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Quantum Motion on Shape Space and the Gauge Dependent Emergence of Dynamics and Probability in Absolute Space and Time

  • Detlef DürrEmail author
  • Sheldon Goldstein
  • Nino Zanghí
Article

Abstract

Relational formulations of classical mechanics and gravity have been developed by Julian Barbour and collaborators. Crucial to these formulations is the notion of shape space. We indicate here that the metric structure of shape space allows one to straightforwardly define a quantum motion, a Bohmian mechanics, on shape space. We show how this motion gives rise to the more or less familiar theory in absolute space and time. We find that free motion on shape space, when lifted to configuration space, becomes an interacting theory. Many different lifts are possible corresponding in fact to different choices of gauges. Taking the laws of Bohmian mechanics on shape space as physically fundamental, we show how the theory can be statistically analyzed by using conditional wave functions, for subsystems of the universe, represented in terms of absolute space and time.

Keywords

Shape space dynamics Quantum mechanics Bohmian mechanics Typicality analysis on shape space 

Notes

Acknowledgements

We are grateful to Florian Hoffmann for his input to a very early draft of this paper and to Antonio Vassallo for his insights. We thank Sahand Tokasi for stimulating discussions. We thank Eddy Chen and Roderich Tumulka for a careful reading of the manuscript and useful suggestions. The many discussions with Julian Barbour are gratefully acknowledged, especially for sharing with us in his well known enthusiastic way his ideas on shape dynamics. N. Zanghí was supported in part by INFN.

References

  1. 1.
    Anderson, E.: The problem of time and quantum cosmology in the relational particle mechanics arena (2011). arXiv:1111.1472
  2. 2.
    Anderson, E.: The problem of time in quantum gravity. Ann. Phys. 524, 757–786 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barbour, J.: Scale-invariant gravity: particle dynamics. Class. Quantum Gravity 20, 1543 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barbour, J.: Shape dynamics. A Introduction. In: Finster, F., Muller, O., Nardmann, M., Tolksdorf, J., Zeidler, E. (eds.) Quantum Field Theory and Gravity, pp. 257–297. Springer, New York (2012)CrossRefGoogle Scholar
  5. 5.
    Barbour, J.B., Bertotti, B.: Mach’s principle and the structure of dynamical theories. Proc. R. Soc. Lond. A 382, 295–306 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barbour, J., Foster, B.Z., Ó Murchadha, N.: Relativity without relativity. Class. Quantum Gravity 19, 3217 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  8. 8.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of“ hidden” variables. I. Phys. Rev. 85, 166 (1952)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bricmont, J.: Making Sense of Quantum Mechanics. Springer, New York (2016)CrossRefzbMATHGoogle Scholar
  10. 10.
    DeWitt, B.S.: Spacetime as a sheaf of geodesics in superspace. In: Carmeli, M., Fickler, S.I., Witten, L. (eds.) Relativity, pp. 359–374. Springer, New York (1970)CrossRefGoogle Scholar
  11. 11.
    Dürr, D., Goldstein, S., Zanghì, N.: Quantum equilibrium and the origin of absolute uncertainty. J. Stat. Phys. 67, 843–907 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dürr, D., Goldstein, S., Zanghì, N.: Quantum Physics Without Quantum Philosophy. Springer, New York (2012)zbMATHGoogle Scholar
  13. 13.
    Gomes, H., Gryb, S., Koslowski, T.: Einstein gravity as a 3d conformally invariant theory. Class. Quantum Gravity 28, 045005 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gryb, S.: Jacobi’s principle and the disappearance of time. Phys. Rev. D 81, 044035 (2010)ADSCrossRefGoogle Scholar
  15. 15.
    Kendall, D.G.: The diffusion of shape. Adv. Appl. Probab. 9, 428–430 (1977)CrossRefGoogle Scholar
  16. 16.
    Kendall, D.G., Barden, D., Carne, T.K., Le, H.: Shape and Shape Theory. John, New York (2009)zbMATHGoogle Scholar
  17. 17.
    Koslowski, T.: Quantum inflation of classical shapes. Found. Phys. 47, 625–639 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Le, H., Kendall, D.G.: The Riemannian structure of Euclidean shape spaces: a novel environment for statistics. Ann. Stat. 21, 1225–1271 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Marques, F.C.: Scalar curvature, conformal geometry, and the Ricci flow with surgery. In: Proceedings of the International Congress of Mathematicians 2010, pp. 811–829. World Scientific, Singapore (2010)Google Scholar
  20. 20.
    Mercati, F.: Shape Dynamics: Relativity and Relationalism. Oxford University Press, Oxford (2018)CrossRefzbMATHGoogle Scholar
  21. 21.
    Small, C.G.: The Statistical Theory of Shape. Springer, New York (2012)Google Scholar
  22. 22.
    Vassallo, A., Ip, P.H.: On the conceptual issues surrounding the notion of relational Bohmian dynamics. Found. Phys. 46, 943–972 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematisches Institut der Universität MünchenMunichGermany
  2. 2.Departments of Mathematics and Physics, Hill CenterRutgers UniversityPiscatawayUSA
  3. 3.Dipartimento di FisicaUniversità di GenovaGenoaItaly
  4. 4.Istituto Nazionale di Fisica Nucleare (Sezione di Genova)GenoaItaly

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