
  17. Baratin A., Oriti D., Group field theory and simplicial gravity path integrals: a model for Holst-Plebanski gravity, Phys. Rev. D 85 (2012), 044003, 15 pages, arXiv:1111.5842.
  18. Baratin A., Oriti D., Ten questions on group field theory (and their tentative answers), J. Phys. Conf. Ser. 360 (2012), 012002, 10 pages, arXiv:1112.3270.
  19. Barbero G. J.F., Real Ashtekar variables for Lorentzian signature space-times, Phys. Rev. D 51 (1995), 5507-5510, gr-qc/9410014.
  20. Barbieri A., Quantum tetrahedra and simplicial spin networks, Nuclear Phys. B 518 (1998), 714-728, gr-qc/9707010.
  21. Barceló C., Liberati S., Visser M., Analogue gravity, Living Rev. Relativ. 14 (2011), 3, 159 pages, gr-qc/0505065.
  22. Barrett J.W., Dowdall R.J., Fairbairn W.J., Gomes H., Hellmann F., Asymptotic analysis of the Engle-Pereira-Rovelli-Livine four-simplex amplitude, J. Math. Phys. 50 (2009), 112504, 30 pages, arXiv:0902.1170.
  23. Ben Geloun J., Renormalizable models in rank $d\geq 2$ tensorial group field theory, Comm. Math. Phys. 332 (2014), 117-188, arXiv:1306.1201.
  24. Ben Geloun J., Gurau R., Rivasseau V., EPRL/FK group field theory, Europhys. Lett. 92 (2010), 60008, 6 pages, arXiv:1008.0354.
  25. Ben Geloun J., Martini R., Oriti D., Functional renormalization group analysis of a tensorial group field theory on $\mathbb{R}^3$, Europhys. Lett. 112 (2015), 31001, 6 pages, arXiv:1508.01855.
  26. Ben Geloun J., Martini R., Oriti D., Functional renormalization group analysis of tensorial group field theories on $\mathbb{R}^d$, Phys. Rev. D 94 (2016), 024017, 45 pages, arXiv:1601.08211.
  27. Benedetti D., Ben Geloun J., Oriti D., Functional renormalisation group approach for tensorial group field theory: a rank-3 model, J. High Energy Phys. 2015 (2015), no. 3, 084, 40 pages, arXiv:1411.3180.
  28. Bianchi E., Doná P., Speziale S., Polyhedra in loop quantum gravity, Phys. Rev. D 83 (2011), 044035, 17 pages, arXiv:1009.3402.
  29. Bianchi E., Haggard H.M., Discreteness of the volume of space from Bohr-Sommerfeld quantization, Phys. Rev. Lett. 107 (2011), 011301, 4 pages, arXiv:1102.5439.
  30. Bianchi E., Rovelli C., Vidotto F., Towards spinfoam cosmology, Phys. Rev. D 82 (2010), 084035, 8 pages, arXiv:1003.3483.
  31. Bodendorfer N., Quantum reduction to Bianchi I models in loop quantum gravity, Phys. Rev. D 91 (2015), 081502, 6 pages, arXiv:1410.5608.
  32. Bojowald M., Loop quantum cosmology, Living Rev. Relativity 11 (2008), 4, 131 pages, gr-qc/0601085.
  33. Bojowald M., Canonical gravity and applications: cosmology, black holes, and quantum gravity, Cambridge University Press, Cambridge, 2010.
  34. Bojowald M., Mathematical structure of loop quantum cosmology: homogeneous models, SIGMA 9 (2013), 082, 43 pages, arXiv:1206.6088.
  35. Bojowald M., Quantum cosmology: a review, Rep. Progr. Phys. 78 (2015), 023901, 21 pages, arXiv:1501.04899.
  36. 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.
  37. Bojowald M., Chinchilli A.L., Simpson D., Dantas C.C., Jaffe M., Nonlinear (loop) quantum cosmology, Phys. Rev. D 86 (2012), 124027, 13 pages, arXiv:1210.8138.
  38. Bombelli L., Corichi A., Winkler O., Semiclassical quantum gravity: statistics of combinatorial Riemannian geometries, Ann. Phys. 14 (2005), 499-519, gr-qc/0409006.
  39. Bombelli L., Corichi A., Winkler O., Semiclassical quantum gravity: obtaining manifolds from graphs, Classical Quantum Gravity 26 (2009), 245012, 15 pages, arXiv:0905.3492.
  40. Bonzom V., Gurau R., Ryan J.P., Tanasa A., The double scaling limit of random tensor models, J. High Energy Phys. 2014 (2014), no. 9, 051, 49 pages, arXiv:1404.7517.
  41. Boulatov D.V., A model of three-dimensional lattice gravity, Modern Phys. Lett. A 7 (1992), 1629-1646, hep-th/9202074.
  42. Breuer H.P., Petruccione F., The theory of open quantum systems, Oxford University Press, New York, 2002.
  43. Calcagni G., Loop quantum cosmology from group field theory, Phys. Rev. D 90 (2014), 064047, 13 pages, arXiv:1407.8166.
  44. Calcagni G., Gielen S., Oriti D., Group field cosmology: a cosmological field theory of quantum geometry, Classical Quantum Gravity 29 (2012), 105005, 22 pages, arXiv:1201.4151.
  45. Calcagni G., Oriti D., Thürigen J., Dimensional flow in discrete quantum geometries, Phys. Rev. D 91 (2015), 084047, 11 pages, arXiv:1412.8390.
  46. Carrozza S., Group field theory in dimension $4-\varepsilon$, Phys. Rev. D 91 (2015), 065023, 10 pages, arXiv:1411.5385.
  47. Carrozza S., Oriti D., Rivasseau V., Renormalization of tensorial group field theories: Abelian ${\rm U}(1)$ models in four dimensions, Comm. Math. Phys. 327 (2014), 603-641, arXiv:1207.6734.
  48. Carrozza S., Oriti D., Rivasseau V., Renormalization of a ${\rm SU}(2)$ tensorial group field theory in three dimensions, Comm. Math. Phys. 330 (2014), 581-637, arXiv:1303.6772.
  49. de Cesare M., Gargiulo M.V., Sakellariadou M., Semiclassical solutions of generalized Wheeler-DeWitt cosmology, Phys. Rev. D 93 (2016), 024046, 15 pages, arXiv:1509.05728.
  50. De Pietri R., Freidel L., Krasnov K., Rovelli C., Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space, Nuclear Phys. B 574 (2000), 785-806, hep-th/9907154.
  51. Di Francesco P., Ginsparg P., Zinn-Justin J., $2$D gravity and random matrices, Phys. Rep. 254 (1995), 1-133, hep-th/9306153.
  52. Diaz-Polo J., Pranzetti D., Isolated horizons and black hole entropy in loop quantum gravity, SIGMA 8 (2012), 048, 58 pages, arXiv:1112.0291.
  53. Dittrich B., From the discrete to the continuous: towards a cylindrically consistent dynamics, New J. Phys. 14 (2012), 123004, 27 pages, arXiv:1205.6127.
  54. Dittrich B., The continuum limit of loop quantum gravity - a framework for solving the theory, arXiv:1409.1450.
  55. Dittrich B., Geiller M., A new vacuum for loop quantum gravity, Classical Quantum Gravity 32 (2015), 112001, 13 pages, arXiv:1401.6441.
  56. Dittrich B., Martin-Benito M., Steinhaus S., Quantum group spin nets: refinement limit and relation to spin foams, Phys. Rev. D 90 (2014), 024058, 15 pages, arXiv:1312.0905.
  57. Dittrich B., Steinhaus S., Time evolution as refining, coarse graining and entangling, New J. Phys. 16 (2014), 123041, 42 pages, arXiv:1311.7565.
  58. Dittrich B., Thiemann T., Are the spectra of geometrical operators in loop quantum gravity really discrete?, J. Math. Phys. 50 (2009), 012503, 11 pages, arXiv:0708.1721.
  59. Dreyer O., The world is discrete, arXiv:1307.6169.
  60. Dupuis M., Livine E.R., Lifting ${\rm SU}(2)$ spin networks to projected spin networks, Phys. Rev. D 82 (2010), 064044, 11 pages, arXiv:1008.4093.
  61. Engle J., Embedding loop quantum cosmology without piecewise linearity, Classical Quantum Gravity 30 (2013), 085001, 14 pages, arXiv:1301.6210.
  62. Engle J., Livine E., Pereira R., Rovelli C., LQG vertex with finite Immirzi parameter, Nuclear Phys. B 799 (2008), 136-149, arXiv:0711.0146.
  63. Fairbairn W.J., Livine E.R., 3D spinfoam quantum gravity: matter as a phase of the group field theory, Classical Quantum Gravity 24 (2007), 5277-5297, gr-qc/0702125.
  64. Fleischhack C., Kinematical foundations of loop quantum cosmology, arXiv:1505.04400.
  65. Flori C., Thiemann T., Semiclassical analysis of the loop quantum gravity volume operator. I. Flux coherent states, arXiv:0812.1537.
  66. Freidel L., Oriti D., Ryan J., A group field theory for 3d quantum gravity coupled to a scalar field, gr-qc/0506067.
  67. Gielen S., Quantum cosmology of (loop) quantum gravity condensates: an example, Classical Quantum Gravity 31 (2014), 155009, 20 pages, arXiv:1404.2944.
  68. Gielen S., Identifying cosmological perturbations in group field theory condensates, J. High Energy Phys. 2015 (2015), no. 8, 010, 23 pages, arXiv:1505.07479.
  69. Gielen S., Perturbing a quantum gravity condensate, Phys. Rev. D 91 (2015), 043526, 11 pages, arXiv:1411.1077.
  70. Gielen S., Oriti D., Quantum cosmology from quantum gravity condensates: cosmological variables and lattice-refined dynamics, New J. Phys. 16 (2014), 123004, 11 pages, arXiv:1407.8167.
  71. Gielen S., Oriti D., Sindoni L., Cosmology from group field theory formalism for quantum gravity, Phys. Rev. Lett. 111 (2013), 031301, 4 pages, arXiv:1303.3576.
  72. Gielen S., Oriti D., Sindoni L., Homogeneous cosmologies as group field theory condensates, J. High Energy Phys. 2014 (2014), no. 6, 013, 69 pages, arXiv:1311.1238.
  73. Giesel K., Thiemann T., Algebraic quantum gravity (AQG). I. Conceptual setup, Classical Quantum Gravity 24 (2007), 2465-2497, gr-qc/0607099.
  74. Giesel K., Thiemann T., Algebraic quantum gravity (AQG). II. Semiclassical analysis, Classical Quantum Gravity 24 (2007), 2499-2564, gr-qc/0607100.
  75. Girelli F., Livine E.R., Oriti D., Four-dimensional deformed special relativity from group field theories, Phys. Rev. D 81 (2010), 024015, 14 pages, arXiv:0903.3475.
  76. Goroff M.H., Sagnotti A., The ultraviolet behavior of Einstein gravity, Nuclear Phys. B 266 (1986), 709-736.
  77. Guedes C., Oriti D., Raasakka M., Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups, J. Math. Phys. 54 (2013), 083508, 31 pages, arXiv:1301.7750.
  78. Gurau R., Colored group field theory, Comm. Math. Phys. 304 (2011), 69-93, arXiv:0907.2582.
  79. Gurau R., The $1/N$ expansion of tensor models beyond perturbation theory, Comm. Math. Phys. 330 (2014), 973-1019, arXiv:1304.2666.
  80. Gurau R., Ryan J.P., Colored tensor models - a review, SIGMA 8 (2012), 020, 78 pages, arXiv:1109.4812.
  81. Han M., Zhang M., Asymptotics of spinfoam amplitude on simplicial manifold: Lorentzian theory, Classical Quantum Gravity 30 (2013), 165012, 57 pages, arXiv:1109.0499.
  82. Henson J., The causal set approach to quantum gravity, in Approaches to Quantum Gravity, Editor D. Oriti, Cambridge University Press, Cambridge, 2009, 393-413, gr-qc/0601121.
  83. Höhn P.A., Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces, J. Math. Phys. 55 (2014), 083508, 45 pages, arXiv:1401.6062.
  84. Hu B.L., Can spacetime be a condensate?, Internat. J. Theoret. Phys. 44 (2005), 1785-1806, gr-qc/0503067.
  85. Ijjas A., Steinhardt P.J., Loeb A., Inflationary paradigm in trouble after Planck2013, Phys. Lett. B 723 (2013), 261-266, arXiv:1304.2785.
  86. Jacobson T., Thermodynamics of spacetime: the Einstein equation of state, Phys. Rev. Lett. 75 (1995), 1260-1263, gr-qc/9504004.
  87. Kiefer C., Quantum gravity, International Series of Monographs on Physics, Vol. 155, 3rd ed., Oxford University Press, Oxford, 2012.
  88. Kiefer C., Krämer M., Quantum gravitational contributions to the cosmic microwave background anisotropy spectrum, Phys. Rev. Lett. 108 (2012), 021301, 4 pages, arXiv:1103.4967.
  89. Kiefer C., Polarski D., Starobinsky A.A., Quantum-to-classical transition for fluctuations in the early universe, Internat. J. Modern Phys. D 7 (1998), 455-462, gr-qc/9802003.
  90. Koslowski T.A., Dynamical quantum geometry (DQG programme), arXiv:0709.3465.
  91. Krajewski T., Group field theories, PoS Proc. Sci. (2011), PoS(QGQGS2011), 005, 58 pages, arXiv:1210.6257.
  92. Lehners J.-L., Classical inflationary and ekpyrotic universes in the no-boundary wavefunction, Phys. Rev. D 91 (2015), 083525, 18 pages, arXiv:1502.00629.
  93. 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.
  94. Livine E.R., Oriti D., Ryan J.P., Effective Hamiltonian constraint from group field theory, Classical Quantum Gravity 28 (2011), 245010, 17 pages, arXiv:1104.5509.
  95. Maldacena J., The large $N$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998), 231-252, hep-th/9711200.
  96. Mandel L., Wolf E., Optical coherence and quantum optics, Cambridge University Press, Cambridge, 1995.
  97. Markopoulou F., Space does not exist, so time can, arXiv:0909.1861.
  98. Marolf D., Emergent gravity requires kinematic nonlocality, Phys. Rev. Lett. 114 (2015), 031104, 5 pages, arXiv:1409.2509.
  99. Misner C.W., Quantum cosmology. I, Phys. Rev. 186 (1969), 1319-1327.
  100. Ooguri H., Topological lattice models in four dimensions, Modern Phys. Lett. A 7 (1992), 2799-2810, hep-th/9205090.
  101. Oriti D., A quantum field theory of simplicial geometry and the emergence of space-time, J. Phys. Conf. Ser. 67 (2007), 012052, 10 pages, hep-th/0612301.
  102. Oriti D., Group field theory as the microscopic description of the quantum spacetime fluid: a new perspective on the continuum in quantum gravity, PoS Proc. Sci. (2007), PoS(QG-Ph), 030, 38 pages, arXiv:0710.3276.
  103. Oriti D., The microscopic dynamics of quantum space as a group field theory, in Foundations of Space and Time: Reflections on Quantum Gravity, Editors J. Murugan, A. Weltman, G. Ellis, Cambridge University Press, Cambridge, 2012, 257-320, arXiv:1110.5606.
  104. Oriti D., Disappearance and emergence of space and time in quantum gravity, Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys. 46 (2014), 186-199, arXiv:1302.2849.
  105. Oriti D., Group field theory as the second quantization of loop quantum gravity, Classical Quantum Gravity 33 (2016), 085005, 32 pages, arXiv:1310.7786.
  106. Oriti D., Pranzetti D., Ryan J.P., Sindoni L., Generalized quantum gravity condensates for homogeneous geometries and cosmology, Classical Quantum Gravity 32 (2015), 235016, 40 pages, arXiv:1501.00936.
  107. Oriti D., Pranzetti D., Sindoni L., Horizon entropy from quantum gravity condensates, Phys. Rev. Lett. 116 (2016), 211301, 6 pages, arXiv:1510.06991.
  108. Oriti D., Ryan J., Group field theory formulation of 3D quantum gravity coupled to matter fields, Classical Quantum Gravity 23 (2006), 6543-6575, gr-qc/0602010.
  109. Oriti D., Ryan J.P., Thürigen J., Group field theories for all loop quantum gravity, New J. Phys. 17 (2015), 023042, 46 pages, arXiv:1409.3150.
  110. Oriti D., Sindoni L., Toward classical geometrodynamics from the group field theory hydrodynamics, New J. Phys. 13 (2011), 025006, 44 pages, arXiv:1010.5149.
  111. Oriti D., Sindoni L., Wilson-Ewing E., Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates, arXiv:1602.05881.
  112. Oriti D., Sindoni L., Wilson-Ewing E., Bouncing cosmologies from quantum gravity condensates, arXiv:1602.08271.
  113. Ousmane Samary D., Closed equations of the two-point functions for tensorial group field theory, Classical Quantum Gravity 31 (2014), 185005, 29 pages, arXiv:1401.2096.
  114. Pachner U., P.L. homeomorphic manifolds are equivalent by elementary shellings, European J. Combin. 12 (1991), 129-145.
  115. Padmanabhan T., Emergent gravity paradigm: recent progress, Modern Phys. Lett. A 30 (2015), 1540007, 21 pages, arXiv:1410.6285.
  116. Peldán P., Actions for gravity, with generalizations: a review, Classical Quantum Gravity 11 (1994), 1087-1132, gr-qc/9305011.
  117. Perez A., The spin foam approach to quantum gravity, Living Rev. Relativ. 16 (2013), 3, 128 pages, arXiv:1205.2019.
  118. Pitaevskii L., Stringari S., Bose-Einstein condensation, International Series of Monographs on Physics, Vol. 116, The Clarendon Press, Oxford University Press, Oxford, 2003.
  119. Pithis A.G.A., Sakellariadou M., Tomov P., Impact of nonlinear effective interactions on GFT quantum gravity condensates, arXiv:1607.06662.
  120. Ponzano G., Regge T., Semiclassical limit of Racah coefficients, in Spectroscopy and Group Theoretical Methods in Physics, Editor F. Block, North Holland, Amsterdam, 1968, 1-58.
  121. Reisenberger M.P., Rovelli C., Spacetime as a Feynman diagram: the connection formulation, Classical Quantum Gravity 18 (2001), 121-140, gr-qc/0002095.
  122. Rivasseau V., The tensor track, III, Fortschr. Phys. 62 (2014), 81-107, arXiv:1311.1461.
  123. Rivasseau V., Why are tensor field theories asymptotically free?, Europhys. Lett. 111 (2015), 60011, 6 pages, arXiv:1507.04190.
  124. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  125. Rovelli C., Zakopane lectures on loop gravity, PoS Proc. Sci. (2011), PoS(QGQGS2011), 003, 60 pages, arXiv:1102.3660.
  126. Rovelli C., Smolin L., Discreteness of area and volume in quantum gravity, Nuclear Phys. B 442 (1995), 593-619, Erratum, Nuclear Phys. B 456 (1995), 753-754, gr-qc/9411005.
  127. Sahlmann H., On loop quantum gravity kinematics with a non-degenerate spatial background, Classical Quantum Gravity 27 (2010), 225007, 17 pages, arXiv:1006.0388.
  128. Sindoni L., Emergent models for gravity: an overview of microscopic models, SIGMA 8 (2012), 027, 45 pages, arXiv:1110.0686.
  129. Sindoni L., Effective equations for GFT condensates from fidelity, arXiv:1408.3095.
  130. Sorkin R.D., Spacetime and causal sets, in Relativity and Gravitation: Classical and Quantum (Cocoyoc, 1990), World Sci. Publ., River Edge, NJ, 1991, 150-173.
  131. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007, gr-qc/0110034.
  132. Unruh W.G., Experimental black-hole evaporation?, Phys. Rev. Lett. 46 (1981), 1351-1353.
  133. Wands D., Malik K.A., Lyth D.H., Liddle A.R., New approach to the evolution of cosmological perturbations on large scales, Phys. Rev. D 62 (2000), 043527, 8 pages, astro-ph/0003278.
  134. Wiltshire D.L., An introduction to quantum cosmology, in Proceedings of 8th Physics Summer School on Cosmology: The Physics of the Universe (Canberra, 1995), Editors B. Robson, N. Visvanathan, W.S. Woolcock, World Sci., Singapore, 1996, 473-531, gr-qc/0101003.
  135. Zanardi P., Paunković N., Ground state overlap and quantum phase transitions, Phys. Rev. E 74 (2006), 031123, 6 pages, quant-ph/0512249.

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