Skip to main content

Quantum Mechanics and the Principle of Maximal Variety


Quantum mechanics is derived from the principle that the universe contain as much variety as possible, in the sense of maximizing the distinctiveness of each subsystem. The quantum state of a microscopic system is defined to correspond to an ensemble of subsystems of the universe with identical constituents and similar preparations and environments. A new kind of interaction is posited amongst such similar subsystems which acts to increase their distinctiveness, by extremizing the variety. In the limit of large numbers of similar subsystems this interaction is shown to give rise to Bohm’s quantum potential. As a result the probability distribution for the ensemble is governed by the Schroedinger equation. The measurement problem is naturally and simply solved. Microscopic systems appear statistical because they are members of large ensembles of similar systems which interact non-locally. Macroscopic systems are unique, and are not members of any ensembles of similar systems. Consequently their collective coordinates may evolve deterministically. This proposal could be tested by constructing quantum devices from entangled states of a modest number of quits which, by its combinatorial complexity, can be expected to have no natural copies.

This is a preview of subscription content, access via your institution.


  1. If the ontology posited by the [611] papers may seem extravagant, their proposal had the virtue of a simple form for the inter-ensemble interactions. This inspired me to seek to use such a simple dynamics in the real ensemble idea. In particular, an important insight contained in [11] is that if there are N particles on a line with positions, \(x_i\), with \(i =1, \dots , N\), the density at the k’th point can be approximated by

    $$\begin{aligned} \rho (x_k ) \approx \frac{1}{N (x_{k+1} - x_k )} \end{aligned}$$

    This motivates the choose of a ultraviolet cutoff, in equation (27) below.

  2. Some possibly related approaches are [1416].

  3. For a different approach to causal sets, see [36, 37].


  1. Unger, R.M., Smolin, L.: The Singular Universe and the Reality of Time. Cambridge University Press, Cambridge (2014)

    Book  MATH  Google Scholar 

  2. Smolin, L.: Time Reborn. Houghton Mifflin Harcourt, Boston (2013)

    Google Scholar 

  3. Smolin, L.: Temporal naturalism. Invited contribution for a special Issue of Studies in History and Philosophy of Modern Physics, on Time and Cosmology, edited by Emily Grosholz. arXiv:1310.8539

  4. Smolin, L.: A real ensemble interpretation of quantum mechanics. Found. Phys. (2012). doi:10.1007/s10701-012-9666-4

  5. Smolin, L.: Precedence and freedom in quantum physics. Int. J. Quantum Found. 1, 44–56 (2015). arXiv:1205.3707

  6. Holland, P.: Computing the wavefunction from trajectories: particle and wave pictures in quantum mechanics and their relation. Ann. Phys. 315, 505 (2005)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. Poirier, B.: Bohmian mechanics without pilot waves. Chem. Phys. 370, 4 (2010)

    ADS  Article  Google Scholar 

  8. Parlant, G., Ou, Y.C., Park, K., Poirier, B.: ClassicalLike trajectory simulations for accurate computation of quantum reactive scattering probabilities. Comput. Theor. Chem. 990, 3 (2012)

    Article  Google Scholar 

  9. Schiff, J., Poirier, B.: Quantum Mechanics without Wavefunctions. J. Chem. Phys. 136, 031102 (2012)

    ADS  Article  Google Scholar 

  10. Sebens, C.: Quantum mechanics as classical physics. arXiv:1403.0014 [quant-ph]

  11. Hall, M.J.W., Deckert, D.-A., Wiseman, H.M.: Quantum phenomena modelled by interactions between many classical worlds. Phys. Rev. X 4, 041013 (2014). arXiv:1402.6144

  12. Barbour, J., Smolin, L.: Variety, complexity and cosmology. arXiv:hep-th/9203041

  13. Barbour, J.: The deep and suggestive principles of Leibnizian philosophy. Harv. Rev. Philos. 11, 45–58 (2003)

    Article  Google Scholar 

  14. Caticha, A., Dynamics, E.: Time and quantum theory. J. Phys. A 44, 225303 (2011). arXiv:1005.2357

  15. Caticha, A., Bartolomeo, D., Reginatto, M.: Entropic Dynamics: from Entropy and information geometry to hamiltonians and quantum mechanics. arXiv:1412.5629

  16. Frieden, B.R., Soffer, B.H.: Lagrangians of physics and the game of Fisher-information transfer. Phys. Rev. E 52, 2274 (1995)

    ADS  Article  Google Scholar 

  17. Leibniz, G.W.: The monadology (1698), translated by Robert Latta.

  18. Leibniz, G.W., Woolhouse, R.S., Francks, R.: Oxford Philosophical Texts. Oxford University Press, Oxford (1999)

    Google Scholar 

  19. Alexander, H.G.: The Leibniz-Clarke Correspondence, Manchester University Press (1956), for an annotated selection, see

  20. Barbour, J.: Leibnizian time, machian dynamics, and quantum gravity. In: Oxford 1984, Proceedings, Quantum Concepts In Space and Time, pp. 236–246

  21. Stachel, J.: Einsteins search for general covariance, 1912–15. In: Howard, D., Stachel, J. (eds.) Einstein and the History of General Relativity vol 1 of Einstein Studies. Birkhauser, Boston (1989)

    Google Scholar 

  22. Mach, E.: The Science of Mechanics. Open Court, Chicago (1893)

    MATH  Google Scholar 

  23. Barbour, J.: Relative or absolute motion: the discovery of dynamics, CUP (1989)

  24. Barbour, J.: The Discovery of Dynamics: A Study from a Machian Point of View of the Discovery and the Structure of Dynamical Theories. Oxford University Press, Oxford (2001)

    Google Scholar 

  25. Smolin, L.: Space and time in the quantum universe. In: Ashtekar, A., Stachel, J. (eds.) Conceptual Problems in Quantum Gravity, Gravity edn. Birkhauser, Boston (1991)

    Google Scholar 

  26. Kuchar, K.: Dynamics of tensor fields in hyperspace. Iii. J. Math. Phys. 17, 792 (1976)

    ADS  MathSciNet  Article  Google Scholar 

  27. Kuchar, K.: Conditional symmetries in parametrized field theories. J. Math. Phys. 23, 1647 (1982)

    ADS  MathSciNet  Article  Google Scholar 

  28. Bombelli, L., Lee, J., Meyer, D., Sorkin, R.D.: Spacetime as a causal set. Phys. Rev. Lett. 59, 521–524 (1987)

    ADS  MathSciNet  Article  Google Scholar 

  29. Smolin, L.: The Life of the Cosmos. Oxford University Press (in the USA), Weidenfeldand Nicolson (in the United Kingdom) (1997)

  30. Smolin, L.: Did the universe evolve? Class. Quantum Gravity 9, 173–191 (1992)

    ADS  MathSciNet  Article  Google Scholar 

  31. Peirce, C.S.: The architecture of theories. Monist I(2), 161–176 (1891)

    Article  Google Scholar 

  32. Markopoulou, F., Smolin, L.: Disordered locality in loop quantum gravity states. Class. Quantum Gravity 24, 3813–3824 (2007). arXiv:gr-qc/0702044

  33. Markopoulou, F., Smolin, L.: Quantum theory from quantum gravity. arXiv:gr-qc/0311059

  34. Amelino-Camelia, G., Freidel, L., Kowalski-Glikman, J., Smolin, L.: The principle of relative locality. Phys. Rev. D 84, 084010 (2011). arXiv:1101.0931 [hep-th]

  35. Cortês, M., Smolin, L.: The universe as a process of unique Events. Phys. Rev. D 90, 084007 (2014). arXiv:1307.6167 [gr-qc]

  36. Furey, C.: Notes on algebraic causal sets, unpublished notes (2011)

  37. Furey, C.: Cambridge Part III research essay (2006)

  38. Cortês, M., Smolin, L.: Quantum energetic causal sets. Phys. Rev. D 90, 044035 (2014). doi:10.1103/PhysRevD.90.044035

  39. Takabayashi, T.: Prog. Theor. Phys. 8, 143 (1952)

  40. Wallstrom, T.C.: Inequivalence between the Schrodinger equation and the Madelung hydrodynamic equations. Phys. Rev. A 49, 1613–1617 (1994)

    ADS  MathSciNet  Article  Google Scholar 

  41. Takabayashi, T.: Prog. Theor. Phys. 8, 143 (1953)

  42. Smolin, L.: Quantum fluctuations and inertia. Phys. Lett. 113A, 408 (1986)

    ADS  MathSciNet  Article  Google Scholar 

  43. Valentini, A.: Signal-locality in hidden-variables theories. Phys. Lett. A 297, 273–278 (2002) arXiv:quant-ph/0106098

  44. Gomes, H., Gryb, S., Koslowski, T.: Einstein gravity as a 3D conformally invariant theory. Class. Quantum Gravity 28, 045005 (2011). arXiv:1010.2481

  45. Barbour, J.: Shape dynamics. An Introduction. arXiv:1105.0183

Download references


It is a pleasure, first of all, to thank Julian Barbour for our collaboration in the invention of the idea of maximal variety [12, 13], and for many years of conversations and friendship since. This work represents a step in a research program which builds on a critique of the role of time in cosmological theories developed with Roberto Mangabeira Unger [1] and explored with Marina Cortes and, most recently Henrique Gomes. This work develops a specific idea that emerged from that critique, which is that ensembles in quantum theory must refer to real systems, that exist somewhere in the universe [4, 5]. I am grateful to Lucien Hardy, Rob Spekkens and Antony Valentini for criticism of my original real ensemble formulation, as well as to Jim Brown, Ariel Caticha, Marina Cortes, Dirk - Andrei Deckert, Michael Friedman, Laurent Freidel, Henrique Gomes, Michael Hall, Marco Masi, Djorje Minic, Wayne Myrvold, John Norton, Antony Valentini and Elie Wolfe for comments on the present draft or related talks.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Lee Smolin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Smolin, L. Quantum Mechanics and the Principle of Maximal Variety. Found Phys 46, 736–758 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Measurement problem
  • Quantum foundations
  • Hidden variables