The Classical Limit of a Physical Theory and the Dimensionality of Space

  • Borivoje Dakić
  • Časlav BruknerEmail author
Part of the Fundamental Theories of Physics book series (FTPH, volume 181)


In the operational approach to general probabilistic theories one distinguishes two spaces, the state space of the “elementary systems” and the physical space in which “laboratory devices” are embedded. Each of those spaces has its own dimension—the minimal number of real parameters (coordinates) needed to specify the state of system or a point within the physical space. Within an operational framework to a physical theory, the two dimensions coincide in a natural way under the following “closeness” requirement: the dynamics of a single elementary system can be generated by the invariant interaction between the system and the “macroscopic transformation device” that itself is described from within the theory in the macroscopic (classical) limit. Quantum mechanics fulfils this requirement since an arbitrary unitary transformation of an elementary system (spin-1/2 or qubit) can be generated by the pairwise invariant interaction between the spin and the constituents of a large coherent state (“classical magnetic field”). Both the spin state space and the “classical field” are then embedded in the Euclidean three-dimensional space. Can we have a general probabilistic theory, other than quantum theory, in which the elementary system (“generalized spin”) and the “classical fields” generating its dynamics are embedded in a higher-dimensional physical space? We show that as long as the interaction is pairwise, this is impossible, and quantum mechanics and the three-dimensional space remain the only solution. However, having multi-particle interactions and a generalized notion of “classical field” may open up such a possibility.


Coherent State Global Parameter Single Spin Trivial Representation Generalize Spin 
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.
    I. Bengtsson, Why is space three-dimensional?,
  2. 2.
    I.M. Freeman, Why is space three-dimensional? Based on W. Büchel: “Warum hat der Raum drei Dimensionen?,” Physikalische Blätter, Vol. 19(12), pp. 547–549 (December 1963). Am. J. Phys. 37, 1222 (1969)Google Scholar
  3. 3.
    P. Ehrenfest, Proc. Amst. Acad. 20, 200 (1917)Google Scholar
  4. 4.
    I.F. Herbut, Majorana mass, time reversal symmetry, and the dimension of space. Phys. Rev. D 87, 085002 (2013)ADSCrossRefGoogle Scholar
  5. 5.
    T. Kaluza, Zum Unitätsproblem in der Physik, Akad. Wiss. Berlin. (Math. Phys.), 966–972 (1921)Google Scholar
  6. 6.
    O. Klein, Quantentheorie und fünfdimensionale Relativitätstheorie. Zeitschrift für Physik A 37(12), 895–906 (1926)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    L. Randal, Warped Passages: Unraveling the Mysteries of the Universe’s Hidden Dimensions (Harper Perennial, New York, 2006)zbMATHGoogle Scholar
  8. 8.
    I. Antoniadis, A possible new dimension at a few TeV. Phys. Lett. B 246, 377–384 (1990)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    N. Arkani-Hamed, S. Dimopoulos, G. Dvali, The Hierarchy problem and new dimensions at a millimeter. Phys. Lett. B 429(3–4), 263–272 (1998)ADSCrossRefGoogle Scholar
  10. 10.
    K. Agashe, A. Pomarol, Focus on extra space dimensions. New J. Phys. 12, 075010 (2010)ADSCrossRefGoogle Scholar
  11. 11.
    J. Barrett, Information processing in general probabilistic theories. Phys. Rev. A. 75, 032304 (2007)ADSCrossRefGoogle Scholar
  12. 12.
    H. Barnum, A. Wilce, Information processing in convex operational theories. Electron. Notes Theor. Comput. Sci. 270(1), 3–15 (2011)CrossRefGoogle Scholar
  13. 13.
    L. Hardy, Quantum theory from five reasonable axioms (2001). arXiv:quant-ph/0101012
  14. 14.
    H. Barnum, J. Barrett, M. Leifer, A. Wilce, A general no-cloning theorem. Phys. Rev. Lett. 99, 240501 (2007)ADSCrossRefGoogle Scholar
  15. 15.
    J.S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics 1, 195–200 (1964); reprinted in J.S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987)Google Scholar
  16. 16.
    S. Popescu, D. Rohrlich, Quantum nonlocality as an axiom. Found. Phys. 24, 379 (1994)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    D.I. Fivel, How interference effects in mixtures determine the rules of quantum mechanics. Phys. Rev. A 59, 2108 (1994)ADSCrossRefGoogle Scholar
  18. 18.
    C.A. Fuchs, Quantum mechanics as quantum information (and only a little more), in Quantum Theory: Reconstruction of Foundations, ed. by A. Khrenikov (Växjo University Press, Växjo, 2002)Google Scholar
  19. 19.
    R. Clifton, J. Bub, H. Halvorson, Characterizing quantum theory in terms of information-theoretic constraints. Found. Phys. 33(11), 1561 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Č. Brukner, A. Zeilinger, Information and fundamental elements of the structure of quantum theory, in Time, Quantum, Information, ed. by L. Castell, O. Ischebeck (Springer, Berlin, 2003)Google Scholar
  21. 21.
    A. Grinbaum, Elements of information-theoretic derivation of the formalism of quantum theory. Int. J. Quant. Inf. 1(3), 289 (2003)CrossRefzbMATHGoogle Scholar
  22. 22.
    G.M. D’Ariano, Operational axioms for quantum mechanics. AIP Conf. Proc. 889, 79–105 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    A. Grinbaum, Reconstruction of quantum theory. Br. J. Philos. Sci. 8, 387 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    P. Goyal, Information-geometric reconstruction of quantum theory. Phys. Rev. A 78, 052120 (2008)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Č. Brukner, A. Zeilinger, Information invariance and quantum probabilities. Found. Phys. 39, 677 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    B. Dakić, Č. Brukner, Quantum theory and beyond: is entanglement special, in Deep Beauty: Understanding the Quantum World through Mathematical Innovation, ed. by H. Halvorson (Cambridge University Press, Cambridge, 2011)Google Scholar
  27. 27.
    L. Masanes, M. Müller, A derivation of quantum theory from physical requirements. New J. Phys. 13, 063001 (2011)ADSCrossRefGoogle Scholar
  28. 28.
    L. Hardy, Reformulating and reconstructing quantum theory (2011). arXiv:1104.2066
  29. 29.
    L. Masanes, M.P. Müller, D.P. Garcia, R. Augusiak, Entangling dynamics beyond quantum theory (2011). arXiv:1111.4060
  30. 30.
    J. Rau, Measurement-based quantum foundations. Found. Phys. 41(3), 380–388 (2011)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    G. Chiribella, G.M. D’Ariano, P. Perinotti, Informational derivation of quantum theory. Phys. Rev. A 84, 012311 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    H. Barnum, Quantum knowledge, quantum belief, quantum reality: notes of a QBist fellow traveler (2010). arXiv:1003.4555v1
  33. 33.
    C.F. von Weizsäcker, in Quantum theory and the structures of time and space, Eds. L. Castell, M. Drieschner, C.F. von Weizsäcker (Hanser, München, 1975). Papers presented at a conference held in Feldafing, July (1974)Google Scholar
  34. 34.
    R. Penrose, Angular momentum: an approach to combinatorial space-time, in Quantum Theory and Beyond, ed. by T. Bastin (Cambridge University Press, Cambridge, 1971)Google Scholar
  35. 35.
    W.K. Wootters, The acquisition of information from quantum measurements, Ph.D. thesis, University of Texas at Austin (1980)Google Scholar
  36. 36.
    A. Einstein, W.J. de Haas, Experimenteller Nachweis des Ampéreschen Molekularströme. Naturwissenschaften 3, 237–238 (1915)ADSCrossRefGoogle Scholar
  37. 37.
    S.J. Barnett, Magnetization by rotation. Phys. Rev. 6, 239–270 (1915)ADSCrossRefGoogle Scholar
  38. 38.
    As noted by A. Peres, in Quantum Theory: Conpcepts and Methods (Kluwer Academic Publishers, 2002): “Even if quantum theory is universal, it is not closed. A distinction must be made between endophysical systems—those which are described by the theory—and exophysical ones, which lie outside the domain of the theory (for example, the telescopes and photographic plates used by astronomers for verifying the laws of celestial mechanics). While quantum theory can in principle describe anything, a quantum description cannot include everything. In every physical situation something must remain unanalyzed.”Google Scholar
  39. 39.
    P.W. Atkins, J.C. Dobson, Angular momentum coherent states. Proc. R. Soc. A 321, 321 (1971)Google Scholar
  40. 40.
    J.M. Radcliffe, Some properties of coherent spin states. J. Phys. A: Gen. Phys. 4, 313 (1971)Google Scholar
  41. 41.
    J. Kofler, Č. Brukner, Classical world arising out of quantum physics under the restriction of coarse-grained measurements. Phys. Rev. Lett. 99, 180403 (2007)ADSCrossRefGoogle Scholar
  42. 42.
    S.D. Bartlett, T. Rudolph, R.W. Spekkens, Reference frames, superselection rules, and quantum information. Rev. Mod. Phys. 79, 555–606 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    M. Dickson, A view from nowhere: quantum reference frames and uncertainty. Stud. Hist. Philos. Mod. Phys. 35, 195–220 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Y. Aharonov, T. Kaufherr, Quantum frames of reference. Phys. Rev. D 30, 368 (1984)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    D. Poulin, J. Yard, Dynamics of a quantum reference frame. New J. Phys. 9, 156 (2007)ADSCrossRefGoogle Scholar
  46. 46.
    D. Poulin, Toy model for a relational formulation of quantum theory (2005). arXiv:0505081v2
  47. 47.
    Č. Brukner, In the Kreisgang between classical and quantum physics, UniMolti modi della filosofia 2008/2, arXiv:0905.3363
  48. 48.
    D.C. Brody, E.M. Graefe, Six-dimensional space-time from quaternionic quantum mechanics. Phys. Rev. D 84, 125016 (2011)ADSCrossRefGoogle Scholar
  49. 49.
    T. Paterek, B. Dakić, Č. Brukner, Theories of systems with limited information content. New J. Phys. 12, 053037 (2010)ADSCrossRefGoogle Scholar
  50. 50.
    G.V. Steeg, S. Wehner, Relaxed uncertainty relations and information processing. Quantum Inf. Comput. 9(9–10), 0801–0832 (2009)MathSciNetzbMATHGoogle Scholar
  51. 51.
    M.P. Müller, L. Masanes, Three-dimensionality of space and the quantum bit: how to derive both from information-theoretic postulates (2012). arXiv:1206.0630
  52. 52.
    H. Araki, On a characterization of the state space of quantum mechanics. Commun. Math. Phys. 75, 1–24 (1980)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    S. Bergia, F. Cannata, A. Cornia, R. Livi, On the actual measurability of the density matrix of a decaying system by means of measurements on the decay products. Found. Phys. 10, 723–730 (1980)ADSCrossRefGoogle Scholar
  54. 54.
    W.K. Wootters, Local accessibility of quantum states, in Complexity, Entropy and the Physics of Information, ed. by W.H. Zurek (Addison-Wesley, Boston, 1990)Google Scholar
  55. 55.
    G. Chiribella, G.M. D’Ariano, P. Perinotti, Informational derivation of quantum theory. Phys. Rev. A 81, 062348 (2010)ADSCrossRefGoogle Scholar
  56. 56.
    L. Hardy, W.K. Wootters, Limited Holism and real-vector-space quantum theory (2010). arXiv:1005.4870
  57. 57.
    E.C.G. Stueckelberg, Quantum theory in real hilbert space. Helv. Phys. Acta 33, 727–752 (1960)MathSciNetzbMATHGoogle Scholar
  58. 58.
    M. Pawlowski, A. Winter, Hyperbits: the information quasiparticles. Phys. Rev. A 85, 022331 (2012)ADSCrossRefGoogle Scholar
  59. 59.
    D. Montgomery, H. Samelson, Transformation groups of spheres. Ann. Math. 44, 454–470 (1943)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    A. Borel, Some remarks about Lie groups transitive on spheres and tori. Bull. A.M.S. 55, 580–587 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    R.E. Behrends, J. Dreitlein, C. Fronsdal, W. Lee, Simple groups and strong interaction symmetries. Rev. Mod. Phys. 34, 1–40 (1962)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    J.C. Baez, The octonions. Bull. Am. Math. Soc. 39, 145–205 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    N.W. Ashcroft, N.D. Mermin, Solid State Physics (Harcourt College Publishers, San Diego, 1976)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Vienna Center for Quantum Science and Technology (VCQ), Faculty of PhysicsUniversity of ViennaViennaAustria
  2. 2.Centre for Quantum TechnologiesNational University of SingaporeSingaporeSingapore
  3. 3.Institute of Quantum Optics and Quantum Information (IQOQI), Austrian Academy of SciencesViennaAustria

Personalised recommendations