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Quantum Studies: Mathematics and Foundations

, Volume 6, Issue 4, pp 481–517 | Cite as

On classical systems and measurements in quantum mechanics

  • Erik DeumensEmail author
Regular Paper
  • 99 Downloads

Abstract

The recent rigorous derivation of the Born rule from the dynamical law of quantum mechanics Allahverdyan et al. (Phys Rep 525:1–166.  https://doi.org/10.1016/j.physrep.2012.11.001, 2013) is taken as incentive to reexamine whether quantum mechanics has to be an inherently probabilistic theory. It is shown, as an existence proof, that an alternative perspective on quantum mechanics is possible where the fundamental ontological element, the ket, is not probabilistic in nature and in which the Born rule can also be derived from the dynamics. The probabilistic phenomenology of quantum mechanics follows from a new definition of statistical state in the form of a probability measure on the Hilbert space of kets that is a replacement for the von Neumann statistical operator to address the lack of uniqueness in recovering the pure states included in mixed states, as was pointed out by Schrödinger. From the statistical state of a quantum system, classical variables are defined as collective variables with negligible dispersion. In this framework, classical variables can be chosen to define a derived classical system that obeys, by Ehrenfest’s theorem, the laws of classical mechanics and that describes the macroscopic behavior of the quantum system. The Born rule is derived from the dynamics of the statistical state of the quantum system composed of the observed system interacting with the measurement system and the role of the derived classical system in the process is exhibited. The approach suggests to formulate physical systems in second quantization in terms of local quantum fields to ensure conceptually equivalent treatment of space and time. A real double-slit experiment, as opposed to a thought experiment, is studied in detail to illustrate the measurement process.

Keywords

Quantum mechanics Measurement theory Quantum statistical mechanics 

Mathematics Subject Classification

81P15 28C20 82C10 60G15 

Notes

Acknowledgements

My deepest appreciation goes to the anonymous reviewers for their thorough analysis of the paper and numerous substantive comments and suggestions that resulted in a very much improved end result. My thanks go to Henk Monkhorst and several anonymous referees for their valuable comments and suggestions on earlier expositions of the ideas presented in this paper.

Supplementary material

References

  1. 1.
    Aharonov, Y., Albert, D.Z.: States and observables in quantum field theories. Phys. Rev. D 21(12), 3316–3324 (1980)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aharonov, Y., Albert, D.Z.: Can we make sense out of the measurement process in relativistic quantum mechanics? Phys. Rev. D 24(2), 359–370 (1981)CrossRefGoogle Scholar
  3. 3.
    Allahverdyan, A.E., Balian, R., Nieuwenhuizen, T.M.: Understanding quantum measurement from the solution of dynamical model. Phys. Rep. 525(1), 1–166 (2013).  https://doi.org/10.1016/j.physrep.2012.11.001 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Allahverdyan, A.E., Balian, R., Nieuwenhuizen, T.M.: A sub-ensemble theory of ideal quantum measurement processes. Ann. Phys. 376, 324–352 (2017).  https://doi.org/10.1016/j.aop.2016.11.001 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Araki, H.: Mathematical Theory of Quantum Fields. No. 101 in International Series of Monographs on Physics. Oxford University Press, Oxford: Japanese Edition: Ryoshiba no Suri, p. 1993. Iwanami Shoten Publishers, Tokyo (1999)Google Scholar
  6. 6.
    ATLAS Collaboration: Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B 716, 1–29 (2012)Google Scholar
  7. 7.
    Ballentine, L.E.: The Statistical Interpretation of Quantum Mechanics. Rev. Mod. Phys. 42(4), 358–381 (1970)CrossRefGoogle Scholar
  8. 8.
    Bayfield, J.E.: Quantum Evolution: An Introduction to Time-dependent Quantum Mechanics. Wiley, New York (1999)Google Scholar
  9. 9.
    Beard, A., Fong, R.: No-interaction theorem in classical relativistic mechanics. Phys. Rev. 182(5), 1397–1399 (1969).  https://doi.org/10.1103/PhysRev.182.1397 CrossRefGoogle Scholar
  10. 10.
    Bell, J.: On the Einstein–Podolsky–Rosen paradox. Physics 1, 195–200 (1964)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bogachev, V.I.: Guassian Measures, Mathematical Surveys and Monographs, vol. 62. American Mathematical Society, Providence (1998)Google Scholar
  12. 12.
    Bogolubov, N.N., Bogolubov Jr., N.N.: Introduction to Quantum Statistical Mechanics, 2nd edn. World Scientific, Singapore (2010)zbMATHGoogle Scholar
  13. 13.
    Bogolubov, N.N., Logunov, A.A., Todorov, I.T.: Introduction to Axiomatic Quantum Field Theory. Advanced Book Program. W.A. Benjamin, Reading (1975)Google Scholar
  14. 14.
    Bohr, N.: On the Constitution of Atoms and Molecules Part I. Philos. Mag. Ser. 6 26(151), 1–25 (1913)Google Scholar
  15. 15.
    Bohr, N.: On the constitution of atoms and molecules. Part II systems containing only a single nucleus. Philos. Mag. Ser. 6 26(153), 476–502 (1913)Google Scholar
  16. 16.
    Bohr, N.: On the constitution of atoms and molecules. Part III systems containing several nuclei. Philos. Mag. Ser. 6 26(155), 857–875 (1913)Google Scholar
  17. 17.
    Bohr, N.: The Quantum Postulate and the Recent Development of Atomic Theory. Nature pp. 580–590 (1928). Content of lecture delivered on Sept. 16, 1927 at the Volta celebration in ComoGoogle Scholar
  18. 18.
    Born, M.: Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik 37, 863–867 (1926)CrossRefGoogle Scholar
  19. 19.
    Born, M.: Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik 38, 803–827 (1926)CrossRefGoogle Scholar
  20. 20.
    Born, M.: The Mechanics of the Atom. International Text Books of Exact Science. G. Bell and Sons, Ltd., London (1927). German original “Vorlesungen Über Atommechanik”, Springer (1925). English edition reproduced by ULAN PressGoogle Scholar
  21. 21.
    Born, M., Heisenberg, W., Jordan, P.: On Quantum Mechanics II. In: van der Waerden [89], pp. 95–137. Original: Zeitschrift für Physik 35 (1926) 557–615Google Scholar
  22. 22.
    Busch, P., Grabowski, M., Lahti, P.J.: Operational Quantum Physics, Lecture Notes in Physics, vol. m31. Springer, Berlin (1995)CrossRefGoogle Scholar
  23. 23.
    Busch, P., Lahti, P.J., Mittelsteadt, P.: The Quantum Theory of Measurement, Lecture Notes in Physics, vol. m2. Sprinter, Berlin (1991)CrossRefGoogle Scholar
  24. 24.
    Chaichian, M., Demichev, A.: Path Integrals in Physics Volume I: Stochastic Processes and Quantum Mechanics, Series in Mathematical and Computational Physics, vol. I. Institute of Physics Publishing, Bristol (2001)Google Scholar
  25. 25.
    CMS Collaboration: Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys. Lett. B 817, 30–61 (2012)Google Scholar
  26. 26.
    Currie, D.G.: Interaction contra classical relativistic Hamiltonian particle mechanics. J. Math. Phys 4(12), 1470–1488 (1963).  https://doi.org/10.1063/1.1703928 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    d’ Espagnat, B.: Conceptual Foundations of Quantum Mechanics, 2nd edn., Advanced Book Classics. Addison-Wesley Publishing Company, Reading (1976). Reprinted in 1989Google Scholar
  28. 28.
    d’ Espagnat, B.: Veiled Reality An Analysis of Present-Day Quantum Mechanical Concepts. Frontiers in Physics. Addison-Wesley Publishing Company, Reading (1995)Google Scholar
  29. 29.
    d’ Espagnat, B.: On Physics and Philosophy. Princeton University Press, Princeton (2006)Google Scholar
  30. 30.
    De Raedt, H., Michielsen, K., Hess, K.: Irrelevance of Bell’s theorem for experiments involving correlation in space and time: a specific loophole-free computer-example. ArXiv arxiv:1605.0537v1 (5 pages) (2016)
  31. 31.
    Dimock, J.: Quantum Mechanics and Quantum Field Theory: A Mathematical Primer. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
  32. 32.
    Dirac, P.A.M.: The Fundamental Equations of Quantum Mechanics. In: van der Waerden [89], pp. 307–320. Original: Proc. Roy. Soc. Lond. A 109, 642–653 (1925)Google Scholar
  33. 33.
    Dirac, P.A.M.: The principles of quantum mechanics, 1958 4th edn., No. 27 . In: International Series of Monographs on Physics. Clarendon Press, Oxford (1930)Google Scholar
  34. 34.
    Ehrenfest, P.: Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Zeitschrift für Physik 45(7–8), 455–457 (1927)CrossRefGoogle Scholar
  35. 35.
    Einstein, A.: Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik 17, 132–148 (1905).  https://doi.org/10.1002/andp.19053220607 CrossRefzbMATHGoogle Scholar
  36. 36.
    Einstein, A.: Quanten-Mechanik und Wirklichkeit. Dialectica 2(3-4), 320–324 (1948).  https://doi.org/10.1111/j.1746-8361.1948.tb00704.x. Translated: Quantum Mechanics and Reality
  37. 37.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)CrossRefGoogle Scholar
  38. 38.
    Elze, H.T.: Linear dynamics of quantum-classical hybrids. Phys. Rev. A 85(5), 052,109 (2012).  https://doi.org/10.1103/PhysRevA.85.052109
  39. 39.
    Everett III, H.: “Relative State” Formulation of Quantum Mechanics. In: Wheeler and Zurek [93], chap. II.3, pp. 315–323. Original: Rev. Mod. Phys. 29 (1957) 454–462Google Scholar
  40. 40.
    Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965). Emended edition by Daniel F. Styer published by Dover (2010)Google Scholar
  41. 41.
    Frigg, R.: A Field Guide to Recent Work in the Foundations of Statistical Mechanics. In: Rickless [68], chap. 3, pp. 99–196Google Scholar
  42. 42.
    Gilder, L.: The Age of Entanglement when Quantum Physics was Reborn. Vintage Books, New York (2008)Google Scholar
  43. 43.
    Goldstein, H.: Classical Mechanics, 2nd edn. Addison-Wesley, Reading (1980)zbMATHGoogle Scholar
  44. 44.
    Greenstein, G., Zajonc, A.G.: The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics, 2nd edn., Jones and Bartlett Titles in Physics and Astronomy. Jones and Bartlett Publishers, Sudbury (2006)Google Scholar
  45. 45.
    Haag, R.: Local Quantum Physics: Fields, Particles, Algebras. Texts and Monographs in Physics, 2nd edn. Springer, Berlin (1996)CrossRefGoogle Scholar
  46. 46.
    Harshman, N.L., Ranade, K.S.: Observables can be tailored to change the entanglement of any pure state. Phys. Rev. A 84, 012303 (2011).  https://doi.org/10.1103/PhysRevA.84.012303
  47. 47.
    Heisenberg, W.: Quantum Theoretical Re-interpretation of Kinematic and Mechanical Relations. In: van der Waerden [89], pp. 261–276. Original: Zeitschrift für Physik 35 (1926) 557–615Google Scholar
  48. 48.
    Heisenberg, W.: Über den anshaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Phsyik 48(3), 127–198 (1927).  https://doi.org/10.1007/BF01397280 CrossRefGoogle Scholar
  49. 49.
    Heslot, A.: Quantum mechanics as a classical theory. Phys. Rev. D 31(6), 1341–1348 (1985).  https://doi.org/10.1103/PhysRevD.31.1341 MathSciNetCrossRefGoogle Scholar
  50. 50.
    Jammer, M.: The Conceptual Development of Quantum Mechanics. International Series in Pure and Applied Physics. McGraw-Hill, New York (1966)Google Scholar
  51. 51.
    Jauch, J.M.: Foundations of Quantum Mechanics. Addison and Wesley, Reading (1968)zbMATHGoogle Scholar
  52. 52.
    Johnson, G.W., Lapidus, M.L.: The Feynman Integral and Feynman’s Operational Calculus. Oxford Mathematical Monographs. Clarendon Press, Oxford (2000)zbMATHGoogle Scholar
  53. 53.
    José, J.V., Saletan, E.J.: Classical Dynamics: A Contemporary Approach. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  54. 54.
    van Kampe, N.G.: Ten theorems about quantum mechanical measurements. Physica A 153(1), 97–113 (1988).  https://doi.org/10.1016/0378-4371(88)90105-7 MathSciNetCrossRefGoogle Scholar
  55. 55.
    Khrennikov, A.Y.: Classical probability model for Bell inequality. J. Phys. Conf. Ser. 504, 012019 (2014).  https://doi.org/10.1088/1742-6596/504/1/012019. EMQ13: Emergent Quantum Mechanics 2013
  56. 56.
    Kupczynski, M.: Bell inequlities, experimental protocols and contextuality. Found. Phys. 45(7), 735–753 (2015).  https://doi.org/10.1007/s10701-014-9863-4 MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Le Bellac, M.: Non equilibrium statistical mechanics. In: DEA Cours aux Houches, August 2007. HAL archives ouvertes (2007). https://cel.archives-ouvertes.fr/cel-00176063
  58. 58.
    Ledoux, M.: The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs, vol. 89. AMS, Providence (2001)Google Scholar
  59. 59.
    Lundeen, J.S., Sutherland, B., Patel, A., Stewart, C., Bamber, C.: Direct measurement of the quantum wavefunction. Nature 474, 188–191 (2011).  https://doi.org/10.1038/nature10120 CrossRefGoogle Scholar
  60. 60.
    de Muynck, W.M.: Foundations of Quantum Mechanics, An Empiricist Approach. Kluwer Academic Publishers, Dordrecht (2002)CrossRefGoogle Scholar
  61. 61.
    von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932). Reprinted 1968Google Scholar
  62. 62.
    Nieuwenhuizen, T.M.: Is the contextuality loophole fatal for the derivation of Bell inequalities? Found. Phys. 41(3), 580–591 (2011).  https://doi.org/10.1007/s10701-010-9461-z MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Park, J.L.: Nature of quantum states. Am. J. Phys. 36, 211–226 (1968).  https://doi.org/10.1119/1.1974484 CrossRefGoogle Scholar
  64. 64.
    Peres, A.: Can we undo quantum measurements? In: Wheeler and Zurek [93], chap. V.7, pp. 692–696. Original: Phys. Rev. D22 (1980) 879–883Google Scholar
  65. 65.
    Pusey, M.F., Barrett, J., Rudolph, T.: On the reality of the quantum state. Nat. Phys. 8, 475–479 (2012).  https://doi.org/10.1038/nphys2309 CrossRefGoogle Scholar
  66. 66.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics: Functional Analysis, vol. I. Academic Press, New York (1972)zbMATHGoogle Scholar
  67. 67.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics: Fourier Analysis, Self-Adjointness, vol. II. Academic Press, New York (1975)zbMATHGoogle Scholar
  68. 68.
    Rickles, D. (ed.): The Ashgate Companion to Contemporary Philosophy of Physics. Ashgate Publishing Company, Burlington (2008)Google Scholar
  69. 69.
    Schatz, G.C., Ratner, M.A.: Quantum Mechanics in Chemistry. Prentice Hall, Engelwood Cliffs (1993)Google Scholar
  70. 70.
    Schmidt, E.: Zur Theorie der linearen und nichtlinearen Integralgleichungen. I. Teil: Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener. Math. Ann. 63, 433–476 (1907)Google Scholar
  71. 71.
    Schmidt, E.: Zur Theorie der linearen und nichtlinearen Integralgleichungen. Zweite Abhandlung: Auflösung der allgemeinen linearen Integralgleichungen. Math. Ann. 64, 161–174 (1907)Google Scholar
  72. 72.
    Schrödinger, E.: Quantisation as a problem of proper values (part I). In: Collected papers on Wave Mechanics [78], pp. 1–12. Original: Annalen der Physik 79 (1926) 489–527Google Scholar
  73. 73.
    Schrödinger, E.: Quantisation as a problem of proper values (Part III: perturbation theory, with application to the stark effect of the Balmer lines. In: Collected papers on Wave Mechanics [78], pp. 62–101. Original: Annalen der Physik 80 (1926) 29–82Google Scholar
  74. 74.
    Schrödinger, E.: Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinem – On the relation between the quantum mechanics of Heisenberg, Born, and Jordan and that of Schrödinger. Annalen der Physik 384, 734–756 (1926).  https://doi.org/10.1002/andp.19263840804. Original in volume 79CrossRefzbMATHGoogle Scholar
  75. 75.
    Schrödinger, E.: Discussion of probability relations between separated systems. Math. Proc. Camb. Philos. Soc. 31(4), 555–563 (1935).  https://doi.org/10.1017/S0305004100013554 CrossRefzbMATHGoogle Scholar
  76. 76.
    Schrödinger, E.: Probability relations between separated systems. Math. Proc. Camb. Philos. Soc. 32(3), 446–452 (1936).  https://doi.org/10.1017/S0305004100019137 CrossRefzbMATHGoogle Scholar
  77. 77.
    Schrödinger, E.: Statistical Thermodynamics, 2nd edn. Cambridge University Press, Cambridge (1952). Reprinted 1989 by Dover PublicationsGoogle Scholar
  78. 78.
    Schrödinger, E.: Collected Papers on Wave Mechanics. Chelsea Publishing Company, New York (1982)Google Scholar
  79. 79.
    Schwartz, M.D.: Quantum Field Theory and the Standard Model. Cambridge University Press, New York (2014)Google Scholar
  80. 80.
    Skorohod, A.V.: Integration in Hilbert Space. No. 79 in Ergebnisse der Mathematik und ihre Grenzgebiete. Springer, Berlin (1974). Russian edition: Integrirovanie v gilbetovyh prostranstvah, Nauka Izdatel’stvo, Moscow, 1974, translated by Kenneth WickwireGoogle Scholar
  81. 81.
    Sohrab, H.H.: Basic Real Analysis. Birkhauser, Boston (2003)CrossRefGoogle Scholar
  82. 82.
    Sommerfeld, A.: Atombau und Spektrallinien. Friedrich Vieweg und Sohn, Braunschweig (1921). First edition 1919. Reproduced by Nabu Public Domain ReprintsGoogle Scholar
  83. 83.
    Srednicki, M.: Quantum Field Theory. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  84. 84.
    Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics, and All That. Princeton Landmarks in Physics. Princeton University Press, Princeton (1964). Original: W.A Benjamin 1964, 1978, 1980, Paperback: Princeton 2000Google Scholar
  85. 85.
    Thaller, B.: The Dirac Equation. Texts and Monographs in Physics. Springer, Berlin (1992)Google Scholar
  86. 86.
    Tonomura, A., Endo, J., Matsuda, T., Kawasaki, T., Exawa, H.: Demonstration of single-electron buildup of an interference pattern. Am. J. Phys. 57(2), 117–120 (1989)CrossRefGoogle Scholar
  87. 87.
    Trefethen, L.N.: Cubature, approximation, and isotropy in the hypercube. SIAM Rev. 59(3), 469–491 (2017)MathSciNetCrossRefGoogle Scholar
  88. 88.
    Varadarajan, V.S.: The geometry of quantum theory, 2nd edn. Springer, New York (1985)zbMATHGoogle Scholar
  89. 89.
    van der Waerden, B.L. (ed.): Sources of Quantum Mechanics, No. V in Classics of Science. Dover Publications, New York (1968)Google Scholar
  90. 90.
    van der Waerden, B.L.: Group Theory and Quantum Mechanics, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 214. Springer, Berlin (1974). Translation of the German edition: Springer, 1932Google Scholar
  91. 91.
    Wallace, D.: Philosophy of Quantum Mechanics. In: Rickless [68], chap. 2, pp. 16–98Google Scholar
  92. 92.
    Weinberg, S.: The Quantum Theory of Fields: Foundations, vol. I. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  93. 93.
    Wheeler, J.A., Zurek, W.H. (eds.): Quantum Theory and Measurement. Princeton Series in Physics. Princeton University Press, Princeton (1983)Google Scholar
  94. 94.
    Wiener, N.: Differential-space. J. Math. Phys. 2, 131–174 (1923).  https://doi.org/10.1002/sapm192321131 CrossRefGoogle Scholar
  95. 95.
    Wightman, A.S., Gårding, L.: Fields as operator-valued distributions. Arkiv för Fysik 28, 129–189 (1964). Published by Royal Swedish Academy of SciencesGoogle Scholar
  96. 96.
    Zurek, W.H.: Quantum darwinism, classical reality, and the randomness of quantum jumps. Phys. Today 67(10), 44–50 (2014)CrossRefGoogle Scholar
  97. 97.
    Zwanzig, R.: Nonequilibrium Statistical Mechanics. Oxford University Press, Oxford (2001)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Quantum Theory Project, Department of ChemistryUniversity of FloridaGainesvilleUSA
  2. 2.Quantum Theory Project, Department of PhysicsUniversity of FloridaGainesvilleUSA

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