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Introduction

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

Abstract

Ensembles on configuration space have wide applicability. They may be used to describe classical, quantum and hybrid quantum-classical systems, physical systems that are deterministic or subject to uncertainty, discrete systems, particles and fields. They also lead to novel reconstructions of quantum theory from physical and geometric axioms. We introduce the basic elements of the theory, discuss a number of classical and quantum examples, and provide an overview of the many generalizations and applications that form the subjects of later chapters. The approach introduces very few physical and mathematical assumptions. The basic building blocks are the configuration space of the physical system, an ensemble of configurations, and dynamics generated from an action principle. An important role is played by the ensemble Hamiltonian which determines the equations of motion. It must satisfy certain requirements which we discuss in detail. We provide examples of classical and quantum systems and show that the primary difference between quantum and classical evolution lies in the choice of the ensemble Hamiltonian.

Keywords

Configuration Space Hamiltonian Equation Classical Particle Classical Ensemble Homogeneity Property 
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.
    Goldstein, H.: Classical Mechanics. Addison-Wesley, New York (1950)zbMATHGoogle Scholar
  2. 2.
    Synge, J.L.: Classical Dynamics. In: Flügge, S. (ed.) Encyclopedia of Physics, vol. III/1, pp. 1–225. Springer, Berlin (1960)Google Scholar
  3. 3.
    Landauer, R.: Path concepts in Hamilton–Jacobi theory. Am. J. Phys. 20, 363–367 (1952)ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Zakharov, V.E., Kuznetsov, E.A.: Hamiltonian formalism for nonlinear waves. Physics Uspekhi 40, 1087–1116 (1997)ADSCrossRefGoogle Scholar
  5. 5.
    Madelung, E.: Quantentheorie in hydrodynamischer Form. Z. Physik 40, 322–326 (1926)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Beichelt, F.: Stochastic Processes in Science, Engineering and Finance, chapter 5. Taylor & Francis, Boca Raton (2006)CrossRefzbMATHGoogle Scholar
  7. 7.
    Hall, M.J.W.: Superselection from canonical constraints. J. Phys. A 27, 7799–7811 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bell, J.S.: Beables for quantum field theory. In: Bell, M., Gottfried, K., Veltman, M., John, S. (eds.) Bell on the Foundations of Quantum Mechanics, pp. 159–166. World Scientific, Singapore (2001)CrossRefGoogle Scholar
  9. 9.
    Gambetta J., Wiseman, H.M.: Modal dynamics for positive operator measures. Found. Phys. 34, 419–448 (2004) (see also references therein)Google Scholar
  10. 10.
    Haag, R., Bannier, U.: Comments on Mielnik’s generalized (non linear) quantum mechanics. Commun. Math. Phys. 60, 1–6 (1978)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Kibble, T.W.B.: Relativistic models of nonlinear quantum Mechanics. Commun. Math. Phys. 64, 73–82 (1978)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Weinberg, S.: Testing quantum mechanics. Ann. Phys. (N.Y.) 194, 336–386 (1989)Google Scholar
  13. 13.
    Hall, M.J.W., Reginatto, M.: Interacting classical and quantum ensembles. Phys. Rev. A 72, 062109 (2005)ADSCrossRefGoogle Scholar
  14. 14.
    Hall, M.J.W.: Consistent classical and quantum mixed dynamics. Phys. Rev. A 78, 042104 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hall, M.J.W., Reginatto, M.: Schrödinger equation from an exact uncertainty principle. J. Phys. A 35, 3289–3303 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hall, M.J.W., Reginatto, M.: Quantum mechanics from a Heisenberg-type equality. Fortschr. Phys. 50, 646–651 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hall, M.J.W., Kumar, K., Reginatto, M.: Bosonic field equations from an exact uncertainty principle. J. Phys. A 36, 9779–9794 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Reginatto, M.: Exact uncertainty principle and quantization: implications for the gravitational field. Braz. J. Phys. 35, 476–480 (2005)ADSCrossRefGoogle Scholar
  19. 19.
    Hall, M.J.W.: Exact uncertainty approach in quantum mechanics and quantum gravity. Gen. Relativ. Gravit. 37, 1505–1515 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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
  21. 21.
    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
  22. 22.
    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)ADSGoogle Scholar
  23. 23.
    Reginatto, M.: The geometrical structure of quantum theory as a natural generalization of information geometry. In: Mohammad-Djafari. A., Barbaresco, F. (eds) Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Clos Lucé, Amboise, France, 21–26 Sep 2014. AIP Conference Proceedings, vol. 1641, American Institute of Physics, Melville, New York (2015)Google Scholar
  24. 24.
    Reginatto, M., Hall, M.J.W.: Quantum-classical interactions and measurement: a consistent description using statistical ensembles on configuration space. J. Phys.: Conf. Ser. 174, 012038 (2009)ADSGoogle Scholar
  25. 25.
    Chua, A.J.K., Hall, M.J.W., Savage, C.M.: Interacting classical and quantum particles. Phys. Rev. A 85, 022110 (2011)ADSCrossRefGoogle Scholar
  26. 26.
    Hall, M.J.W., Reginatto, M., Savage, C.M.: Nonlocal signaling in the configuration space model of quantum-classical interactions. Phys. Rev. A 86, 054101 (2012)ADSCrossRefGoogle Scholar
  27. 27.
    Albers, M., Kiefer, C., Reginatto, M.: Measurement analysis and quantum gravity. Phys. Rev. D 78, 064051 (2008)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Reginatto, M.: Cosmology with quantum matter and a classical gravitational field: the approach of configuration-space ensembles. J. Phys.: Conf. Ser. 442, 012009 (2013)ADSGoogle 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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