Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
Bell's Theorem and Quantum Realism

Part of the book series: SpringerBriefs in Physics ((SpringerBriefs in Physics))

  • 1273 Accesses

Abstract

Whatever position one takes on the subject, quantum theory is certainly surprising in its radical break from other fields of physics. Not only does it exhibit indeterminism, but the theory entails an essential denial of objectivity, an abandonment of realism. The latter issue stems from the fact that quantum theory offers little description of physical systems apart from what takes place during measurement processes.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Even the description of measurement lacks clarity, as it depends upon vaguely defined notions of ‘system’ and ‘apparatus.’ We submit that as theoretical physicists, it is our duty to seek out sharp, objective definitions of all physical concepts that come into play.

  2. 2.

    Some may regard Einstein’s position as being summed up by the quotation “God does not play dice.” However, this statement was made quite early, and he later became concerned with other difficulties of the theory. In a letter (see [2, p. 221]) to Max Born, Pauli writes: “ \(\ldots\) I was unable to recognize Einstein whenever you talked about him either in your letter or your manuscript. It seemed to me as if you had erected some dummy Einstein for yourself, which you then knocked down with great pomp. In particular Einstein does not consider the concept of ‘determinism’ to be as fundamental as it is frequently held to be (as he told me emphatically many times) \(\ldots\) he disputes that he uses as a criterion for the admissibility of a theory the question ‘Is it rigorously deterministic?’ ”

  3. 3.

    The page numbers given in citations of this work always refer to the English translation. Chaps. 4 and 6 of the book are reprinted in Wheeler and Zurek’s collection [Wheeler, J.A., Zurek, W.H. (eds.) Quantum Theory and Measurement. Princeton University Press, Princeton (1983)] on pages 549–647, although the theorem of von Neumann is not.

  4. 4.

    As will be discussed in Chap. 2, Kochen and Specker’s theorem is closely related to yet another mathematical result that relates to the quantum observables in a similar way. This is Gleason’s theorem [5].

  5. 5.

    This work is based on a doctoral dissertation by Douglas Hemmick. See [10].

  6. 6.

    See Bell [11, 12], Dürr et al. [13, (Sect. 8)], Maudlin [14, 15], Norsen [16–18] and Wiseman [19].

  7. 7.

    For the view that the theorem does not constitute a proof of nonlocality see Jarrett [20] and also Evans et al. [21]. Some proceed by citing “counterfactual definiteness,” as an attack on the premises that go into the EPR analysis. This position is set forth for example, by Stapp [22] and by Redhead [23]. For more on counterfactuals, see Maudlin [15]. Another issue is what we call the possibility of a “superdeterministic” interpretation (see Bell [24]). This will be discussed in Sect. 3.4.3. One of the most complete sources on all issues surrounding quantum nonlocality is the book by Maudlin [14].

  8. 8.

    Please see Schrödinger in [25–27].

  9. 9.

    Readers should also have been reassured that none of the other “no-hidden-variables theorems” imply any true constraint upon realistic quantum as such, but serve only to draw attention to features of quantum theory itself.

  10. 10.

    See also Bassi and Ghiradi [28], Tumulka [29] and also Goldstein et al. in [30].

  11. 11.

    See [31–33]. A summary of the early development of the theory is given by de Broglie [34].

  12. 12.

    See [35]. See also Bell [36, 37] Books on Bohmian mechanics include Holland’s 1993 work [38] also Dürr and Teufel’s more recent effort published in 2009 [39]. A book which addresses the theory and its general relationship to foundations of quantum mechanics is the 1996 Cushing et al. [40].

  13. 13.

    This is the motivation mentioned by von Neumann in his no-hidden variables proof. See Ref. [4].

  14. 14.

    For his part, Bell was also intrigued by the theory put forth by Ghirardi et al. [42]. See Bell [43, 44].

  15. 15.

    Contextuality, as we will see, is associated with the fact that measurement procedures for each observable are often not unique. Hidden variable theories must take account of the possibility that different measurements are possible for a single observable, and so we must not expect a “one-to-one” mapping from observables to values.

  16. 16.

    To be precise, von Neumann’s theorem falls into a different category than the others. However, examining this classic argument offers an excellent introduction which will be quite useful when we proceed to the more advanced theorems in Chap. 2.

  17. 17.

    This discussion allows us to understand contextual and noncontextual hidden variables. This will be of great utility not only in making clear the Einstein–Podolsky–Rosen paradox and Bell’s Theorem, but we will also find these concepts playing an important role in the discussion of the nonlocality theorems arising from the Schrödinger paradox.

  18. 18.

    See also Ghirardi [46, pp. 213–217].

  19. 19.

    A work by Daumer et al. [47] argues for the same conclusion.

  20. 20.

    See Greenberger et al. [49], and also Mermin [50].

  21. 21.

    Arguments falling into the same category as the Schrödinger proofs have also been found by Brown and Svetlichny [51] and by Heywood and Redhead [52]. More recent such results were found by Aravind [53], and by Cabello [54].

  22. 22.

    It is the existence of such rules in the quantum mechanical formalism that marks its departure from an objective theory, as we discussed above.

  23. 23.

    In the absence of spin.

  24. 24.

    We ignore here the possibility of an external magnetic field.

  25. 25.

    In the following description, we use the terms observable and operator interchangeably.

  26. 26.

    A measurement may be classified either as ideal or non-ideal. Unless specifically stated, it will be assumed whenever we refer to a measurement process, what is said will apply just as well to either of these situations.

  27. 27.

    See the discussion of Bohmian mechanics above. The success of this theory is that it explains quantum phenomena without such features. The general issue of hidden variables is discussed in Bell [36, 55, 56]. Many of the works by Bell which are of concern to us may be found in [Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, Cambridge (1987)]. See also [Bell, M., Gottfried, K., Veltman, M., John S. Bell on the Foundations of Quantum Mechanics. World Scientific Publishing Company, Singapore (2001)] and [Bell, M., Gottfried, K., Veltman, M., (eds.) Quantum Mechanics, High Energy Physics and Accelerators: Selected Papers of John S. Bell (with commentary). World Scientific Publishing Company, Singapore (1995)]. The latter two are complete collections containing all of Bell’s papers on quantum foundations. See also Bohm [35, 58], Belinfante [59], Hughes [60] and Jammer [61].

  28. 28.

    The original work is [4]. Discussions of von Neumann’s hidden variables analysis may be found within [36, 55, 62, 63]

  29. 29.

    Within Chap. 4 of the present work, we shall discuss a 1935 analysis by Erwin Schrödinger [25]. This is the paper in which the ‘Schrödinger’s cat’ paradox first appeared, but it contains also other significant results such as Schrödinger’s generalization of the Einstein–Podolsky–Rosen paradox.

  30. 30.

    See Jammer’s book [61, p. 272] for a discussion of an early work by Grete Hermann that addresses the impact of von Neumann’s Theorem.

  31. 31.

    When generalized to the case of a mixed state, this becomes

    $$ E(O)=\hbox{Tr}(UO) $$
    (1.16)

    where U is a positive operator with the property \(\hbox{Tr}(U)=1.\) Here U is known as the “density matrix”. See for example, [65, p. 378]

  32. 32.

    Of course, this raises the question of whether it is possible to prepare an ensemble of states with fixed \(\lambda.\) Whether the answer to this question is yes or no, the existence of dispersion free states still implies that it is possible to write a value map function \(V^{\psi}_{\lambda}(O)\) on the observables. For fixed \(\psi,\) it is the variation in \(\lambda\) which is considered responsible for the variation in results of measurement. If, on the other hand, we fix both \(\psi\) and \(\lambda\) then we are dealing with just one member of the ensemble and there should be just a single value assigned to each observable. This is the meaning of the value map E(O) in the case of dispersion free states.

  33. 33.

    A proof of the theorem may be found in von Neumann’s original work [4]. Albertson presented a simplification of this proof in 1961 [62]. What we present here is a further simplification.

  34. 34.

    The equality \(\hbox{Tr}(UP_{\chi})= \langle \chi | U | \chi \rangle\) in (1.28) is seen as follows. The expression \(\hbox{Tr}(UP_{\chi})\) is independent of the orthonormal basis \(\phi_{n}\) in terms of which the matrix representations of U and \(P_{\chi}\) are expressed, so that one may choose an orthonormal basis of which \(| \chi\rangle\) itself is a member. Since \(P_{\chi} = | \chi \rangle \langle \chi |,\) and \(P_{\chi} | \phi_{n} \rangle = 0\) for all \(| \phi_{n} \rangle\) except \(| \chi \rangle,\) we have \(\hbox{Tr}(UP_{\chi})=\langle \chi | U | \chi \rangle.\)

  35. 35.

    It should be noted that the same result may be proven without use of (1.30) since the fact that \(V^{\psi}_{\lambda}(P_{\phi})\) must be either 0 or 1 follows simply from the observation that these are the eigenvalues of \(P_{\phi}.\)

  36. 36.

    It is not difficult to show that \(\sigma^{\prime}\) defined in this way is the spin component along an axis which is in the x, y plane and lies at \(45^{\circ}\) from both the x and y axis.

  37. 37.

    Reference to such a remark by Einstein is also found in Gilder’s recent book. See [67, pp. 160–161].

  38. 38.

    In fact, if Schrödinger had interpreted his result this way, this—in light of his own generalization of the EPR paradox presented in the same paper–would have allowed him to reach a further and very striking conclusion. We shall discuss this in Chap. 4.

  39. 39.

    See Max Jammer’s book [61, p. 265].

References

  1. Bell, J.S.: The theory of local beables. Epistemol. Lett. 9 (1976). Reprinted in [68].

    Google Scholar 

  2. Born, M. (eds.): The Born–Einstein Letters. Macmillan, London (1971)

    Google Scholar 

  3. Schilpp, P.A.: Albert Einstein: Philosopher-Scientist. Harper and Row, New York (1949)

    Google Scholar 

  4. von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932) English translation: Beyer, R.: Mathematical Foundations of Quantum Mechanics, pp. 305–324 (trans: Princeton University Press, Princeton) (1955)

    Google Scholar 

  5. Gleason, A.M.: J. Math. Mech. 6, 885 (1957)

    Google Scholar 

  6. Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59 (1967)

    Google Scholar 

  7. Bell, J.S.: On the Einstein Podolsky Rosen paradox. Physics 1, 195–200 (1964). Reprinted in [57], p. 14 and in [69], p. 403

    Google Scholar 

  8. Conway, J., Kochen, S.: The free will theorem. Found. Phys. 36(10), 1441–1473 (2006). Physics archives arXiv:quantph/0604079

    Google Scholar 

  9. Conway, J., Kochen, S.: The strong free will theorem. Notices AMS 56(2), 226–232 (2009)

    Google Scholar 

  10. Hemmick, D.: Doctoral dissertation, Rutgers University, Department of Physics (1996). Research supervised by Goldstein, S., Lebowitz, J. See arXiv:quant-ph/0412011v1

    Google Scholar 

  11. Bell, J.S.: Atomic cascade photons and quantum mechanical nonlocality. Comm. At. Mol. Phys. 9, 121–126 (1980). Reprinted in [68], p. 782

    Google Scholar 

  12. Bell, J.S.: Bertlemann’s socks and the nature of reality. J. de Physique Colloque C2. Tome 42 (suppl. au numero 3) C2 41–C2 61 (1981). Reprinted in [57], p. 139

    Google Scholar 

  13. Dürr, D., Goldstein, S., Zanghí, N.: Quantum equilibrium and the role of operators as observables in quantum theory. J. Stat. Phys. 116, 959–1055 (2004)

    Google Scholar 

  14. Maudlin, T.: Quantum Non-locality and Relativity: Metaphysical Intimations of Modern Physics, 3rd edn. Wiley-Blackwell, Oxford (2011)

    Google Scholar 

  15. Maudlin, T.: Space-time in the quantum world. In: Fine, A., Goldstein, S., Cushing, J. (eds.) Bohmian Mechanics and Quantum Theory: An Appraisal, pp. 285–307. Kluwer, Dordrecht (1996)

    Google Scholar 

  16. Norsen, T.: Found. Phys. 37, 311–340 (2007)

    Google Scholar 

  17. Norsen, T.: Bell locality and the nonlocal character of nature. Found. Phys. Lett. 19(7), 633–655 (2006)

    Google Scholar 

  18. Norsen, T.: Local causality and completeness: Bell vs. Jarrett. Found. Phys. 39(3), 273 (2009)

    Google Scholar 

  19. Wiseman, H.M.: From Einstein’s theorem to Bell’s theorem: A history of quantum nonlocality. Contemp. Phys. 47, 79–88 (2006)

    Google Scholar 

  20. Jarrett, J.: On the physical significance of the locality conditions in the bell argument. Nous 18, 569–589 (1984)

    Google Scholar 

  21. Evans, P., Price, H., Wharton, K.B.: New slant on the EPR-Bell experiment. The Physics archives arXiv:1001.5057v3 [quant-ph]

    Google Scholar 

  22. Stapp, H.: Quantum Nonlocality and the Description of Nature. In: Cushing J., McMullin, E. (eds.) Philosophical Consequences of Quantum Theory: Reflections on Bell’s Theorem, pp.154–174. University of Notre Dame Press, Notre Dame

    Google Scholar 

  23. Redhead, M.: Incompletness, Nonlocality and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics. Clarendon Press, Oxford (1987)

    Google Scholar 

  24. Davies, P.C.W., Brown, J.R.: John bell (interview) p. 45. In: The Ghost in the Atom. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  25. Schrödinger, E.: Naturwissenschaften 23, 807–812, 823–828, 844–849 (1935). English translation: Schrödinger, E.: The present situation in quantum mechanics. Proc. Cambridge Phil. Soc. 124, 323–338 (1980) (trans: Drimmer, J.) and can also be found in [69]

    Google Scholar 

  26. Schrödinger, E.: Discussion of probability relations between separated systems. Proc. Cambridge Phil. Soc. 31, 555 (1935)

    Google Scholar 

  27. Schrödinger, E.: Probability relations between separated systems. Proc. Cambridge Phil. Soc. 32, 446 (1936)

    Google Scholar 

  28. Bassi, A., Ghirardi, G.C.: The Conway-Kochen argument and relativistic GRW Models. Found. Phys. 37, 169 (2007). Physics archives: arXiv:quantph/0610209.

    Google Scholar 

  29. Tumulka, R.: Comment on ‘The Free will Theorem’. Found. Phys. 37, 186–197 (2007). Physics Archives arXiv:quant-ph/0611283.

    Google Scholar 

  30. Goldstein, S., Tausk, D., Tumulka, R., Zanghí, N.: What does the free will theorem actually prove? Notices of the AMS 57(11), 1451–1453, arXiv:0905.4641v1 [quant-ph] (2010)

    Google Scholar 

  31. de Broglie, L.: Sur la possibilite de relier les phenomenes d’interference et de diffraction a la theorie des quanta de lumiere. Compt. Rend. 183, 447–448 (1926)

    Google Scholar 

  32. de Broglie, L.: Sur la possibilite de mettre en accord la theorie electomagnetique avec la nouvelle. Compt. Rend. 185, 380–382 (1927)

    Google Scholar 

  33. de Broglie, L.: Rapport au V’ieme Congres de Physique Solvay Gauthier–Villars, Paris (1930)

    Google Scholar 

  34. de Broglie, L.: Physicien et Penseur, p. 465, Gauthier-Villars, Paris (1953).

    Google Scholar 

  35. Bohm, D.: A suggested interpretation of the quantum theory in terms of “Hidden” variables. Phys. Rev. 85, 166, 180 (1952)

    Google Scholar 

  36. Bell, J.S.: On the impossible pilot wave. Found. Phys. 12, 989–999 (1982). Reprinted in [57], p. 159

    Google Scholar 

  37. Bell, J.S.: Six possible worlds of quantum mechanics. Proceedings of the Nobel Symposium 65: Possible Worlds in Arts and Sciences. Stockholm, 11–15 August 1986. Reprinted in [57], p. 181

    Google Scholar 

  38. Holland, P.R.: The Quantum Theory of Motion: An account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics. Cambridge University Press, Cambridge (1993). Reprinted in paperback 1995

    Google Scholar 

  39. Dürr, D., Teufel, S.: Bohmian Mechanics: The Physics and Mathematics of Quantum Theory. Springer, Berlin (2009). Reprinted 2010 in softcover

    Google Scholar 

  40. Cushing, J.T., Fine, A., Goldstein, S. (eds.): Bohmian Mechanics and Quantum Theory: An Appraisal. Springer, Berlin (1996). Reprinted in softcover 2010

    Google Scholar 

  41. Bell, J.S.: De Broglie-Bohm, delayed choice double slit experiment, and density matrix. Int. J. Quant. Chem. Quantum Chemistry Symposium. 14, 155–159 (1980). Reprinted in [57], p. 111

    Google Scholar 

  42. Ghirardi, G.C, Rimini, A., Weber, T.: Unified dynamics for microscopic and macroscopic systems. Phys. Rev. D 34, 470–491 (1986)

    Google Scholar 

  43. Bell, J.S.: Against ‘Measurement’. Phys. World 3, 33 (1990). This is reprinted in [68], p. 902

    Google Scholar 

  44. Bell, J.S.: Are there quantum jumps? Schrödinger, Centenary of a Polymath. Cambridge University Press, Cambridge (1987). Reprinted in [68], p. 866

    Google Scholar 

  45. Albert, David: Quantum Mechanics and Experience. Harvard University Press, Cambridge (1992)

    Google Scholar 

  46. Ghirardi, G.C.: Sneaking a Look at God’s Cards Princeton. University Press, Princeton (2005)

    Google Scholar 

  47. Daumer, M., Dürr, D., Goldstein, S., Zanghí, N.: Naive realism about operators. In: Costantini, D., Gallavotti, M.C. (eds.) Erkenntnis, 1996, Special Issue in Honor of Prof. R. Jeffrey, Proceedings of the International Conference “Probability, Dynamics and Causality", Luino, Italy 15–17 June 1995.

    Google Scholar 

  48. Greenberger, D.M., Horne, M.A., Zeilinger, A. Going beyond Bell’s theorem. in [70], p. 69

    Google Scholar 

  49. Greenberger, D.M., Horne, M.A., Shimony, A., Zeilinger, A.: Bell’s theorem without inequalities. Am. J. Phys. 58, 1131–1143 (1990)

    Google Scholar 

  50. Mermin, N.D.: What’s wrong with these elements of reality? Phys. Today 43, 9 (1990)

    Google Scholar 

  51. Brown, H.R., Svetlichny, G.: Nonlocality and Gleason’s lemma. Part I. Deterministic theories. Found. Phys. 20, 1379 (1990)

    Google Scholar 

  52. Heywood, P., Redhead, M.L.G.: Nonlocality and the Kochen-Specker paradox. Found. Phys. 13, 481 (1983)

    Google Scholar 

  53. Aravind, P.K.: Bell’s theorem without inequalities and only two distant observers. Found. Phys. Lett. 15, 399–405 (2002)

    Google Scholar 

  54. Cabello, A.: Bell’s theorem without inequalities and without probabilities for two observers. Phys. Rev. Lett. 86, 1911–1914 (2001)

    Google Scholar 

  55. Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447–452 (1966). Reprinted in [57], p. 1 and [69]

    Google Scholar 

  56. Bell, J.S.: Introduction to the hidden-variable question. In: Foundations of Quantum Mechanics. Proceedings of the International School of Physics ‘Enrico Fermi’ course IL, New York, Academic, pp. 171–181 (1971). This article is reprinted in [57], p. 29

    Google Scholar 

  57. Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987)

    Google Scholar 

  58. Bohm, D., Hiley, B.J.: The Undivided Universe: An Ontological Interpretation of Quantum Theory. Routledge, New York (1993)

    Google Scholar 

  59. Belinfante, F.J.: A Survey of Hidden-Variables Theories. Pergamon Press, New York (1973)

    Google Scholar 

  60. Hughes, R.I.G.: The Structure and Interpretation of Quantum Mechanics. Harvard University Press, Cambridge (1989)

    Google Scholar 

  61. Jammer, M.: The Philosophy of Quantum Mechanics. Wiley, New York (1974)

    Google Scholar 

  62. Albertson, J.: von Neumann’s hidden - parameter proof. Am. J. Phys. 29, 478 (1961)

    Google Scholar 

  63. Jauch, J.M., Piron, C.: Can hidden variables be excluded in quantum mechanics? Helv. Phys. Acta 36, 827 (1963)

    Google Scholar 

  64. Einstein A., Podolsky B., Rosen N.: Can quantum mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935). Reprinted in [69], p. 138

    Google Scholar 

  65. Schiff, L.: Quantum Mechanics. McGraw-Hill, New York (1955)

    Google Scholar 

  66. Shimony, A.: Search for a Naturalistic World View, vol. 2, p. 89. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  67. Gilder, L.: The Age of Entanglement: When Quantum Physics Was Reborn. Alfred A Knopf, New York (2008)

    Google Scholar 

  68. Bell, M., Gottfried, K., Veltman, M., (eds.): Quantum Mechanics, High Energy Physics and Accelerators: Selected Papers of John S. Bell (with commentary). World Scientific Publishing Company, Singapore (1995), p. 744

    Google Scholar 

  69. Wheeler, J.A., Zurek, W.H. (eds.): Quantum Theory and Measurement. Princeton University Press, Princeton (1983), p. 397

    Google Scholar 

  70. Kafatos, M. (ed.): Bell’s Theorem: Quantum Theory and Conceptions of the Universe. Kluwer Academic, Dordrecht (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Boston Drive, Berlin, MD, 6621811, USA

    Douglas L. Hemmick

  2. Department of Physics, Salisbury University, Camden Avenue 1101, Salisbury, MD, 21801, USA

    Asif M. Shakur

Authors
  1. Douglas L. Hemmick
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Asif M. Shakur
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Douglas L. Hemmick .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 The Author(s)

About this chapter

Cite this chapter

Hemmick, D.L., Shakur, A.M. (2012). Introduction. In: Bell's Theorem and Quantum Realism. SpringerBriefs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23468-2_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-23468-2_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23467-5

  • Online ISBN: 978-3-642-23468-2

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation