# Two Approaches to Fractional Statistics in the Quantum Hall Effect: Idealizations and the Curious Case of the Anyon

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## Abstract

This paper looks at the nature of idealizations and representational structures appealed to in the context of the fractional quantum Hall effect, specifically, with respect to the emergence of anyons and fractional statistics. Drawing on an analogy with the Aharonov–Bohm effect, it is suggested that the standard approach to the effects—(what we may call) the *topological approach *to fractional statistics—relies essentially on problematic idealizations that need to be revised in order for the theory to be explanatory. An alternative *geometric approach* is outlined and endorsed. Roles for idealizations in science, as well as consequences for the debate revolving around so-called essential idealizations, are discussed.

## Keywords

Idealization Approximation Emergence Reduction Aharonov–Bohm effect Quantum Hall effect Anyons Fractional Statistics Representation Explanation## Notes

### Acknowledgments

Parts of this paper were presented at the “Bucharest Colloquium in Analytic Philosophy: New Directions in the Philosophy of Physics” conference at University of Bucharest. I am grateful to the audience for stimulating discussion, to Iulian Toader for editing this volume of *Foundations of Physics*, and to two anonymous referees for helpful comments. I am especially grateful to John Earman and John D. Norton for comments on earlier versions of this paper, numerous insightful discussions, and their constant support. This paper is heavily in debt to work done by John Earman, who initially drew my attention to the main issues discussed in this paper, and has been especially kind in helping me work through the finer details. Special thanks to Naharin Shech for her help with figures.

## References

- 1.Aharonov, Y., Anandan, J.: Phase change during a cyclic quantum evolution. Phys. Rev. Lett.
**58**, 1593–1596 (1987)MathSciNetADSCrossRefGoogle Scholar - 2.Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev.
**115**, 485–91 (1959)MATHMathSciNetADSCrossRefGoogle Scholar - 3.Aharonov, Y., Bohm, D.: Remarks on the possibility of quantum electrodynamics without potentials. Phys. Rev.
**125**, 2192 (1962)MATHMathSciNetADSCrossRefGoogle Scholar - 4.Aharonov, Y., Bohm, D.: Further discussion of the role of electromagnetic potentials in the quantum theory. Phys. Rev.
**130**, 1625 (1963)MATHMathSciNetADSCrossRefGoogle Scholar - 5.Ando, T., Fowler, A.B., Stern, F.: Electronic properties of two-dimensional systems. Rev. Mod. Phys.
**54**, 437–672 (1982)ADSCrossRefGoogle Scholar - 6.Arovas, D.P.: Topics in fractional statistics. In: Shapere, A., Wilczek, F. (eds.) Geometric Phases in Physics. World Scientific, Singapore (1989)Google Scholar
- 7.Arovas, D., Schrieffer, J.R., Wilczek, F.: Fractional statistics and the Quantum Hall effect. Phys. Rev. Lett.
**53**, 722–723 (1984)ADSCrossRefGoogle Scholar - 8.Artin, E.: Theory of braids. Ann. Math.
**48**(1), 101–126 (1947)MATHMathSciNetCrossRefGoogle Scholar - 9.Babiker, M., Loudon, R.: Gauge invariance of the Aharonov–Bohm effect. J. Phys. A
**17**, 2973–2982 (1984)MathSciNetADSCrossRefGoogle Scholar - 10.Ballentine, L.E.: Quantum Mechanics: A Modern Development. World Scientific, Singapore (1998)MATHCrossRefGoogle Scholar
- 11.Ballesteros, M., Weder, R.: The Aharonov–Bohm effect and Tonomura et al.: experiments. Rigorous results. J. Math. Phys.
**50**, 122108 (2009)MathSciNetADSCrossRefGoogle Scholar - 12.Ballesteros, M., Weder, R.: High-velocity estimates for the scattering operator and Aharonov–Bohm effect in three dimensions. Commun. Math. Phys.
**285**, 345–398 (2009)MATHMathSciNetADSCrossRefGoogle Scholar - 13.Ballesteros, M., Weder, R.: Aharonov–Bohm effect and high-velocity estimates of solutions to the Schrodinger equation. Commun. Math. Phys.
**303**(1), 175–211 (2011)MATHMathSciNetADSCrossRefGoogle Scholar - 14.Batterman, R.: The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence. Oxford University Press, London (2002)Google Scholar
- 15.Batterman, R.: Falling cats, parallel parking, and polarized light. Stud. Hist. Philos. Mod. Phys.
**34**, 527–557 (2003)MATHMathSciNetCrossRefGoogle Scholar - 16.Batterman, R.: Critical phenomena and breaking drops: infinite idealizations in physics. Stud. Hist. Philos. Mod. Phys.
**36B**, 225–244 (2005)MathSciNetCrossRefGoogle Scholar - 17.Bransden, B.H., Joachaim, J.C.: Quantum Mechanics. Prentice Hall, New York (2000)Google Scholar
- 18.Birman, J. S., Brendle, T. E.: Braids: A survey. Available via arXiv:math/0409205 (2005)
- 19.Belot, G.: Understanding electromagnetism. Br. J. Philos. Sci.
**49**(4), 531–555 (1998)MATHMathSciNetCrossRefGoogle Scholar - 20.Berry, M.V.: The Aharonov-Bohm effect is real physics not ideal physics. In: Gorini, V., Frigerio, A. (eds.) Fundamental Aspects of Quantum Theory, vol. 144, pp. 319–320. Plenum, New York (1986)CrossRefGoogle Scholar
- 21.Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Lond.
**A392**, 45–57 (1984)ADSCrossRefGoogle Scholar - 22.Bokulich, A.: Re-examining the Quantum-Classical Relation: Beyond Reductionism and Pluralism. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
- 23.Born, M., Fock, V.A.: Beweis des Adiabatensatzes. Z. Phys. A
**51**(3—-4), 165–180 (1928)MATHCrossRefGoogle Scholar - 24.Butterfield, J.: Less is different: emergence and reduction reconciled. Found. Phys.
**41**, 1065–1135 (2011)MATHMathSciNetADSCrossRefGoogle Scholar - 25.Callender, C.: Taking thermodynamics too seriously. Stud. Hist. Philos. Mod. Phys.
**32**(4), 539–553 (2001)MATHMathSciNetCrossRefGoogle Scholar - 26.Camino, F.E., Zhou, W., Goldman, V.J.: Realization of a Laughlin quasiparticle interferometer: observation of fractional statistics. Phys. Rev. B
**72**, 075342 (2005)ADSCrossRefGoogle Scholar - 27.Canright, G.S., Johnson, M.D.: Fractional statistics: \(\alpha \) to \(\beta \). J. Phys. A Math. Gen.
**27**, 3579–3598 (1994)ADSCrossRefGoogle Scholar - 28.Caprez, A., Barwick, B.B., Batelaan, H.: Macroscopic test of the Aharonov–Bohm effect. Phys. Rev. Lett.
**99**, 210401 (2007)ADSCrossRefGoogle Scholar - 29.Chakraborty, T., Pietilinen, P.: The Quatnum Hall Effects. Springer, Berlin (1995)CrossRefGoogle Scholar
- 30.Chambers, R.G.: Shift of an electron interference pattern by enclosed magnetic flux. Phys. l Rev. Lett.
**5**, 3–5 (1960)ADSCrossRefGoogle Scholar - 31.Chen, Y.R., Wilczek, F., Witten, E., Halperin, B.I.: On anyon superconductivity. Int. J. Mod. Phys. B
**3**, 1001 (1989)MathSciNetADSCrossRefGoogle Scholar - 32.de Oliveira, C.R., Pereira, M.: Mathematical justification of the Aharonov–Bohm hamiltonion. J. Stat. Phys.
**133**, 1175–1184 (2008)MATHMathSciNetADSCrossRefGoogle Scholar - 33.de Oliveira, C.R., Pereira, M.: Scattering and Self-adjoint extensions of the Aharonov–Bohm Hamiltonian. J. Phys. A
**43**, 1–29 (2010)CrossRefGoogle Scholar - 34.de Oliveira, C.R., Pereira, M.: Impenetrability of Aharonov–Bohm solenoids: proof of norm resolvent convergence. Lett. Math. Phys.
**95**, 41–51 (2011)MATHMathSciNetADSCrossRefGoogle Scholar - 35.Douçot, B., Pasquier, V., Duplantier, B., Rivasseau, V. (eds.): The Quantum Hall Effect Poincaré Seminar. Birkhäuser, Berlin (2004)Google Scholar
- 36.Dresden, M.: The existence and significance of parastatistics. In: Hayakawa, H. (ed.) Lectures on astrophysics and weak interactions, pp. 377–469. Brandeis University, Waltham (1964)Google Scholar
- 37.Duck, I., Sudarshan, E.C.G. (eds.): Pauli and the sping-statistics theorem. World Scientific, Singapore (1997)Google Scholar
- 38.Ehrenberg, W., Siday, R.W.: The refractive index in electron optics and the principles of dynamics. Proc. Phys. Soc. Lond.
**B62**, 8–21 (1949)ADSCrossRefGoogle Scholar - 39.Earman, J.: Curie’s principle and spontaneous symmetry breaking. Int. Stud. Philos. Sci.
**18**(2–3), 173–198 (2004)MATHMathSciNetCrossRefGoogle Scholar - 40.Earman, J.: J. Understanding permutation invariance in quantum mechanics. Unpublished preprint (2010)Google Scholar
- 41.Ezawa, Z.F.: Quantum Hall Effects. World Scientific, Singapore (2013)CrossRefGoogle Scholar
- 42.Fadell, E., Neuwirth, L.: Configuration spaces. Math. Scand.
**10**, 111–118 (1962)MATHMathSciNetGoogle Scholar - 43.Fox, R., Neuwirth, L.: The Braid groups. Math Scand.
**10**, 119–126 (1962)MATHMathSciNetGoogle Scholar - 44.Griffiths, D.J.: Introduction to Quantum Mechanics. Pearson Prentice Hall, New Jersey (2005)Google Scholar
- 45.Haldane, F.D.M.: Fractional quantization of the hall effect: a hierarchy of incompressible quantum fluid states. Phys. Rev. Lett.
**51**, 605 (1983)MathSciNetADSCrossRefGoogle Scholar - 46.Halperin, B.I.: Statistics of quasiparticles and the hierarchy of fractional quantized hall states. Phys. Rev. Lett.
**52**, 1583–1586 (1984)ADSCrossRefGoogle Scholar - 47.Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
- 48.Healey, R.: Nonlocality and the Aharonov–Bohm effect. Philos. Sci.
**64**, 18–41 (1997)MathSciNetCrossRefGoogle Scholar - 49.Healey, R.: On the reality of gauge potentials. Philos. Sci.
**68**(4), 432–455 (2001)MathSciNetCrossRefGoogle Scholar - 50.Healey, R.A.: Gaugin What’s Real: The Conceptual Foundatinos of Contemporary Gauge Theories. Oxford University Press, New York (2007)CrossRefGoogle Scholar
- 51.Huang, W.-H.: Boson–Fermion Transmutation and the Statistics of Anyons. Available via arXiv:hep-th/0308095
- 52.Jackiw, R.: Dynamical symmetry of the magnetic vortex. Ann. Phys.
**201**, 83–116 (1990)MathSciNetADSCrossRefGoogle Scholar - 53.Jackiw, R., Tempelton, S.: How super-renormalizable interactions cure their infrared divergences. Phys. Rev. D
**23**, 2291 (1981)ADSCrossRefGoogle Scholar - 54.Jones, N. J.: Ineliminable Idealizations, Phase Transitions and Irreversibility. Dissertation, Ohio State University (2006)Google Scholar
- 55.Kaplan, I.G.: Symmetry of many-particle systems. In: Bonchev, D., Rouvray, D.H. (eds.) Chemical Group Theory: Introduction and Fundamentals, pp. 209–254. Gordon and Breach Science Publishers, Amsterdam (1994)Google Scholar
- 56.Katanaev, M.O.: On geometric interpretation of the Aharonov–Bohm effect. Russ. Phys. J.
**54**(5), 507–514 (2011)MathSciNetCrossRefGoogle Scholar - 57.Katanaev, M.O.: On geometric interpretation of the Berry phase. Russ. Phys. J.
**54**(10), 1082–1092 (2012)MATHMathSciNetCrossRefGoogle Scholar - 58.Khare, A.: Fractional Statistics and Quantum Theory. World Scientific, Singapore (2005)MATHCrossRefGoogle Scholar
- 59.Kitzing, K.V., Dorda, G., Pepper, M.: New method for high-accuracy determination of the fine-structure constant based on quantized hall. Phys. Rev. Lett.
**45**, 494 (1980)ADSCrossRefGoogle Scholar - 60.Klitzing, K.V.: 25 Years of quantum hall effect (QHE): a personal view on the discovery, physica and application of this quantum effect. In: Douçot, B., Pasquier, V., Duplantier, B., Rivasseau, V. (eds.) he Quantum Hall Effect Poincaré Seminar, vol. 45, pp. 1–23. Birkhäuser, Basel, Berlin (2004)Google Scholar
- 61.Kretzschmar, M.: Aharonov-Bohm scattering of a wave packet of finite extension. Z. Phys.
**185**, 84–96 (1965)MATHADSCrossRefGoogle Scholar - 62.Laidlaw, M.G., DeWitt, C.M.: Feyman functional integrals for system of indistinguishable particles. Phys. Rev. D
**3**, 1375–1378 (1971)ADSCrossRefGoogle Scholar - 63.Landsman, N. P.: Quantization and superselection sectors III: multiply connected spaces and indistinguishable particles. Available via arXiv:1302.3637 (2013)
- 64.Laughlin, R.B.: Quantized motion of three two-dimensional electrons in a strong magnetic field. Phys. Rev. B
**27**, 3383–3389 (1983)ADSCrossRefGoogle Scholar - 65.Laughlin, R.B.: Anomalous quantum hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett.
**50**, 1395–1398 (1983)ADSCrossRefGoogle Scholar - 66.Laughlin, R.B.: Fractional statistics in the quantum hall effect. In: Wilczek, F. (ed.) Fractional Statistics and Anyon Superconductivity. World Scientific, Singapore (1990)Google Scholar
- 67.Leeds, S.: Gauges: Aharonov, Bohm, Yang, Healey. Philos. Sci.
**66**, 606–627 (1999)MathSciNetCrossRefGoogle Scholar - 68.Leinaas, J.M., Myrheim, J.: On the theory of identical particles. Nuovo Cimento B
**37**, 1–23 (1977)ADSCrossRefGoogle Scholar - 69.Lerda, A.: Anyons: Quantum Mechanics of Particles with Fractional Statistics. Springer, Berlin (1992)MATHGoogle Scholar
- 70.Lyre, H.: The principles of gauging. Philos. Sci.
**68**(3), S371–S381 (2001)MathSciNetCrossRefGoogle Scholar - 71.Lyre, H.: Holism and structuralism in U(1) Gauge Theory. Stud. Hist. Philos. Mod. Phys.
**35**, 643–670 (2004)MATHMathSciNetCrossRefGoogle Scholar - 72.Magni, C., Valz-Gris, F.: Can elementary quantum mechanics explain the Aharonov–Bohm effect? J. Math. Phys.
**36**(1), 177–186 (1995)MATHMathSciNetADSCrossRefGoogle Scholar - 73.Mattingly, J.: Which gauge matters. Stud. Hist. Philos. Mod. Phys.
**37**, 243–262 (2006)MATHMathSciNetCrossRefGoogle Scholar - 74.Maudlin, T.: Discussion: healey on the Aharonov–Bohm effect. Philos. Sci.
**65**, 361–368 (1998)MathSciNetCrossRefGoogle Scholar - 75.Messiah, A.M., Greenberg, O.W.: Symmetrization postulate and its experimental foundation. Phys. Rev. B
**136**, 248–267 (1964)MathSciNetADSCrossRefGoogle Scholar - 76.Morandi, G.: The Role of Topology in Classical and Quantum Mechanics. Springer, Berlin (1992)Google Scholar
- 77.Morandi, G., Menossie, E.: Path-integrals in multiply connected spaces and the Aharonov–Bohm effect. Eur. J. Phys.
**5**, 49–58 (1984)CrossRefGoogle Scholar - 78.Morrison, M.: Emergence, reduction and theoretical principles: rethinking fundamentalism. Philos. Sci.
**73**, 876–887 (2006)CrossRefGoogle Scholar - 79.Mourik, V., Zuo, K., Frolov, S.M., Plissard, S.R., Bakkers, E.P.A.M., Kouwenhoven, L.P.: Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices. Science
**336**, 1003–1007 (2012)ADSCrossRefGoogle Scholar - 80.Munkres, J.R.: Topology. Prentice Hall, Upper Saddle River (2000)MATHGoogle Scholar
- 81.Nakhara, M.: Geometry. Topology and Physics. Institute of Physics Publishing, Bristol (1990)CrossRefGoogle Scholar
- 82.Nash, C., Sen, S.: Topology and Geometry for Physicists. Academic, New York (1983)MATHGoogle Scholar
- 83.Nayak, C., Simon, S., Stern, A., Freedman, M., Das Sarma, S.: Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys
**80**, 1083 (2008)MATHMathSciNetADSCrossRefGoogle Scholar - 84.Norton, J.D.: Approximations and idealizations: why the difference matters. Philos. Sci.
**79**, 207–232 (2012)MathSciNetCrossRefGoogle Scholar - 85.Nounou, A.M.: A fourth way to the Aharonov–Bohm effect. In: Bradind, K., Castellani, E. (eds.) Symmetries in Physics: Philosophical Replections. Cambridge University Press, Cambridge (2003)Google Scholar
- 86.Pachos, J.K.: Introduction to Topological Quantum Computation. Cambridge University Press, Cambridge (2012)MATHCrossRefGoogle Scholar
- 87.Pauli, W.: The connection between spin and statistics. Phys. Rev.
**58**, 716–722 (1940)ADSCrossRefGoogle Scholar - 88.Peshkin, M., Tonomura, A.: The Aharonov–Bohm Effect. Lecture Notes in Physics, vol. 340. Springer, Berlin (1989)CrossRefGoogle Scholar
- 89.Prange, R.E., Girvin, S. (eds.): The Quantum Hall Effect. Springer, New York (1987)Google Scholar
- 90.Rao, S.: An Anyon Primer. Available via arXiv:hep-th/9209066 (2001)
- 91.Roy, S.M.: Condition for nonexistence of Aharonov–Bohm effect. Phys. Rev. Lett.
**44**, 111–114 (1980)MathSciNetADSCrossRefGoogle Scholar - 92.Ruetsche, L.: Interpreting Quantum Theories: The Art of the Possible. Oxford University Press, Oxford (2011)CrossRefGoogle Scholar
- 93.Ryder, L.H.: Quantum Field Theory. Cambridge University Press, Cambridge (1996)MATHCrossRefGoogle Scholar
- 94.Schulman, L.: A path integral for spin. Phys. Rev.
**176**, 1558–1569 (1971)MathSciNetADSCrossRefGoogle Scholar - 95.Shapere, A., Wilczek, F. (eds.): Geometric Phases in Physics. World Scientific, Singapore (1989)MATHGoogle Scholar
- 96.Shech, E.: Scientific misrepresentation and guides to ontology: the need for representational code and contents. Synthese. (2014). doi: 10.1007/s11229-014-0506-2
- 97.Shech, E.: Assume a Spherical Cow: Studies on Representation and Idealizations. Doctoral Dissertation. University of Pittsburgh, Pittsburgh (2015)Google Scholar
- 98.Shech, E.: What is the paradox of phase transitions? Philos. Sci.
**80**, 1170–1181 (2013)MathSciNetCrossRefGoogle Scholar - 99.Shrivastava, K.N.: Quantum Hall Effect: Expressions. Nova Science Publishers, New York (2005)Google Scholar
- 100.Stern, A.: Anyons and the quantum hall effect-a pedagogical review. Ann. Phys.
**323**, 204–249 (2008)MATHADSCrossRefGoogle Scholar - 101.Tonomura, A.: Electron Holography. Springer, Berlin (1999)MATHCrossRefGoogle Scholar
- 102.Tonomura, A.: The AB effect and its expanding applications. J. Phys. A
**43**, 1–13 (2010)CrossRefGoogle Scholar - 103.Tonomura, A., Osakabe, N., Matsuda, T., Kawasaki, T., Endo, J., Yano, S., Yamada, H.: Observation of the Aharonov–Bohm effect by electron holography. Phys. Rev. Lett.
**48**, 1443 (1982)ADSCrossRefGoogle Scholar - 104.Tonomura, A., Osakabe, N., Matsuda, T., Kawasaki, T., Endo, J., Yano, S., Yamada, H.: Evidence for Aharonov–Bohm effect with magnetic field completely shielded from electron wave. Phys. Rev. Lett.
**56**, 792–795 (1986)ADSCrossRefGoogle Scholar - 105.Tsui, D.C., Stormer, H.L., Gossard, A.C.: Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett.
**48**(22), 1559 (1982)ADSCrossRefGoogle Scholar - 106.Weisskopf, V.F. In: Brittin, W.E. (ed.) Lectures in Theoretical Physics, vol. III, pp. 67–70. Interscience, New York (1961)Google Scholar
- 107.Wilczek, F.: Magnetic flux, angular momentum and statistics. Phys. Rev. Lett.
**48**, 1144–1146 (1982)MathSciNetADSCrossRefGoogle Scholar - 108.Wilczek, F.: Quantum mechanics of fractional-spin particles. Phys. Rev. Lett.
**49**, 957–959 (1982)MathSciNetADSCrossRefGoogle Scholar - 109.Wilczek, F. (ed.): Fractional Statistics and Anyon Superconductivity. World Scientific, Singapore (1990)Google Scholar
- 110.Wilczek, F., Zee, A.: Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett.
**52**, 2111–2114 (1984)MathSciNetADSCrossRefGoogle Scholar - 111.Wu, Y.S.: General theory for quantum statistics in two dimensions. Phys. Rev. Lett.
**52**, 2103–2106 (1984)MathSciNetADSCrossRefGoogle Scholar - 112.Wu, T.T., Yang, C.N.: Concept of nonintegrable phase factors and global formulation of gauge fields. Phys. Rev. D
**12**, 3845 (1975)MathSciNetADSCrossRefGoogle Scholar - 113.Yoshioka, D.: Quantum Hall Effect. Springer, Berlin (2002)MATHCrossRefGoogle Scholar