Consistency of Hybrid Quantum-Classical Ensembles

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


The formalism of ensembles on configuration space allows for a general description of interactions between quantum and classical ensembles. In this chapter, we consider such hybrid ensembles and focus on consistency requirements for models of quantum-classical interactions. We show how the configuration ensemble approach is able to satisfy desirable properties such as a Lie algebra of observables and Ehrenfest relations, while evading no-go theorems based in part on such properties. We then discuss issues concerning locality. It is found that, in principle, noninteracting ensembles of quantum and classical particles can be associated with nonlocal energy flows and nonlocal signaling. However, it is shown that such effects can be suppressed by a requirement of ‘classicality’, that localised classical systems have a very large number of degrees of freedom. Measurement aspects are also discussed and again ‘classicality’ plays an important role, this time ensuring an effective and irreversible decoherence. Finally, comparisons are briefly made with elements of the mean-field approach to quantum-classical interactions.


Poisson Bracket Classical Particle Strong Separability Quantum Observable Quantum 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.
    Elze, H.-T.: Linear dynamics of quantum-classical hybrids Phys. Rev. A 85, 052109 (2012)CrossRefGoogle Scholar
  4. 4.
    Koopman, B.O.: Hamiltonian systems and transformations in Hilbert space. Proc. Natl. Acad. Sci. U.S.A. 17, 315–318 (1931)ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Sudarshan, E.C.G.: Interaction between classical and quantum systems and the measurement of quantum observables. Pramana 6, 117–126 (1976)ADSCrossRefGoogle Scholar
  6. 6.
    Sherry, T.N., Sudarshan, E.C.G.: Interaction between classical and quantum systems: a new approach to quantum measurement. I. Phys. Rev. D 18, 4580–4589 (1978)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Sherry, T.N., Sudarshan, E.C.G.: Interaction between classical and quantum systems: a new approach to quantum measurement. II. Theoretical considerations. Phys. Rev. D 20, 857–868 (1979)ADSCrossRefGoogle Scholar
  8. 8.
    Gautam, S.R., Sherry, T.N., Sudarshan, E.C.G.: Interaction between classical and quantum systems: a new approach to quantum measurement. III. Illustration. Phys. Rev. D 20, 3081–3094 (1979)ADSCrossRefGoogle Scholar
  9. 9.
    Peres, A., Terno, D.R.: Hybrid classical-quantum dynamics. Phys. Rev. A 63, 022101 (2001)ADSCrossRefGoogle Scholar
  10. 10.
    Terno, D.R.: Inconsistency of quantum-classical dynamics, and what it implies. Found. Phys. 36, 102–111 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Diosi, L., Gisin, N., Strunz, W.T.: Quantum approach to coupling classical and quantum dynamics. Phys. Rev. A 61, 022108 (2000)ADSCrossRefGoogle Scholar
  12. 12.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “Hidden” variables. II. Phys. Rev. 85, 180–193 (1952)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  14. 14.
    Holland, P.R.: The Quantum Theory of Motion. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
  15. 15.
    Gindensperger, E., Meier, C., Beswick, J.A.: Mixing quantum and classical dynamics using Bohmian trajectories. J. Chem. Phys. 113, 9369–9372 (2000)ADSCrossRefGoogle Scholar
  16. 16.
    Prezhdo, O.V., Brooksby, C.: Quantum backreaction through the Bohmian particle. Phys. Rev. Lett. 86, 3215–3219 (2001)ADSCrossRefGoogle Scholar
  17. 17.
    Burghardt, I., Parlant, G.: On the dynamics of coupled Bohmian and phase-space variables: a new hybrid quantum-classical approach. J Chem Phys. 120, 3055–3058 (2004)ADSCrossRefGoogle Scholar
  18. 18.
    Salcedo, L.L.: Comment on “Quantum Backreaction through the Bohmian Particle”. Phys. Rev. Lett. 90, 118901 (2003)ADSCrossRefGoogle Scholar
  19. 19.
    Prezhdo, O., Brooksby, C.: Comment on “Quantum Backreaction through the Bohmian Particle.” Prezhdo and Brooksby Reply. Phys. Rev. Lett. 90, 118902 (2003)ADSCrossRefGoogle Scholar
  20. 20.
    Salcedo, L.L.: Absence of classical and quantum mixing. Phys. Rev. A 54, 3657–3660 (1996)ADSCrossRefGoogle Scholar
  21. 21.
    Caro, J., Salcedo, L.L.: Impediments to mixing classical and quantum dynamics. Phys. Rev. A 60, 842–852 (1999)ADSCrossRefGoogle Scholar
  22. 22.
    Sahoo, D.: Mixing quantum and classical mechanics and uniqueness of Planck’s constant. J. Phys. A 37, 997–1010 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hall, M.J.W.: Consistent classical and quantum mixed dynamics. Phys. Rev. A 78, 042104 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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
  25. 25.
    Agostini, F., Caprara, S., Ciccotti, G.: Do we have a consistent non-adiabatic quantum-classical mechanics? Europhys. Lett. 78, 30001 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Aleksandrov, I.V.: The statistical dynamics of a system consisting of a classical and a quantum subsystem. Z. Naturforsch. A 36, 902–908 (1981)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Prezhdo, O.V.: A quantum-classical bracket that satisfies the Jacobi identity. J. Chem. Phys. 124, 201104 (2006)ADSCrossRefGoogle Scholar
  28. 28.
    Goldstein, H.: Classical Mechanics. Addison-Wesley, New York (1950)zbMATHGoogle Scholar
  29. 29.
    Merzbacher, E.: Quantum Mechanics, 3rd edn. Wiley, New York (1998)zbMATHGoogle Scholar
  30. 30.
    Salcedo, L.L.: Statistical consistency of quantum-classical hybrids. Phys. Rev. A 85, 022127 (2012)ADSCrossRefGoogle Scholar
  31. 31.
    Hall, M.J.W., Reginatto, M.: Interacting classical and quantum ensembles. Phys. Rev. A 72, 062109 (2005)ADSCrossRefGoogle Scholar
  32. 32.
    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
  33. 33.
    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
  34. 34.
    Peres, A.: Can we undo quantum measurements? Phys. Rev. D 22, 879–883 (1980)ADSCrossRefGoogle Scholar
  35. 35.
    Gamow, G.: Zur Quantentheorie des Atomkernes. Z. Physik 51, 204–212 (1928)ADSCrossRefzbMATHGoogle Scholar
  36. 36.
    Gurney, R.W., Condon, E.U.: Wave mechanics and radioactive disintegration. Nature 122, 439 (1928)ADSCrossRefzbMATHGoogle Scholar
  37. 37.
    Gurney, R.W., Condon, E.U.: Quantum mechanics and radioactive disintegration. Phys. Rev 33, 127–140 (1929)ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    Mott, N.F.: The wave mechanics of \(\alpha \)-ray tracks. Proc. Roy. Soc. A 126, 79–84 (1929)ADSCrossRefzbMATHGoogle Scholar
  39. 39.
    Heisenberg, W.: The Physical Principle of the Quantum Theory. Dover, New York (1949)Google Scholar
  40. 40.
    Zhang, Q., Wu, B.: General approach to quantum-classical hybrid systems and geometric forces. Phys. Rev. Lett. 97, 190401 (2006)ADSCrossRefGoogle Scholar
  41. 41.
    Alonso, J.L., Castro, A., Clemente-Gallardo, J., Cuchí, J.C., Echenique, P., Falceto, F.: Statistics and Nosé formalism for Ehrenfest dynamics. J. Phys. A 44, 395004 (2011)MathSciNetCrossRefzbMATHGoogle 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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