Advertisement

Synthese

, Volume 195, Issue 11, pp 4839–4863 | Cite as

Idealizations, essential self-adjointness, and minimal model explanation in the Aharonov–Bohm effect

  • Elay Shech
Article

Abstract

Two approaches to understanding the idealizations that arise in the Aharonov–Bohm (AB) effect are presented. It is argued that a common topological approach, which takes the non-simply connected electron configuration space to be an essential element in the explanation and understanding of the effect, is flawed. An alternative approach is outlined. Consequently, it is shown that the existence and uniqueness of self-adjoint extensions of symmetric operators in quantum mechanics have important implications for philosophical issues. Also, the alleged indispensable explanatory role of said idealizations is examined via a minimal model explanatory scheme. Last, the idealizations involved in the AB effect are placed in a wider philosophical context via a short survey of part of the literature on infinite and essential idealizations.

Keywords

Idealization Aharonov–Bohm effect Representation Models Explanation Topology Emergence and reduction 

Notes

Acknowledgements

I am extremely grateful to John Earman, John D. Norton, Laura Ruetsche, Robert W. Batterman, James Woodward, Mark Wilson, Giovanni Valente, Nicholaos Jones, Bryan W. Roberts, Aaron Novick, and Samuel C. Fletcher, as well as audiences in numerous workshops and conferences, for insightful conversations and many constructive comments on earlier versions of this paper going back to 2014 under the title of “Topological Idealization, Asymptotic-Minimal Model Explanation, and the Aharonov–Bohm Effect.” Also thanks to Narin Shech for help with figures, and to Michel Smith and Cesar R. de Oliveira for assistance with technical issues with this version of the paper. Special thanks to John Earman for his guidance and mentorship regarding the details and issues discussed in this paper and over the years. Needless to say, my mistakes are my own, and I refer the reader to John Earman’s own more recent and excellent contribution in his “The Role of Idealization in the Aharonov–Bohm Effect” (http://philsci-archive.pitt.edu/12696/).

References

  1. Aharonov, Y., & Bohm, D. (1959). Significance of electromagnetic potentials in the quantum theory. Physical Review, 115, 485–491.CrossRefGoogle Scholar
  2. Bain, J. (2016). Emergence and the mechanism in the fractional quantum Hall effect. Studies in History and Philosophy of Modern Physics, 56, 27–38.CrossRefGoogle Scholar
  3. Ballentine, L. E. (1998). Quantum mechanics: A modern development. Singapore: World Scientific.CrossRefGoogle Scholar
  4. Ballesteros, M., & Weder, R. (2009). The Aharonov–Bohm effect and Tonomura et al. experiments: Rigorous results. Journal of Mathematical Physics, 50, 122108.CrossRefGoogle Scholar
  5. Ballesteros, M., & Weder, R. (2011). Aharonov–Bohm effect and high-velocity estimates of solutions to the Schrodinger equation. Communications in Mathematical Physics, 303(1), 175–211.CrossRefGoogle Scholar
  6. Bangu, S. (2009). Understanding thermodynamic singularities: Phase transitions, date and phenomena. Philosophy of Science, 76, 488–505.CrossRefGoogle Scholar
  7. Batterman, R. (2002). The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence. London: Oxford University Press.Google Scholar
  8. Batterman, R. (2003). Falling cats, parallel parking, and polarized light. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 34, 527–557.CrossRefGoogle Scholar
  9. Batterman, R. (2005). Critical phenomena and breaking drops: Infinite idealizations in physics. Studies in History and Philosophy of Modern Physics, 36, 225–244.CrossRefGoogle Scholar
  10. Batterman, R. (2015). Autonomy and scales. In B. Falkenburg & M. Morrison (Eds.), Why more is different: Philosophical issues in condensed matter physics and complex systems (pp. 115–136). Heidelberg: Springer.Google Scholar
  11. Batterman, R. (2017). Autonomy of theories: An explanatory problem. Noûs. doi: 10.1111/nous.12191.
  12. Batterman, R., & Rice, C. (2014). Minimal model explanations. Philosophy of Science, 81(3), 349–376.CrossRefGoogle Scholar
  13. Belot, G. (1998). Understanding electromagnetism. British Journal for Philosophy of Science, 49(4), 531–555.CrossRefGoogle Scholar
  14. Berry, M. V. (1986). The Aharonov–Bohm effect is real physics not ideal physics. In V. Gorini & A. Frigerio (Eds.), Fundamental aspects of quantum theory (Vol. 144, pp. 319–320). New York: Plenum.CrossRefGoogle Scholar
  15. Bocchieri, P., & Loinger, A. (1978). Nonexistence of the Aharonov–Bohm effect. Nuovo Cimento, 47A(4), 475–482.CrossRefGoogle Scholar
  16. Bokulich, A. (2008). Re-examining the quantum-classical relation: Beyond reductionism and pluralism. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  17. Borrmann, P., Mülken, O., & Harting, J. (2000). Classification of phase transitions in small systems. Physical Review Letters, 84, 3511–3514.CrossRefGoogle Scholar
  18. Boyer, T. H. (2008). Comment on the experiments related to the Aharonov–Bohm phase shift. Foundations of Physics, 38, 398–505.CrossRefGoogle Scholar
  19. Bub, J. (1988). How to solve the measurement problem in quantum mechanics. Foundations of Physics, 18, 701–722.CrossRefGoogle Scholar
  20. Butterfield, J. (2011). Less is different: Emergence and reduction reconciled. Foundations of Physics, 41(6), 1065–1135.CrossRefGoogle Scholar
  21. Caprez, A., Barwick, B. B., & Batelaan, H. (2007). Macroscopic test of the Aharonov–Bohm effect. Physical Review Letters, 99, 210401.CrossRefGoogle Scholar
  22. Chambers, R. G. (1960). Shift of an electron interference pattern by enclosed magnetic flux. Physical Review Letter, 5(1), 3–5.CrossRefGoogle Scholar
  23. Chomaz, P., Gulminelli, F., & Duflot, V. (2001). Topology of event distributions as a generalized definition of phase transitions in finite systems. Physical Review E, 64, 046114.CrossRefGoogle Scholar
  24. Cooper, J. L. B. (1947). One-parameter semigroups of isometric operators in Hilbert space. Annals of Mathematics, 48(4), 827–842.CrossRefGoogle Scholar
  25. Cooper, J. L. B. (1948). Symmetric operators in Hilbert space. Proceedings of the London Mathematical Society, 2(1), 11–55.CrossRefGoogle Scholar
  26. de Oliveira, C. R., & Pereira, M. (2008). Mathematical justification of the Aharonov–Bohm Hamiltonian. Journal of Statistical Physics, 133, 1175–1184.CrossRefGoogle Scholar
  27. de Oliveira, C. R., & Pereira, M. (2010). Scattering and self-adjoint extensions of the Aharonov–Bohm Hamiltonian. Journal of Physics A: Mathematical and Theoretical, 43, 1–29.CrossRefGoogle Scholar
  28. de Oliveira, C. R., & Pereira, M. (2011). Impenetrability of Aharonov–Bohm solenoids: Proof of norm resolvent convergence. Letters in Mathematical Physics, 95, 41–51.CrossRefGoogle Scholar
  29. Earman, J. (2004). Curie’s principle and spontaneous symmetry breaking. International Studies in the Philosophy of Science, 18(2–3), 173–198.CrossRefGoogle Scholar
  30. Earman, J. (2008). Superselection rules for philosophers. Erkenntnis, 69, 377–414.CrossRefGoogle Scholar
  31. Earman, J. (2009). Essential self-adjointness: Implications for determinism and the classical-quantum correspondence. Synthese, 169, 2750.Google Scholar
  32. Earman, J. (2010). Understanding permutation invariance in quantum mechanics. https://www.youtube.com/watch?v=xciuUhnsx1k (unpublished preprint).
  33. Earman, J. (2016). The role of idealization in the Aharonov–Bohm effect. http://philsci-archive.pitt.edu/12696/.
  34. Ehrenberg, W., & Siday, R. W. (1949). The refractive index in electron optics and the principles of dynamics. Proceedings of the Physical Society London: Section B, 62(1), 8–21.CrossRefGoogle Scholar
  35. Ellis, B. (1992). Idealizations in science. In C. Dilworth (Ed.), Idealization IV: Intelligibility in science. Amsterdam: Rodopi.Google Scholar
  36. Emch, G. (2006). Quantum statistical physics. In J. Butterfield, & J. Earman (Eds.), Philosophy of physics, part B, a volume of the handbook of the philosophy of science (pp. 1075–1182). North Holland.Google Scholar
  37. Eskin, G. (2013). A simple proof of magnetic and electric Aharonov–Bohm effects. Communications in Mathematical Physics, 321(3), 747–767.CrossRefGoogle Scholar
  38. Ezawa, Z. F. (2013). Quantum Hall effects: Recent theoretical and experimental developments. Singapore: World Scientific.CrossRefGoogle Scholar
  39. Fisher, R. A. (1930). The genetical theory of natural selection. Oxford: Clarendon.CrossRefGoogle Scholar
  40. Fodor, J. (1974). Special sciences, or the disunity of sciences as a working hypothesis. Synthese, 28, 97–115.CrossRefGoogle Scholar
  41. Franzosi, R., & Pettini, M. (2004). Theorem on the origin of phase transitions. Physical Review Letters, 92, 060601.CrossRefGoogle Scholar
  42. Franzosi, R., Pettini, M., & Spinelli, L. (2000). Topology and phase transitions: Paradigmatic evidence. Physical Review Letters, 84, 2774–2777.CrossRefGoogle Scholar
  43. Fraser, J. D. (2016). Spontaneous symmetry breaking in finite systems. Philosophy of Science, 83(4), 585–605.CrossRefGoogle Scholar
  44. Gelfert, A. (2016). How to do science with models: A philosophical primer. Cham: Springer.CrossRefGoogle Scholar
  45. Gross, D. H. E., & Votyakov, E. V. (2000). Phase transitions in “small” systems. The European Physical Journal B-Condensed Matter and Complex Systems, 15, 115–126.CrossRefGoogle Scholar
  46. Healey, R. (1997). Nonlocality and the Aharonov–Bohm effect. Philosophy of Science, 64, 18–41.CrossRefGoogle Scholar
  47. Healey, R. (1999). Quantum analogies: A reply to Maudlin. Philosophy of Science, 66, 440–447.CrossRefGoogle Scholar
  48. Healey, R. A. (2007). Gauging what’s real: The conceptual foundations of contemporary gauge theories. New York: Oxford University Press.CrossRefGoogle Scholar
  49. Hiley, B. J. (2013). The early history of the AB effect. arXiv:1304.4736.
  50. Kadanoff, L. P. (2000). Statistical physics: Statics, dynamics and renormalization. Singapore: World Scientific.CrossRefGoogle Scholar
  51. Landsman, N. P. (2013). Spontaneous symmetry breaking in quantum systems: Emergence or reduction? Studies in History and Philosophy of Modern Physics, 44, 379–394.CrossRefGoogle Scholar
  52. Landsman, N. P. (2016). Quantization and superselection III: Mutliply connected spaces and indistinguishable particles. Reviews in Mathematical Physics, 28, 1650019.CrossRefGoogle Scholar
  53. Lebowitz, J. L. (1999). Statistical mechanics: A selective review of two central issues. Reviews of Modern Physics, 71(2), S346–S357.CrossRefGoogle Scholar
  54. Liu, C., & Emch, G. G. (2005). Explaining quantum spontaneous symmetry breaking. Studies in History and Philosophy of Modern Physics, 36, 137–163.CrossRefGoogle Scholar
  55. Lui, C. (1999). Explaining the emergence of cooperative phenomena. Philosophy of Science, 66, S92–S106.CrossRefGoogle Scholar
  56. Lyre, H. (2001). The principles of gauging. Philosophy of Science, 68(3), S371–S381.CrossRefGoogle Scholar
  57. Lyre, H. (2009). Aharonov–Bohm effect. In D. Greengerger, C. Hentschel, & F. Weinert (Eds.), Compendium of quantum mechanics (pp. 1–3). Berlin: Springer.Google Scholar
  58. Magni, C., & Valz-Gris, F. (1995). Can elementary quantum mechanics explain the Aharonov–Bohm effect? Journal of Mathematical Physics, 36(1), 177–186.CrossRefGoogle Scholar
  59. Maudlin, T. (1998). Healey on the Aharonov–Bohm effect. Philosophy of Science, 65, 361–368.CrossRefGoogle Scholar
  60. Menon, T., & Callender, C. (2013). Turn and face the strange.. ch-ch-changes: Philosophical questions raised by phase transitions. In R. W. Batterman (Ed.), The Oxford handbook of philosophy of physics. Oxford: Oxford University Press.Google Scholar
  61. Möllenstedt, G., & Bayh, W. (1962). Kontinuierliche Phasenschiebung von Elektronenwellen im kraftfeldfreien Raum durch das magnetische Vektorpotential eines Solenoids. Zeitschrift für Physik, 169, 263.CrossRefGoogle Scholar
  62. Morrison, M. (2012). Emergent physics and micro-ontology. Philosophy of Science, 79, 141–166.CrossRefGoogle Scholar
  63. Morrison, M. (2015). Why is more different? In B. Falkenburg & M. Morrison (Eds.), Why more is different: Philosophical issues in condensed matter physics and complex systems (pp. 91–114). Heidelberg: Springer.Google Scholar
  64. Norton, J. D. (2012). Approximations and idealizations: Why the difference matters. Philosophy of Science, 79, 207–232.CrossRefGoogle Scholar
  65. Nounou, A. M. (2003). A fourth way to the Aharonov–Bohm effect. In K. Bradind & E. Castellani (Eds.), Symmetries in physics: Philosophical reflections. Cambridge: Cambridge University Press.Google Scholar
  66. Peshkin, M., & Tonomura, A. (1989). The Aharonov–Bohm effect. Berlin: Springer.CrossRefGoogle Scholar
  67. Prigogine, I. (1997). The end of certainty. New York: The Free Press.Google Scholar
  68. Reed, M., & Simon, B. (1980). Methods of modern mathematical physics (Vol. I-IV). San Diego: Academic Press Inc.Google Scholar
  69. Roberts, B. (2016). Observables, disassembled. http://philsci-archive.pitt.edu/12478/.
  70. Rueger, A. (2000). Physical emergence, diachronic and synchronic. Synthese, 124, 297–322.CrossRefGoogle Scholar
  71. Rueger, A. (2006). Functional reduction and emergence in the physical sciences. Synthese, 151, 335–346.CrossRefGoogle Scholar
  72. Ruetsche, L. (2003). A matter of degree: Putting unitary inequivalence to work. Philosophy of Science, 70(5), 1329–1342.CrossRefGoogle Scholar
  73. Ruetsche, L. (2011). Interpreting quantum theories. Oxford: Oxford University Press.CrossRefGoogle Scholar
  74. Ruijsenaars, S. N. M. (1983). The Aharonov–Bohm effect and scattering theory. Annals of Physics, 146, 1–34.CrossRefGoogle Scholar
  75. Ryder, L. H. (1996). Quantum field theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  76. Schulman, L. (1971). A path integral for spin. Physical Review, 176, 1558–1569.CrossRefGoogle Scholar
  77. Shech, E. (2013). What is the ‘paradox of phase transitions? Philosophy of Science, 80, 1170–1181.CrossRefGoogle Scholar
  78. Shech, E. (2014). Scientific misrepresentation and guides to ontology: The need for representational code and contents. Synthese, 192(11), 3463–3485.CrossRefGoogle Scholar
  79. Shech, E. (2015). Two approaches to fractional statistics in the quantum Hall effect: Idealizations and the curious case of the anyon. Foundations of Physics, 45(9), 1063–1110.CrossRefGoogle Scholar
  80. Shech, E. (2016). Fiction, depiction, and the complementarity thesis in art and science. The Monist, 99(3), 311–332.CrossRefGoogle Scholar
  81. Shech, E.,  & Gelfert, A. (2016). The exploratory role of idealizations and limiting cases in models (preprint).Google Scholar
  82. Tonomura, A. (1999). Electron holography. Berlin: Springer.CrossRefGoogle Scholar
  83. Tonomura, A. (2010). The AB effect and its expanding applications. Journal of Physics A, 43(35), 1–13.CrossRefGoogle Scholar
  84. Tonomura, A., Matsuda, T., Suzuki, R., Fukuhara, A., Osakabe, N., Umezaki, H., Endo, J., Shinagawa, K., Sugita, Y., & Fujiwara, H. (1982). Observation of the Aharonov-Bohm effect by electron holography. Physical Review Letters, 48, 1443.Google Scholar
  85. Tonomura, A., Osakabe, N., Matsuda, T., Kawasaki, T., Endo, J., Yano, S., & Yamada, H. (1986). Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave. Physical Review Letter, 56, 792–795.Google Scholar
  86. Wales, D. J., & Berry, R. S. (1994). Coexistence in finite systems. Physical Review Letters, 73, 2875–2878.CrossRefGoogle Scholar
  87. Wallace, D. (2014). Deflating the Aharonov–Bohm effect. https://arxiv.org/abs/1407.5073.
  88. Wayne, A. (2009). Emergence and singular limits. Synthese, 184(3), 341–356.CrossRefGoogle Scholar
  89. Weisskopf, V. F. (1961). Selected topics in theoretical physics. In W. Brittin (Ed.), Lectures in theoretical physics (Vol. III, pp. 67–70). New York: Interscience.Google Scholar
  90. Wilczek, F. (1982). Quantum mechanics of fractional-spin particles. Physical Review Letters, 49(14), 957–959.Google Scholar
  91. Wilczek, F. (Ed.). (1990). Fractional statistics and anyon superconductivity. Singapore: World Scientific.Google Scholar
  92. Wu, T. T., & Yang, C. N. (1975). Concept of nonintegrable phase factors and global formulation of gauge fields. Physical Review D, 12, 3845.CrossRefGoogle Scholar
  93. Yang, N. C., & Lee, T. D. (1952). Statistical theory of equations of state and phase transitions. I. Theory of condensation. Physical Review, 97, 404.CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2017

Authors and Affiliations

  1. 1.Department of Philosophy, 6080 Haley CenterAuburn UniversityAuburnUSA

Personalised recommendations