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The Geometry of Ensembles on Configuration Space

  • Michael J. W. HallEmail author
  • Marcel Reginatto
Chapter
Part of the Fundamental Theories of Physics book series (FTPH, volume 184)

Abstract

A description of ensembles on configuration space incorporates at least two geometrical structures which arise in a natural way: a metric structure, which derives from the natural geometry associated with a space of probabilities, and a symplectic structure, which derives from the symplectic geometry associated with a Hamiltonian description of motion. We show that these two geometrical structures give rise to a Kähler geometry. We first consider probabilities P and introduce the information metric. This leads to information geometry, a Riemannian geometry defined on the space of probabilities. We then bring in dynamics via a Hamiltonian formalism defined on a phase space with canonically conjugate coordinates P and S. This leads to more geometrical structure, a symplectic geometry defined on this phase space. The next step is to extend the information metric, which is defined over the space of probabilities only, to a metric over the full phase space. This requires satisfying certain conditions which ensure the compatibility of the metric and symplectic structures. These conditions are equivalent to requiring that the space have a Kähler structure. In this way, we are led to a Kähler geometry. This rich geometrical structure allows for a reconstruction of the geometric formulation of quantum theory. One may associate a Hilbert space with the Kähler space and this leads to the standard version of quantum theory. Thus the theory of ensembles on configuration space permits a geometric derivation of quantum theory.

Keywords

Theinformation Rich Geometric Structure Symplectic Structure Meta Information Ensemble Hamiltonian 
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.

References

  1. 1.
    Kibble, T.W.B.: Geometrization of quantum mechanics. Commun. Math. Phys. 65, 189–201 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Reginatto, M., Hall, M.J.W.: Quantum theory from the geometry of evolving probabilities. In: Goyal, P., Giffin, A., Knuth, K.H., Vrscay, E. (eds.) Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 31st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Waterloo, Canada, 10–15 July 2011, AIP Conference Proceedings, vol. 1443, American Institute of Physics, Melville, New York (2012)Google Scholar
  3. 3.
    Reginatto, M., Hall, M.J.W.: Information geometry, dynamics and discrete quantum mechanics. In: von Toussaint, U. (ed.) Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 32nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Garching, Germany, 15–20 July 2012. AIP Conference Proceedings, vol. 1553, American Institute of Physics, Melville, New York (2013)Google Scholar
  4. 4.
    Reginatto, M.: From probabilities to wave functions: a derivation of the geometric formulation of quantum theory from information geometry. J. Phys. Conf. Ser. 538, 012018 (2014)ADSCrossRefGoogle Scholar
  5. 5.
    Bhattacharyya, A.: On a measure of divergence between two statistical populations defined by their probability distributions. Bull. Calcutta Math. Soc. 35, 99–109 (1943)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bhattacharyya, A.: On a measure of divergence between two multinomial populations. Sankhya 7, 401–406 (1946)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Good, I.J.: Invariant distance in statistics: some miscellaneous comments. J. Stat. Comput. Simul. 36, 179–186 (1990)CrossRefGoogle Scholar
  8. 8.
    Wootters, W.K.: Statistical distance and Hilbert space. Phys. Rev. D. 23, 357–362 (1981)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Rao, C.R.: Differential metrics in probability spaces. In: Amari, S.-I., Barndorff-Nielsen, O.E., Kass, R.E., Lauritzen, S.L., Rao, C.R. (eds.) Differential Geometry in Statistical Inference, pp. 217–240. Institute of Mathematical Statistics, Hayward (1987)CrossRefGoogle Scholar
  10. 10.
    Jeffreys, H.: An invariant form for the prior probability in estimation problems. Proc. Roy. Soc. A 186, 453–461 (1946)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Čencov, N.N.: Statistical Decision Rules and Optimal Inference. Translations of Mathematical Monographs. American Mathematical Society, Providence (1981)Google Scholar
  12. 12.
    Campbell, L.L.: An extended Čencov characterization of the information metric. Proc. Am. Math. Soc. 98, 135–141 (1986)zbMATHGoogle Scholar
  13. 13.
    Goldberg, S.I.: Curvature and Homology. Dover Publications, New York (1982)Google Scholar
  14. 14.
    Scheifele, G.: On nonclassical canonical systems. Celest. Mech. 2, 296–310 (1970)ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Kurcheeva, I.V.: Kustaanheimo-Stiefel regularization and nonclassical canonical transformations. Celest. Mech. 15, 353–365 (1977)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Doebner, H.-D., Goldin, G.A.: Introducing nonlinear gauge transformations in a family of nonlinear Schrdinger equations. Phys. Rev. A 54, 3764–3771 (1996)Google Scholar
  17. 17.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, Berlin (1978)CrossRefzbMATHGoogle Scholar
  18. 18.
    Hall, M.J.W.: Superselection from canonical constraints. J. Phys. A 37, 7799–7811 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mehrafarin, M.: Quantum mechanics from two physical postulates. Int. J. Theor. Phys. 44, 429–442 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Goyal, P.: Information-geometric reconstruction of quantum theory. Phys. Rev. A 78, 052120 (2008)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Goyal, P.: From information geometry to quantum theory. New J. Phys. 12, 023012 (2010)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Stewart, J.: Advanced General Relativity. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Centre for Quantum DynamicsGriffith UniversityBrisbaneAustralia
  2. 2.Physikalisch-Technische BundesanstaltBraunschweigGermany

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