Foundations of Physics

, Volume 42, Issue 9, pp 1210–1238 | Cite as

The Role of Time in Relational Quantum Theories

  • Sean Gryb
  • Karim Thébault


We propose a solution to the problem of time for systems with a single global Hamiltonian constraint. Our solution stems from the observation that, for these theories, conventional gauge theory methods fail to capture the full classical dynamics of the system and must therefore be deemed inappropriate. We propose a new strategy for consistently quantizing systems with a relational notion of time that does capture the full classical dynamics of the system and allows for evolution parametrized by an equitable internal clock. This proposal contains the minimal temporal structure necessary to retain the ordering of events required to describe classical evolution. In the context of shape dynamics (an equivalent formulation of general relativity that is locally scale invariant and free of the local problem of time) our proposal can be shown to constitute a natural methodology for describing dynamical evolution in quantum gravity and to lead to a quantum theory analogous to the Dirac quantization of unimodular gravity.


Problem of time Relationalism Quantum gravity Constraint quantization Reparametrization invariance Internal time 



We would like to thank Julian Barbour for helpful comments and criticisms and Lee Smolin for encouraging us to look more closely at the Hamilton–Jacobi formalism. We would also like to thank the participants of the 2nd PIAF conference in Brisbane (in particular Hans Westman) for their inspiring questions on the problem of time. Finally, we would like to extend a special thanks to Tim Koslowski for many useful discussions and for making valuable contributions to our overall understanding of this work. Research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MEDT.


  1. 1.
    Barbour, J.: The End of Time: The Next Revolution in Physics. Oxford University Press, London (2000) Google Scholar
  2. 2.
    Barbour, J.: The timelessness of quantum gravity: II. The appearance of dynamics in static configurations. Class. Quantum Gravity 11, 2875–2897 (1994) MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Halliwell, J.J.: Trajectories for the wave function of the universe from a simple detector model. Phys. Rev. D 64, 044008 (2001). arXiv:gr-qc/0008046 MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Rovelli, C.: Quantum mechanics without time: a model. Phys. Rev. D 42, 2638–2646 (1990) ADSCrossRefGoogle Scholar
  5. 5.
    Rovelli, C.: Time in quantum gravity: an hypothesis. Phys. Rev. D 43, 442 (1991) MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Rovelli, C.: Partial observables. Phys. Rev. D 65, 124013 (2002) MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Dittrich, B.: Partial and complete observables for canonical general relativity. Class. Quantum Gravity 23, 6155 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Dittrich, B.: Partial and complete observables for hamiltonian constrained systems. Gen. Relativ. Gravit. 39, 1891 (2007) MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Kiefer, C.: Wave packets in minisuperspace. Phys. Rev. D 38(6), 1761–1772 (1988) MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Halliwell, J., Hartle, J.B.: Wave functions constructed from an invariant sum over histories satisfy constraints. Phys. Rev. D 43(4), 1170–1194 (1991) MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Halliwell, J.J., Thorwart, J.: Life in an energy eigenstate: decoherent histories analysis of a model timeless universe. Phys. Rev. D 65, 104009 (2002). arXiv:gr-qc/0201070 MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Halliwell, J.J.: Probabilities in quantum cosmological models: a decoherent histories analysis using a complex potential. Phys. Rev. D 80, 124032 (2009) MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Anderson, E.: The problem of time in quantum gravity. arXiv:1009.2157 [gr-qc]
  14. 14.
    Anderson, E.: The problem of time and quantum cosmology in the relational particle mechanics arena. arXiv:1111.1472v1
  15. 15.
    Thiemann, T.: Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge (2007) zbMATHCrossRefGoogle Scholar
  16. 16.
    Dirac, P.A.M.: The theory of gravitation in hamiltonian form. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 246, 333–343 (1958) MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Arnowitt, R., Deser, S., Misner, C.W.: Canonical variables for general relativity. Phys. Rev. 117, 1595–1602 (1960). MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Arnowitt, R., Deser, S., Misner, C.W.: The dynamics of general relativity. In: Witten, L. (ed.) Gravitation: An Introduction to Current Research, pp. 227–265. Wiley, New York (1962), Chap. 7 Google Scholar
  19. 19.
    Kuchar, K.: The problem of time in canonical quantization of relativistic systems. In: Ashtekar, A., Stachel, J. (eds.) Conceptual Problems of Quantum Gravity, p. 141. Boston University Press, Boston (1991) Google Scholar
  20. 20.
    Kuchar, K.: Time and interpretations of quantum gravity. In: 4th Canadian Conference on General Relativity…. World Scientific, Singapore (1992) Google Scholar
  21. 21.
    Kuchař, K.V.: Canonical quantum gravity. In: Gleiser, R.J., Kozameh, C.N., Moreschi, O.M. (eds.) General Relativity and Gravitation 1992, p. 119 (1993). arXiv:gr-qc/9304012 Google Scholar
  22. 22.
    Kuchar, K.: The problem of time in quantum geometrodynamics. In: Butterfield, J. (ed.) The Arguments of Time, pp. 169–195 (1991) Google Scholar
  23. 23.
    Thébault, K.P.Y.: Quantisation, representation and reduction; how should we interpret the quantum hamiltonian constraints of canonical gravity? Symmetry 3, 134 (2011) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Gomes, H., Gryb, S., Koslowski, T.: Einstein gravity as a 3D conformally invariant theory. Class. Quantum Gravity 28, 045005 (2011). arXiv:1010.2481 [gr-qc] MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Gomes, H., Koslowski, T.: The link between general relativity and shape dynamics. arXiv:1101.5974 [gr-qc]
  26. 26.
    Dirac, P.A.M.: Lectures on Quantum Mechanics. Dover, New York (1964) Google Scholar
  27. 27.
    Barbour, J.B.: The timelessness of quantum gravity. 1: The evidence from the classical theory. Class. Quantum Gravity 11, 2853–2873 (1994) MathSciNetADSCrossRefGoogle Scholar
  28. 28.
    Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton University Press, Princeton (1992) zbMATHGoogle Scholar
  29. 29.
    Barbour, J., Foster, B.Z.: Constraints and gauge transformations: Dirac’s theorem is not always valid. arXiv:0808.1223 [gr-qc]
  30. 30.
    Faddeev, L.D.: The Feynman integral for singular lagrangians. Theor. Math. Phys. 1(1), 1–13 (1969) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Gotay, M.J.: Constraints, reduction, and quantization. J. Math. Phys. 27, 2051–2066 (1986) MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. 32.
    Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Celestial Mechanics. Springer, Berlin (1988) Google Scholar
  33. 33.
    Souriau, J.: Structure of Dynamical Systems: A Symplectic View of Physics. Birkhäuser, Basel (1997) zbMATHGoogle Scholar
  34. 34.
    Faddeev, L.D.: Feynman integral for singular lagrangians. Theor. Math. Phys. 1, 1–13 (1969) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Guillemin, V., Sternberg, S.: Geometric quantization and multiplicities of group representations. Invent. Math. 67(3), 515–538 (1982) MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. 36.
    Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004) zbMATHCrossRefGoogle Scholar
  37. 37.
    Gerlach, U.H.: Derivation of the ten Einstein field equations from the semiclassical approximation to quantum geometrodynamics. Phys. Rev. 177, 1929–1941 (1969) ADSCrossRefGoogle Scholar
  38. 38.
    Mach, E.: Die Mechanik in Ihrer Entwicklung Historisch-Kritsch Dargestellt. Barth, Leipzig (1883). English transl.: Mach, E., The Science of Mechanics, Open Court, Chicago (1960) (transl. of 1912 German edn.) Google Scholar
  39. 39.
    Mittelstaedt, P.: Der Zeitbegriff in der Physik. B.I.-Wissenschaftsverlag, Mannheim (1976) Google Scholar
  40. 40.
    Barbour, J.B., Pfister, H. (eds.): Mach’s Principle: From Newton’s Bucket to Quantum Gravity. Proceedings, Conference, Tuebingen, Germany, July 26–30, 1993. Birkhäuser, Boston (1995), 536 pp. (Einstein studies. 6) Google Scholar
  41. 41.
    Marolf, D.: Almost ideal clocks in quantum cosmology: a brief derivation of time. Class. Quantum Gravity 12, 2469–2486 (1995). arXiv:gr-qc/9412016 [gr-qc] MathSciNetADSzbMATHCrossRefGoogle Scholar
  42. 42.
    Marolf, D.: Solving the problem of time in mini-superspace: measurement of Dirac observables. Phys. Rev. D 79, 084016 (2009). arXiv:0902.1551 [gr-qc] MathSciNetADSCrossRefGoogle Scholar
  43. 43.
    Giddings, S.B., Marolf, D., Hartle, J.B.: Observables in effective gravity. Phys. Rev. D 74, 064018 (2006). arXiv:hep-th/0512200 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  44. 44.
    Bojowald, M., Hoehn, P.A., Tsobanjan, A.: An effective approach to the problem of time. Class. Quantum Gravity 28, 035006 (2011). arXiv:1009.5953 [gr-qc] ADSCrossRefGoogle Scholar
  45. 45.
    Hilgevoord, J.: Time in quantum mechanics: a story of confusion. Stud. Hist. Philos. Sci. Part B: Stud. Hist. Philos. Mod. Phys. 36(1), 29–60 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Lanczos, C.: The Variational Principles of Mechanics. Dover, New York (1970) zbMATHGoogle Scholar
  47. 47.
    Teitelboim, C.: How commutators of constraints reflect the spacetime structure. Ann. Phys. 79(2), 542–557 (1973). ADSCrossRefGoogle Scholar
  48. 48.
    Thebault, K.P.Y.: Three denials of time in the interpretation of canonical gravity (2011).
  49. 49.
    Brown, J.D., York, J.W.J.: Jacobi’s action and the recovery of time in general relativity. Phys. Rev. D 40, 3312–3318 (1989) MathSciNetADSCrossRefGoogle Scholar
  50. 50.
    Henneaux, M., Teitelboim, C.: The cosmological constant and general covariance. Phys. Lett. B 222, 195–199 (1989) ADSCrossRefGoogle Scholar
  51. 51.
    Unruh, W.G.: A unimodular theory of canonical quantum gravity. Phys. Rev. D 40, 1048 (1989) MathSciNetADSCrossRefGoogle Scholar
  52. 52.
    Unruh, W.G., Wald, R.M.: Time and the interpretation of canonical quantum gravity. Phys. Rev. D 40, 2598 (1989) MathSciNetADSCrossRefGoogle Scholar
  53. 53.
    Barbour, J.: Shape dynamics. An introduction. arXiv:1105.0183
  54. 54.
    Gryb, S.: Shape dynamics and Mach’s principles: gravity from conformal geometrodynamics. Ph.D. thesis, University of Waterloo (2011). Available at
  55. 55.
    Gomes, H., Gryb, S., Koslowski, T., Mercati, F.: The gravity/CFT correspondence. arXiv:1105.0938 [gr-qc]
  56. 56.
    York, J.J.W.: Role of conformal three geometry in the dynamics of gravitation. Phys. Rev. Lett. 28, 1082–1085 (1972) ADSCrossRefGoogle Scholar
  57. 57.
    Gryb, S.B.: A definition of background independence. Class. Quantum Gravity 27, 215018 (2010). arXiv:1003.1973 [gr-qc] MathSciNetADSCrossRefGoogle Scholar
  58. 58.
    Sorkin, R.D.: Forks in the road, on the way to quantum gravity. Int. J. Theor. Phys. 36, 2759–2781 (1997). arXiv:gr-qc/9706002 MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Daughton, A., Louko, J., Sorkin, R.D.: Initial conditions and unitarity in unimodular quantum cosmology. arXiv:gr-qc/9305016
  60. 60.
    Smolin, L.: The quantization of unimodular gravity and the cosmological constant problem. arXiv:0904.4841 [hep-th]
  61. 61.
    Kuchar, K.V.: Does an unspecified cosmological constant solve the problem of time in quantum gravity? Phys. Rev. D 43, 3332–3344 (1991) MathSciNetADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsUtrecht UniversityUtrechtNetherlands
  2. 2.Centre for Time, Department of PhilosophyUniversity of SydneySydneyAustralia

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