The Role of Time in Relational Quantum Theories

Abstract

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.

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Fig. 1
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Notes

  1. 1.

    For an analysis of approaches broadly constituted along these lines, we refer the reader to the recent reviews of Anderson [13, 14]. For a comprehensive modern text on canonical quantum gravity see [15]. The canonical formulation of general relativity was originally developed by Dirac [16] and Arnowitt, Deser and Misner (‘ADM’) [17, 18].

  2. 2.

    Defence of this largely unexplored second option is, to our knowledge, only found within the work of Kuchař [1922].

  3. 3.

    By conventional gauge theory methods we will understand: (i) Dirac quantization, (ii) reduced phase space quantization, and (iii) Faddeev–Popov gauge fixing, as outline in Sect. 2.

  4. 4.

    N.B. What we are proposing is only claimed to be a consistent methodology for the quantization of what is, under our definition, a relational theory. There are, of course, alternative types of quantization, just as there are alternative definitions of relationalism.

  5. 5.

    We are indebted to Tim Koslowski for helping to clarify this key point.

  6. 6.

    Here, and below, by equitable we will mean dependent upon the contributions of all the dynamical variables/subsystems of the system in question.

  7. 7.

    We are again indebted to Tim Koslowski for his valuable insight in regards to this extension procedure.

  8. 8.

    Second class constraints can always be made first class by a suitable redefinition of Ω following Dirac [26].

  9. 9.

    There is also a global requirement that Σ gf only intersects the gauge orbits once. We will assume that this requirement can be satisfied.

  10. 10.

    Crucially, this is the step that can be performed in shape dynamics that is highly non-trivial in the ADM formulation of general relativity.

  11. 11.

    This distinction constitutes a temporal orientation rather than a temporal direction, which would imply an arrow of time.

  12. 12.

    See [28, p. 280] for a analogous case.

  13. 13.

    In this section we will sometimes write the coordinates of p using lower case indices for convenience.

  14. 14.

    For an elegant treatment of both Jacobi and parameterized particle models the reader is referred to [46].

  15. 15.

    In particular, in GR it is still the case that standard gauge theory methods lead to a classical theory without even minimal temporal structure. See §3 of [48].

  16. 16.

    For simplicity, we will assume that Σ is compact without boundary.

  17. 17.

    The solutions must admit at least one foliation where the spatial Cauchy hypersurfaces have constant mean (extrinsic) curvature—i.e. be ‘CMC foliable.’ One may argue that all physically reasonable solutions will satisfy this condition which, in any case, is not dramatically stronger than the global hyperbolicity assumption that is fundamental to ADM GR.

References

  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)

    MathSciNet  ADS  Article  Google 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

    MathSciNet  ADS  Article  Google Scholar 

  4. 4.

    Rovelli, C.: Quantum mechanics without time: a model. Phys. Rev. D 42, 2638–2646 (1990)

    ADS  Article  Google Scholar 

  5. 5.

    Rovelli, C.: Time in quantum gravity: an hypothesis. Phys. Rev. D 43, 442 (1991)

    MathSciNet  ADS  Article  Google Scholar 

  6. 6.

    Rovelli, C.: Partial observables. Phys. Rev. D 65, 124013 (2002)

    MathSciNet  ADS  Article  Google Scholar 

  7. 7.

    Dittrich, B.: Partial and complete observables for canonical general relativity. Class. Quantum Gravity 23, 6155 (2006)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  8. 8.

    Dittrich, B.: Partial and complete observables for hamiltonian constrained systems. Gen. Relativ. Gravit. 39, 1891 (2007)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  9. 9.

    Kiefer, C.: Wave packets in minisuperspace. Phys. Rev. D 38(6), 1761–1772 (1988)

    MathSciNet  ADS  Article  Google 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)

    MathSciNet  ADS  Article  Google 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

    MathSciNet  ADS  Article  Google 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)

    MathSciNet  ADS  Article  Google 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)

    Google 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)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  17. 17.

    Arnowitt, R., Deser, S., Misner, C.W.: Canonical variables for general relativity. Phys. Rev. 117, 1595–1602 (1960). http://link.aps.org/doi/10.1103/PhysRev.117.1595

    MathSciNet  ADS  MATH  Article  Google 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)

    MathSciNet  Article  Google 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]

    MathSciNet  ADS  Article  Google 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)

    MathSciNet  ADS  Article  Google Scholar 

  28. 28.

    Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton University Press, Princeton (1992)

    Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Gotay, M.J.: Constraints, reduction, and quantization. J. Math. Phys. 27, 2051–2066 (1986)

    MathSciNet  ADS  MATH  Article  Google 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)

    Google Scholar 

  34. 34.

    Faddeev, L.D.: Feynman integral for singular lagrangians. Theor. Math. Phys. 1, 1–13 (1969)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Guillemin, V., Sternberg, S.: Geometric quantization and multiplicities of group representations. Invent. Math. 67(3), 515–538 (1982)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  36. 36.

    Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)

    Google 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)

    ADS  Article  Google 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]

    MathSciNet  ADS  MATH  Article  Google 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]

    MathSciNet  ADS  Article  Google 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]

    MathSciNet  ADS  Article  Google 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]

    ADS  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Lanczos, C.: The Variational Principles of Mechanics. Dover, New York (1970)

    Google Scholar 

  47. 47.

    Teitelboim, C.: How commutators of constraints reflect the spacetime structure. Ann. Phys. 79(2), 542–557 (1973). http://www.sciencedirect.com/science/article/pii/0003491673900961

    ADS  Article  Google Scholar 

  48. 48.

    Thebault, K.P.Y.: Three denials of time in the interpretation of canonical gravity (2011). http://philsci-archive.pitt.edu/8774/

  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)

    MathSciNet  ADS  Article  Google Scholar 

  50. 50.

    Henneaux, M., Teitelboim, C.: The cosmological constant and general covariance. Phys. Lett. B 222, 195–199 (1989)

    ADS  Article  Google Scholar 

  51. 51.

    Unruh, W.G.: A unimodular theory of canonical quantum gravity. Phys. Rev. D 40, 1048 (1989)

    MathSciNet  ADS  Article  Google Scholar 

  52. 52.

    Unruh, W.G., Wald, R.M.: Time and the interpretation of canonical quantum gravity. Phys. Rev. D 40, 2598 (1989)

    MathSciNet  ADS  Article  Google 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 http://www.uwspace.uwaterloo.ca/handle/10012/6124

  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)

    ADS  Article  Google Scholar 

  57. 57.

    Gryb, S.B.: A definition of background independence. Class. Quantum Gravity 27, 215018 (2010). arXiv:1003.1973 [gr-qc]

    MathSciNet  ADS  Article  Google 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

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  ADS  Article  Google Scholar 

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Acknowledgements

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.

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Gryb, S., Thébault, K. The Role of Time in Relational Quantum Theories. Found Phys 42, 1210–1238 (2012). https://doi.org/10.1007/s10701-012-9665-5

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Keywords

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