A Momentous Arrow of Time

Part of the Fundamental Theories of Physics book series (FTPH, volume 172)


Quantum cosmology offers a unique stage to address questions of time related to its underlying (and perhaps truly quantum dynamical) meaning as well as its origin. Some of these issues can be analyzed with a general scheme of quantum cosmology, others are best seen in loop quantum cosmology. The latters status is still incomplete, and so no full scenario has yet emerged. Nevertheless, using properties that have a potential of pervading more complicated and realistic models, a vague picture shall be sketched here. It suggests the possibility of deriving a beginning within a beginningless theory, by applying cosmic forgetfulness to an early history of the universe.


Quantum Gravity Poisson Bracket Quantum Correction Friedmann Equation Massless Scalar 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was partially supported by NSF grant PHY0748336 and a grant from the Foundational Questions Institute (FQXi).


  1. 1.
    Ch. Berger, L. Sehgal, CP violation and arrows of time evolution of a neutral Kor Bmeson from an incoherent to a coherent state. Phys. Rev. D 76, 036003 (2007) [arXiv:0704.1232]Google Scholar
  2. 2.
    K.V. Kuchař, Time and interpretations of quantum gravity, in Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, ed. by G. Kunstatter, D. E. Vincent, J.G. Williams, (World Scientific, Singapore, 1992)Google Scholar
  3. 3.
    H.D. Zeh, Open questions regarding the arrow of time, in this volume. (Springer, Berlin) [arXiv:0908.3780]Google Scholar
  4. 4.
    M. Gasperini, M. Giovannini, Quantum squeezing and cosmological entropy production. Class. Quantum Grav. 10, L133–L136 (1993)Google Scholar
  5. 5.
    M. Kruczenski, L.E. Oxman, M. Zaldarriaga, Large squeezing behaviour of cosmological entropy generation. Class. Quantum Grav. 11, 2317–2329 (1994)Google Scholar
  6. 6.
    D. Koks, A. Matacz, B.L. Hu, Entropy and uncertainty of squeezed quantum open systems. Phys. Rev. D 55, 5917–5935 (1997) (Erratum: [7])Google Scholar
  7. 7.
    D. Koks, A. Matacz, B.L. Hu, Phys. Rev. D 56, 5281 (1997)Google Scholar
  8. 8.
    C. Kiefer, D. Polarski, A.A. Starobinsky, Entropy of gravitons produced in the early universe. Phys. Rev. D 62, 043518 (2000)Google Scholar
  9. 9.
    M. Bojowald, B. Sandhöfer, A. Skirzewski, A. Tsobanjan, Effective constraints for quantum systems. Rev. Math. Phys. 21, 111–154 (2009) [arXiv:0804.3365]Google Scholar
  10. 10.
    M. Bojowald, R. Tavakol, Recollapsing quantum cosmologies and the question of entropy. Phys. Rev. D 78, 023515 (2008) [arXiv:0803.4484]Google Scholar
  11. 11.
    L. Mersini-Houghton, Notes on Time’s Enigma [arXiv:0909.2330]Google Scholar
  12. 12.
    W. Kaminski, J. Lewandowski, The flat FRW model in LQC: The self-adjointness. Class. Quant. Grav. 25, 035001 (2008) [arXiv:0709.3120]Google Scholar
  13. 13.
    D.L. Wiltshire, An introduction to quantum cosmology, in Cosmology: The Physics of the Universe, ed. by B. Robson, N. Visvanathan, W.S. Woolcock (World Scientific, Singapore, 1996) pp. 473–531 [gr-qc/0101003]Google Scholar
  14. 14.
    M. Bojowald, Large scale effective theory for cosmological bounces. Phys. Rev. D 75, 081301(R) (2007) [gr-qc/0608100]Google Scholar
  15. 15.
    M. Bojowald, Dynamical coherent states and physical solutions of quantum cosmological bounces. Phys. Rev. D 75, 123512 (2007) [gr-qc/0703144]Google Scholar
  16. 16.
    M. Bojowald, H. Hernández, A. Skirzewski, Effective equations for isotropic quantum cosmology including matter. Phys. Rev. D 76, 063511 (2007) [arXiv:0706.1057]Google Scholar
  17. 17.
    M. Bojowald, A. Tsobanjan, Effective constraints for relativistic quantum systems. Phys. Rev. D 80, 125008 (2009) [arXiv:0906.1772]Google Scholar
  18. 18.
    M. Bojowald, Loop quantum cosmology. Living Rev. Relativity 11, 4 (2nd July 2008) [gr-qc/0601085] http://www.livingreviews.org/lrr-2008-4
  19. 19.
    K. Banerjee, G. Date, Discreteness corrections to the effective hamiltonian of isotropic loop quantum cosmology. Class. Quant. Grav. 22, 2017–2033 (2005) [gr-qc/0501102]Google Scholar
  20. 20.
    M. Bojowald, Loop quantum cosmology and inhomogeneities. Gen. Rel. Grav. 38, 1771–1795 (2006) [gr-qc/0609034]Google Scholar
  21. 21.
    M. Bojowald, The dark side of a patchwork universe. Gen. Rel. Grav. 40, 639–660 (2008) [arXiv:0705.4398]Google Scholar
  22. 22.
    M. Bojowald, Inverse scale factor in isotropic quantum geometry. Phys. Rev. D 64, 084018 (2001) [gr-qc/0105067]Google Scholar
  23. 23.
    T. Thiemann, QSD V: Quantum gravity as the natural regulator of matter quantum field theories. Class. Quantum Grav. 15, 1281–1314 (1998) [gr-qc/9705019]Google Scholar
  24. 24.
    M. Bojowald, D. Cartin, G. Khanna, Lattice refining loop quantum cosmology, anisotropic models and stability. Phys. Rev. D 76, 064018 (2007) [arXiv:0704.1137]Google Scholar
  25. 25.
    W. Nelson, M. Sakellariadou, Lattice refining loop quantum cosmology and inflation. Phys. Rev. D 76, 044015 (2007) [arXiv:0706.0179]Google Scholar
  26. 26.
    W. Nelson, M. Sakellariadou, Lattice refining LQC and the matter hamiltonian. Phys. Rev. D 76, 104003 (2007) [arXiv:0707.0588]Google Scholar
  27. 27.
    M. Bojowald, Consistent loop quantum cosmology. Class. Quantum Grav. 26, 075020 (2009) [arXiv:0811.4129]Google Scholar
  28. 28.
    A. Ashtekar, T. Pawlowski, P. Singh, Quantum nature of the big bang: Improved dynamics. Phys. Rev. D 74, 084003 (2006) [gr-qc/0607039]Google Scholar
  29. 29.
    G.J. Olmo, P. Singh, Covariant effective action for loop quantum cosmology a la Palatini. J. Cosmology Astropart. Phys. 0901, 030 (2009) [arXiv:0806.2783]Google Scholar
  30. 30.
    G. Date, S. Sengupta, Effective actions from loop quantum cosmology: Correspondence with higher curvature gravity. Class. Quant. Grav. 26, 105002 (2009) [arXiv:0811.4023]Google Scholar
  31. 31.
    T.P. Sotiriou, Covariant effective action for loop quantum cosmology from order reduction. Phys. Rev. D 79, 044035 (2009) [arXiv:0811.1799]Google Scholar
  32. 32.
    M. Bojowald, G. Hossain, Cosmological vector modes and quantum gravity effects. Class. Quantum Grav. 24, 4801–4816 (2007) [arXiv:0709.0872]Google Scholar
  33. 33.
    M. Bojowald, G. Hossain, Quantum gravity corrections to gravitational wave dispersion. Phys. Rev. D 77, 023508 (2008) [arXiv:0709.2365]Google Scholar
  34. 34.
    M. Bojowald, G. Hossain, M. Kagan, S. Shankaranarayanan, Anomaly freedom in perturbative loop quantum gravity. Phys. Rev. D 78, 063547 (2008) [arXiv:0806.3929]Google Scholar
  35. 35.
    M. Bojowald, G. Hossain, M. Kagan, S. Shankaranarayanan, Gauge invariant cosmological perturbation equations with corrections from loop quantum gravity. Phys. Rev. D 79, 043505 (2009) [arXiv:0811.1572]Google Scholar
  36. 36.
    M. Bojowald, T. Harada, R. Tibrewala, Lemaitre-Tolman-Bondi collapse from the perspective of loop quantum gravity. Phys. Rev. D 78, 064057 (2008) [arXiv:0806.2593]Google Scholar
  37. 37.
    M. Bojowald, J.D. Reyes, Dilaton gravity, poisson sigma models and loop quantum gravity. Class. Quantum Grav. 26, 035018 (2009) [arXiv:0810.5119]Google Scholar
  38. 38.
    M. Bojowald, J.D. Reyes, R. Tibrewala, Non-marginal LTB-like models with inverse triad corrections from loop quantum gravity. Phys. Rev. D 80, 084002 (2009) [arXiv:0906.4767]Google Scholar
  39. 39.
    J.M. Bardeen, Gauge-invariant cosmological perturbations. Phys. Rev. D 22, 1882–1905 (1980)Google Scholar
  40. 40.
    D. Langlois, Hamiltonian formalism and gauge invariance for linear perturbations in inflation. Class. Quant. Grav. 11, 389–407 (1994)Google Scholar
  41. 41.
    E.J.C. Pinho, N. Pinto-Neto, Scalar and vector perturbations in quantum cosmological backgrounds. Phys. Rev. D 76, 023506 (2007) [arXiv:hep-th/0610192]Google Scholar
  42. 42.
    B. Dittrich, J. Tambornino, A perturbative approach to Dirac observables and their space-time algebra. Class. Quant. Grav. 24, 757–784 (2007) [gr-qc/0610060]Google Scholar
  43. 43.
    B. Dittrich, J. Tambornino, Gauge invariant perturbations around symmetry reduced sectors of general relativity: Applications to cosmology, Class. Quantum Grav. 24, 4543–4585 (2007) [gr-qc/0702093]Google Scholar
  44. 44.
    K. Giesel, S. Hofmann, T. Thiemann, O. Winkler, Manifestly Gauge-Invariant General Relativistic Perturbation Theory: I. Foundations. Class. Quant. Grav. 27, 055005 (2010) [arXiv:0711.0115]Google Scholar
  45. 45.
    K. Giesel, S. Hofmann, T. Thiemann, O. Winkler, Manifestly Gauge-Invariant General Relativistic Perturbation Theory: II. FRW Background and First Order. Class. Quant. Grav. 27, 055006 (2010) [arXiv:0711.0117]Google Scholar
  46. 46.
    F.T. Falciano, N. Pinto-Neto, Scalar perturbations in scalar field Quantum cosmology. Phys. Rev. D 79, 023507 (2009) [arXiv:0810.3542]Google Scholar
  47. 47.
    J. Puchta, Master thesis, Warsaw UniversityGoogle Scholar
  48. 48.
    P. Dzierzak, P. Malkiewicz, W. Piechocki, Turning Big Bang into Big Bounce: I. Classical Dynamics. Phys. Rev. D 80, 104001 (2009) [arXiv:0907.3436]Google Scholar
  49. 49.
    P. Malkiewicz, W. Piechocki, Turning big bang into big bounce: Quantum dynamics. Class. Quant. Grav. 27, 225018 (2010) [arXiv:0908.4029]Google Scholar
  50. 50.
    W.F. Blyth, C.J. Isham, Quantization of a Friedmann universe filled with a scalar field. Phys. Rev. D 11, 768–778 (1975)Google Scholar
  51. 51.
    M. Artymowski, Z. Lalak, L. Szulc, Loop quantum cosmology corrections to inflationary models. J. Cosmology Astropart. Phys. 0901, 004 (2009) [arXiv:0807.0160]Google Scholar
  52. 52.
    J. Mielczarek, The Observational Implications of Loop Quantum Cosmology. Phys. Rev. D 81, 063503 (2010) [arXiv:0908.4329]Google Scholar
  53. 53.
    M. Bojowald, What happened before the big bang? Nat. Phys. 3, 523–525 (2007)Google Scholar
  54. 54.
    M. Bojowald, Harmonic cosmology: How much can we know about a universe before the big bang? Proc. Roy. Soc. A 464, 2135–2150 (2008) [arXiv:0710.4919]Google Scholar
  55. 55.
    A. Corichi, P. Singh, Quantum bounce and cosmic recall. Phys. Rev. Lett. 100, 161302 (2008) [arXiv:0710.4543]Google Scholar
  56. 56.
    M. Bojowald, Comment on Quantum bounce and cosmic recall. Phys. Rev. Lett. 101, 209001 (2008) [arXiv:0811.2790]Google Scholar
  57. 57.
    A. Corichi, P. Singh, Reply to ‘Comment on Quantum Bounce and Cosmic Recall, Phys. Rev. Lett. 101, 209002 (2008) [ arXiv:0811.2983]Google Scholar
  58. 58.
    C. Kiefer, H.D. Zeh, Arrow of time in a recollapsing quantum universe. Phys. Rev. D 51, 4145–4153 (1995) [gr-qc/9402036]Google Scholar
  59. 59.
    A. Aguirre, S. Gratton, Steady-state eternal inflation. Phys. Rev. D 65, 083507 (2002) [astro-ph/0111191]Google Scholar
  60. 60.
    A. Aguirre, S. Gratton, Inflation without a beginning: A null boundary proposal. Phys. Rev. D 67, 083515 (2003) [gr-qc/0301042]Google Scholar
  61. 61.
    S.M. Carroll, J. Chen, Spontaneous Inflation and the Origin of the Arrow of Time (2004) [hep-th/0410270]Google Scholar
  62. 62.
    D. Brizuela, G.A. Mena Marugan, T. Pawlowski, Big Bounce and inhomogeneities (2009). Class. Quant. Grav. 27, 052001 (2010) [arXiv:0902.0697]Google Scholar
  63. 63.
    M. Martín-Benito, L.J. Garay, G.A. Mena Marugán, Hybrid quantum gowdy cosmology: Combining loop and fock quantizations. Phys. Rev. D 78, 083516 (2008) [arXiv:0804.1098]Google Scholar
  64. 64.
    M. Novello, S.E.P. Bergliaffa, Bouncing cosmologies. Phys. Rep. 463, 127–213 (2008)Google Scholar
  65. 65.
    R. Vaas, Time before Time - Classifications of universes in contemporary cosmology, and how to avoid the antinomy of the beginning and eternity of the world (2004) [physics/0408111]; see also R. Vaas, Time after time - big bang cosmology and the arrow of time (2012), this volumeGoogle Scholar
  66. 66.
    M. Bojowald, A. Skirzewski, Effective equations of motion for quantum systems. Rev. Math. Phys. 18, 713–745 (2006) [math-ph/0511043]Google Scholar
  67. 67.
    M. Bojowald, A. Skirzewski, Quantum gravity and higher curvature actions. Int. J. Geom. Meth. Mod. Phys. 4, 25–52 (2007) [hep-th/0606232]; in Proceedings of Current Mathematical Topics in Gravitation and Cosmology (42nd Karpacz Winter School of Theoretical Physics), ed. by A. Borowiec, M. FrancavigliaGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute for Gravitation and the CosmosThe Pennsylvania State UniversityUniversity ParkUSA

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