General Relativity and Gravitation

, Volume 45, Issue 5, pp 911–938 | Cite as

Scale anomaly as the origin of time

  • Julian Barbour
  • Matteo Lostaglio
  • Flavio MercatiEmail author
Editor's Choice (Research Article)


We explore the problem of time in quantum gravity in a point-particle analogue model of scale-invariant gravity. If quantized after reduction to true degrees of freedom, it leads to a time-independent Schrödinger equation. As with the Wheeler–DeWitt equation, time disappears, and a frozen formalism that gives a static wavefunction on the space of possible shapes of the system is obtained. However, if one follows the Dirac procedure and quantizes by imposing constraints, the potential that ensures scale invariance gives rise to a conformal anomaly, and the scale invariance is broken. A behaviour closely analogous to renormalization-group (RG) flow results. The wavefunction acquires a dependence on the scale parameter of the RG flow. We interpret this as time evolution and obtain a novel solution of the problem of time in quantum gravity. We apply the general procedure to the three-body problem, showing how to fix a natural initial value condition, introducing the notion of complexity. We recover a time-dependent Schrödinger equation with a repulsive cosmological force in the ‘late-time’ physics and we analyse the role of the scale invariant Planck constant. We suggest that several mechanisms presented in this model could be exploited in more general contexts.


Quantum gravity Problem of time Shape dynamics Scale invariance \(1/r^2\) potential Scale anomaly 



We would like to thank S. Gryb for his initial input that proved crucial for the beginning of this project. We thank also J. Louko, P. Hoehn, E. Anderson and G. Canevari for useful comments and discussions during the preparation of this paper. M.L. thanks St. Hugh’s College for hospitality when working on his Master Thesis in a joint exchange programme with Collegio Ghislieri; he also thanks the Institute for Advanced Studies of Pavia for partial funding. This work was supported by a grant from the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi Time and Foundations 2010 to the Foundational Questions Institute. It was also made possible in part through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.


  1. 1.
    Kuchař, K.: The Problem of Time in Quantum Geometrodynamics. Oxford University Press, New York (1999)Google Scholar
  2. 2.
    Isham, C.J.: Canonical Quantum Gravity and the Problem of Time. arXiv:gr-qc/9210011Google Scholar
  3. 3.
    Anderson, E.: The Problem of Time in Quantum Gravity. arXiv:1009.2157 [gr-qc]Google Scholar
  4. 4.
    York, J.J.W.: Gravitational degrees of freedom and the initial-value problem. Phys. Rev. Lett. 26, 1656–1658 (1971)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    York, J.J.W.: Role of conformal three geometry in the dynamics of gravitation. Phys. Rev. Lett. 28, 1082–1085 (1972)ADSCrossRefGoogle Scholar
  6. 6.
    York, J.J.W.: Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial value problem of general relativity. J. Math. Phys. 14, 456–464 (1973)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Anderson, E., Barbour, J., Foster, B., O’Murchadha, N.: Scale invariant gravity: geometrodynamics. Class. Quant. Grav. 20, 1571 (2003). arXiv:gr-qc/0211022 [gr-qc]Google Scholar
  8. 8.
    Barbour, J.: Scale-invariant gravity: particle dynamics. Class. Quant. Grav. 20, 1543–1570 (2003). arXiv:gr-qc/0211021Google Scholar
  9. 9.
    Anderson, E.: The Problem of Time and Quantum Cosmology in the Relational Particle Mechanics Arena. arXiv:1111.1472 [gr-qc]Google Scholar
  10. 10.
    Chamon, C., Jackiw, R., Pi, S.-Y., Santos, L.: Conformal quantum mechanics as the \(\text{ CFT }_1\) dual to \(\text{ AdS }_2\). Phys. Lett. B 701, 503–507 (2011). arXiv:1106.0726 [hep-th]Google Scholar
  11. 11.
    Gryb, S., Mercati, F.: Right About Time? arXiv:1301.1538 [gr-qc]Google Scholar
  12. 12.
    Barbour, J.B., Koslowski, T., Mercati, F.: The Solution to the Problem of Time in Shape Dynamics (2013) (in preparation—provisional title)Google Scholar
  13. 13.
    Barbour, J.B., Koslowski, T., Mercati, F.: Complexity and the Arrow of Time in Shape Dynamics (2013) (in preparation—provisional title)Google Scholar
  14. 14.
    Barbour, J.B., Bertotti, B.: Mach’s principle and the structure of dynamical theories. Proc. R. Soc. A 382(1783), 295–306 (1982)MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Dirac, P.A.M.: Lectures on Quantum Mechanics. Dover, New York (1964)Google Scholar
  16. 16.
    Jackiw, R.: What Good are Quantum Field Theory Infinities. arXiv:hep-th/9911071Google Scholar
  17. 17.
    Littlejohn, R., Reinsch, M.: Gauge fields in the separation of rotations and internal motions in the n-body problem. Rev. Mod. Phys. 69, 213 (1997)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Saari, D.G.: Collisions, Rings, and Other Newtonian N-Body Problems. American Mathematical Society, Providence (2005)zbMATHGoogle Scholar
  19. 19.
    Barbour, J.B.: The timelessness of quantum gravity. 2: the appearance of dynamics in static configurations. Class. Quant. Grav. 11, 2875–2897 (1994)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics: Fourier Analysis, Self-Adjointness, Vol. 2. Academic Press, London (1972)Google Scholar
  21. 21.
    Case, K.: Singular potentials. Phys. Rev. 20, 5 (1950)Google Scholar
  22. 22.
    Camblong, H.E., Ordonez, C.R.: Anomaly in conformal quantum mechanics: from molecular physics to black holes. Phys. Rev. D 68, 125013 (2003). arXiv:hep-th/0303166 [hep-th]Google Scholar
  23. 23.
    Inouye, S., Andrews, M.R., Stenger, J., Miesner, H.-J., Stamper-Kurn, D.M., Ketterle, W.: Observation of Feshbach resonances in a Bose–Einstein condensate. Nature 392, 151–154 (1998)ADSCrossRefGoogle Scholar
  24. 24.
    Kraemer, T., Mark, M., Waldburger, P., Danzl, J.G., Chin, C., Engeser, B., Lange, A.D., Pilch, K., Jaakkola, A., Naegerl, H.-C., Grimm, R.: Evidence for Efimov quantum states in an ultracold gas of cesium atoms. Nature 440, 315–318 (2006). arXiv:cond-mat/0512394v2Google Scholar
  25. 25.
    Efimov, V.: Energy levels arising from resonant two-body forces in a three-body system. Phys. Lett. B 33, 563 (1970)Google Scholar
  26. 26.
    Ananos, G.N., Camblong, H.E., Gorrichategui, C., Hernadez, E., Ordonez, C.R.: Anomalous commutator algebra for conformal quantum mechanics. Phys. Rev. D 67, 045018 (2003). arXiv:hep-th/0205191v3Google Scholar
  27. 27.
    Gopalakrishnan, S.: Self-Adjointness and the Renormalization of Singular Potentials. BA thesis, Amherst College (2006)Google Scholar
  28. 28.
    Kaplan, D.B., Lee, J.-W., Son, D.T., Stephanov, M.A.: Conformality lost. Phys. Rev. D 80, 125005 (2009). arXiv:0905.4752 [hep-th]Google Scholar
  29. 29.
    Kolomeisky and Straley, Renormalization-group analysis of the ground-state properties of dilute Bose systems in d spatial dimensions, Phys. Rev. B 46, 12664 (1992)Google Scholar
  30. 30.
    Mueller, E.J., Ho T.-L.: Renormalization Group Limit Cycles in Quantum Mechanical Problems. arXiv:cond-mat/0403283Google Scholar
  31. 31.
    Strominger, A.: Inflation and the dS/CFT correspondence. J. High Energy Phys. 0111, 049 (2001). arXiv:hep-th/0110087 [hep-th]Google Scholar
  32. 32.
    McFadden, P., Skenderis, K.: Holography for cosmology. Phys. Rev. D 81, 021301 (2010). arXiv:0907.5542 [hep-th]Google Scholar
  33. 33.
    Barbour, J.B.: Time and complex numbers in canonical quantum gravity. Phys. Rev. D 47, 5422–5429 (1993)Google Scholar
  34. 34.
    Montgomery, R.: Infinitely many syzygies. Arch. Ration. Mech. Anal. 164, 311–340 (2002)Google Scholar
  35. 35.
    Battye, R., Gibbons, G., Sutcliffe, P.: Central configurations in three dimensions. Proc. R. Soc. A 459, 911–943 (2003). arXiv:hep-th/0201101Google Scholar
  36. 36.
    Barbour, J., O’Murchadha, N.: Classical and Quantum Gravity on Conformal Superspace. arXiv:gr-qc/9911071Google Scholar
  37. 37.
    Barbour, J.B.: The timelessness of quantum gravity. 1: the evidence from the classical theory. Class. Quant. Grav. 11, 2853–2873 (1994)MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Barbour, J.: The End of Time: The Next Revolution in Physics. Oxford University Press, UKGoogle Scholar
  39. 39.
    Barbour, J.: Shape Dynamics. An Introduction. arXiv:1105.0183Google Scholar
  40. 40.
    Gomes, H., Gryb, S., Koslowski, T.: Einstein gravity as a 3D conformally invariant theory. Class. Quant. Grav. 28, 045005 (2011) arXiv:1010.2481 [gr-qc]Google Scholar
  41. 41.
    Gomes, H., Koslowski, T.: The link between general relativity and shape dynamics. Class. Quant. Grav. 29, 075009 (2012). arXiv:1101.5974 [gr-qc]Google Scholar
  42. 42.
    Barbour, J., O’Murchadha, N.: Conformal Superspace: The Configuration Space of General Relativity. arXiv:1009.3559 [gr-qc]Google Scholar
  43. 43.
    Anderson, E., Barbour, J., Foster, B.Z., Kelleher, B., O’Murchadha, N.: The physical gravitational degrees of freedom. Class. Quant. Grav. 22, 1795–1802 (2005). arXiv:gr-qc/0407104Google Scholar
  44. 44.
    Lim, C.C.: Binary trees, symplectic matrices and the Jacobi coordinates of celestial mechanics. Arch. Ration. Mech. Anal. 115(2), 153–165 (1991)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Julian Barbour
    • 1
  • Matteo Lostaglio
    • 2
  • Flavio Mercati
    • 3
    • 4
    Email author
  1. 1.College FarmSouth NewingtonBanburyUK
  2. 2.Department of PhysicsImperial College LondonLondonUK
  3. 3.School of Mathematical SciencesUniversity of NottinghamNottinghamUK
  4. 4.Perimeter Institute for Theoretical PhysicsWaterlooCanada

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