Hybrid Quantum-Classical Ensembles

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


The problem of defining hybrid systems comprising quantum and classical components is highly nontrivial, and the approaches that have been proposed to solve this problem run into various types of fundamental difficulties. The formalism of configuration-space ensembles is able to overcome many of these difficulties, allowing for a general and consistent description of interactions between quantum and classical ensembles. Such hybrid ensembles have a number of desirable features; e.g., quantum-classical interactions do not blur the fundamental distinction between the quantum and classical components; configuration separability is satisfied; and non-relativistic systems are Galilean invariant whenever the interaction potential itself is Galilean invariant. After demonstrating general properties of hybrid ensembles, we consider their application to the description of measurement of a quantum system by a classical apparatus, including examples of position and spin measurement; the scattering of a classical particle from a quantum superposition; and the definitions of Gaussian and coherent ensembles for quantum-classical oscillators. Finally, we generalise quantum Wigner functions to hybrid ensembles.


Configuration Space Wigner Function Quantum Particle Classical Particle Classical Ensemble 
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.


  1. 1.
    Boucher, W., Traschen, J.: Semiclassical physics and quantum fluctuations. Phys. Rev. D 37, 3522–3532 (1988)ADSCrossRefGoogle Scholar
  2. 2.
    Makri, N.: Time-dependent quantum methods for large systems. Ann. Rev. Phys. Chem. 50, 167–191 (1999)ADSCrossRefGoogle Scholar
  3. 3.
    Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199–220 (1975)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Kiefer, C.: Quantum Gravity. Oxford University Press, Oxford (2012)zbMATHGoogle Scholar
  5. 5.
    Bohr, N.: Atomic Physics and Human Knowledge. Wiley, New York (1958)zbMATHGoogle Scholar
  6. 6.
    Heisenberg, W.: Physics and Philosphy. Allen and Unwin, London (1958). Chaps. 3, 8Google Scholar
  7. 7.
    Albers, M., Kiefer, C., Reginatto, M.: Measurement analysis and quantum gravity. Phys. Rev. D 78, 064051 (2008)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Dyson, F.: The world on a string. N. Y. Rev. Books 51(8), 16–19 (2004)Google Scholar
  9. 9.
    Rothman, T., Boughn, S.: Can gravitons be detected? Found. Phys. 36, 1801–1825 (2006)ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Boughn, S., Rothman, T.: Aspects of graviton detection: graviton emission and absorption by atomic hydrogen. Class. Quantum Grav. 23, 5839–5852 (2006)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Hall, M.J.W., Reginatto, M.: Interacting classical and quantum ensembles. Phys. Rev. A 72, 062109 (2005)ADSCrossRefGoogle Scholar
  12. 12.
    Hall, M.J.W.: Consistent classical and quantum mixed dynamics. Phys. Rev. A 78, 042104 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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
  14. 14.
    Caro, J., Salcedo, L.L.: Impediments to mixing classical and quantum dynamics. Phys. Rev. A 60, 842–852 (1999)ADSCrossRefGoogle Scholar
  15. 15.
    Salcedo, L.L.: Comment on "a quantum-classical bracket that satisfies the Jacobi identity" [J. Chem. Phys. 124, 201104 (2006)]. J. Chem. Phys. 126, 057101 (2007)ADSCrossRefGoogle Scholar
  16. 16.
    Peres, A., Terno, D.R.: Hybrid classical-quantum dynamics. Phys. Rev. A 63, 022101 (2001)ADSCrossRefGoogle Scholar
  17. 17.
    Agostini, F., Caprara, S., Ciccotti, G.: Do we have a consistent non-adiabatic quantum-classical mechanics? Europhys. Lett. 78, 30001 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Joos, E., Zeh, H.D., Kiefer, C., Giulini, D.J.W., Kupsch, J., Stamatescu, I.-O.: Decoherence and the appearance of a classical world in quantum theory, 2nd edn. Springer, New York (2003)CrossRefzbMATHGoogle Scholar
  19. 19.
    Diosi, L., Gisin, N., Strunz, W.T.: Quantum approach to coupling classical and quantum dynamics. Phys. Rev. A 61, 022108 (2000)ADSCrossRefGoogle Scholar
  20. 20.
    Takabayasi, T.: On the formulation of quantum mechanics associated with classical pictures. Prog. Theor. Phys. 8, 143–182 (1952)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Takabayasi, T.: Remarks on the formulation of quantum mechanics with classical pictures and on relations between linear scalar fields and hydrodynamical fields. Prog. Theor. Phys. 9, 187–222 (1953)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schiller, R.: Quasi-classical theory of the nonspinning electron. Phys. Rev. 125, 1100–1108 (1962)Google Scholar
  23. 23.
    Page, D.N., Geilker, C.D.: Indirect evidence for quantum gravity. Phys. Rev. Lett. 47, 979–982 (1981)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Chua, A.J.K., Hall, M.J.W., Savage, C.M.: Interacting classical and quantum particles. Phys. Rev. A 85, 022110 (2011)ADSCrossRefGoogle Scholar
  25. 25.
    Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    Hillery, M., O’Connell, R.F., Scully, M.O., Wigner, E.P.: Distribution functions in physics: fundamentals. Phys. Rep. 106, 121–167 (1984)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Hall, M.J.W.: Exact uncertainty relations. Phys. Rev. A 64, 052103 (2001)ADSCrossRefGoogle Scholar
  28. 28.
    Wiseman, H.M., Jones, S.J., Doherty, A.C.: Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox. Phys. Rev. Lett. 98, 140402 (2007)ADSMathSciNetCrossRefzbMATHGoogle 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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