Foundations of Physics

, Volume 48, Issue 3, pp 295–332 | Cite as

What is Quantum Mechanics? A Minimal Formulation



This paper presents a minimal formulation of nonrelativistic quantum mechanics, by which is meant a formulation which describes the theory in a succinct, self-contained, clear, unambiguous and of course correct manner. The bulk of the presentation is the so-called “microscopic theory”, applicable to any closed system S of arbitrary size N, using concepts referring to S alone, without resort to external apparatus or external agents. An example of a similar minimal microscopic theory is the standard formulation of classical mechanics, which serves as the template for a minimal quantum theory. The only substantive assumption required is the replacement of the classical Euclidean phase space by Hilbert space in the quantum case, with the attendant all-important phenomenon of quantum incompatibility. Two fundamental theorems of Hilbert space, the Kochen–Specker–Bell theorem and Gleason’s theorem, then lead inevitably to the well-known Born probability rule. For both classical and quantum mechanics, questions of physical implementation and experimental verification of the predictions of the theories are the domain of the macroscopic theory, which is argued to be a special case or application of the more general microscopic theory.


Quantum mechanics Hilbert space Quantum incompatibility 


  1. 1.
    Friedberg, R., Hohenberg, P.: Compatible quantum theory. Rep. Prog. Phys. 77, 092001 (2014)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Kochen, S.: A reconstruction of quantum mechanics. Found. Phys. 45, 557 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823 (1936)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Griffiths, R.B.: Consistent Quantum Theory. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  5. 5.
    Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447 (1966)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kochen, S., Specker, E.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)MathSciNetMATHGoogle Scholar
  7. 7.
    Mermin, N.D.: Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett. 65, 3373 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885 (1957)MathSciNetMATHGoogle Scholar
  9. 9.
    von Neumann, J.: Mathematical Foundations of Quantum Mechanics, vol. 2. Princeton University Press, Princeton (1996)MATHGoogle Scholar
  10. 10.
    Preskill, J.: Lecture Notes for Physics 219: Quantum Information and Computation (2015).
  11. 11.
    Bub, J., Pitowsky, I.: In: S. Saunders (ed.) Many Worlds?: Everett, Quantum Theory, and Reality, pp. 433–459. Oxford University Press, Oxford (2010)Google Scholar
  12. 12.
    Schrödinger, E.: In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 31, pp. 555–563. Cambridge University Press (1935)Google Scholar
  13. 13.
    Schrödinger, E.: In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 32, pp. 446–452. Cambridge University Press, Cambridge (1936)Google Scholar
  14. 14.
    Hughston, L.P., Jozsa, R., Wootters, W.K.: A complete classification of quantum ensembles having a given density matrix. Phys. Lett. A 183(1), 14 (1993)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Wootters, W.K., Zurek, W.H.: The no-cloning theorem. Phys. Today 62(2), 76 (2009)CrossRefGoogle Scholar
  16. 16.
    Bub, J.: Quantum probabilities as degrees of belief. Stud. Hist. Philos. Sci. Part B 38(2), 232 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. 3. Addison-Wesley, Reading, MA (2006)MATHGoogle Scholar
  18. 18.
    Zeh, H.D.: On the interpretation of measurements in quantum theory. Found. Phys. 1, 69 (1970)ADSCrossRefGoogle Scholar
  19. 19.
    Zurek, W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gell-Mann, M., Hartle, J.B.: Classical equations for quantum systems. Phys. Rev. D 47, 3345 (1993)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Landau, L., Lifshitz, E.: Course of Theoretical Physics: Vol.: 3: Quantum Mechanics: Non-relativistic Theory. Pergamon Press, Oxford (1965)MATHGoogle Scholar
  22. 22.
    Peres, A.: Quantum Theory: Concepts and Methods, vol. 72. Springer, Berlin (1995)MATHGoogle Scholar
  23. 23.
    Bell, J.: Against ‘measurement’. Phys. World 3(8), 33 (1990)CrossRefGoogle Scholar
  24. 24.
    Brukner, Č., Zeilinger, A.: Operationally invariant information in quantum measurements. Phys. Rev. Lett. 83(17), 3354 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Fuchs, C.A.: arXiv:1003.5209 (2010)
  26. 26.
    Fuchs, C.A., Mermin, N.D., Schack, R.: An introduction to QBism with an application to the locality of quantum mechanics. Am. J. Phys. 82(8), 749 (2014)ADSCrossRefGoogle Scholar
  27. 27.
    Dirac, P.A.M.: The Principles of Quantum Mechanics. Oxford University Press, Oxford (1981)MATHGoogle Scholar
  28. 28.
    Everett, H.I.: Relative state formulation of quantum mechanics. Rev. Mod. Phys. 29, 454 (1957)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Griffiths, R.B.: Consistent histories and the interpretation of quantum mechanics. J. Stat. Phys. 36, 219 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Omnès, R.: Consistent interpretations of quantum mechanics. Rev. Mod. Phys. 64(2), 339 (1992)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    de Broglie, L.: The principles of the new undulatory mechanics. J. Phys. Rad. 7, 321 (1926)CrossRefGoogle Scholar
  32. 32.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev. 85, 180 (1952)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Ghirardi, G.C., Rimini, A., Weber, T.: Unified dynamics for microscopic and macroscopic systems. Phys. Rev. D 34(2), 470 (1986)ADSMathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Hartle, J.B.: arXiv preprint arXiv:gr-qc/0508001 (2005)

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsColumbia UniversityNew YorkUSA
  2. 2.Department of PhysicsNew York UniversityNew YorkUSA

Personalised recommendations