Foundations of Physics

, Volume 40, Issue 7, pp 776–792 | Cite as

Connecting Spin and Statistics in Quantum Mechanics

  • Arthur JabsEmail author


The spin-statistics connection is derived in a simple manner under the postulates that the original and the exchange wave functions are simply added, and that the azimuthal phase angle, which defines the orientation of the spin part of each single-particle spin-component eigenfunction in the plane normal to the spin-quantization axis, is exchanged along with the other parameters. The spin factor (−1)2s belongs to the exchange wave function when this function is constructed so as to get the spinor ambiguity under control. This is achieved by effecting the exchange of the azimuthal angle by means of rotations and admitting only rotations in one sense. The procedure works in Galilean as well as in Lorentz-invariant quantum mechanics. Relativistic quantum field theory is not required.


Spin and statistics Spinor Spinor ambiguity Bose and Fermi statistics Pauli exclusion principle Symmetrization 


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  1. 1.
    Fierz, M.: Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin. Helv. Phys. Acta. 12, 3–37 (1939) CrossRefGoogle Scholar
  2. 2.
    Pauli, W.: The connection between spin and statistics. Phys. Rev. 58, 716–722 (1940) zbMATHCrossRefADSGoogle Scholar
  3. 3.
    Jost, R.: Das Pauli-Prinzip und die Lorentz-Gruppe. In: Fierz, M., Weisskopf, V.F. (eds.) Theoretical Physics in the Twentieth Century, pp. 107–136. Interscience, New York (1960) Google Scholar
  4. 4.
    Duck, I., Sudarshan, E.C.G.: Pauli and the Spin-Statistics Theorem. World Scientific, Singapore (1997) zbMATHGoogle Scholar
  5. 5.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. III. Addison-Wesley, Reading (1965) Google Scholar
  6. 6.
    About half of these publications are accessible via the internet under [Title: spin AND statistics]; the others can be traced back from these
  7. 7.
    Hilborn, R.C.: Answer to Question #7 [“The spin-statistics theorem,” Dwight E. Neuenschwander, Am. J. Phys. 62 (11), 972 (1994)]. Am. J. Phys. 63, 298–299 (1995) CrossRefADSGoogle Scholar
  8. 8.
    Duck, I., Sudarshan, E.C.G.: Toward an understanding of the spin-statistics theorem. Am. J. Phys. 66, 284–303 (1998) CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Romer, R.H.: The spin-statistics theorem. Am. J. Phys. 70, 791 (2002) CrossRefADSGoogle Scholar
  10. 10.
    Morgan, J.A.: Spin and statistics in classical mechanics. Am. J. Phys. 72, 1408–1417 (2004). arXiv:quant-ph/0401070 CrossRefADSGoogle Scholar
  11. 11.
    Broyles, A.A.: Derivation of the Pauli exchange principle. arXiv:quant-ph/9906046
  12. 12.
    Broyles, A.A.: Spin and statistics. Am. J. Phys. 44, 340–343 (1976) CrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Peshkin, M.: Reply to “Non-relativistic proofs of the spin-statistics connection”, by Shaji and Sudarshan. arXiv:quant-ph/0402118
  14. 14.
    Peshkin, M.: Reply to “Comment on ‘Spin and statistics in nonrelativistic quantum mechanics: The spin-zero case'''. Phys. Rev. A 68, 046102 (2003) CrossRefADSGoogle Scholar
  15. 15.
    Peshkin, M.: Reply to “No spin-statistics connection in nonrelativistic quantum mechanics”. arXiv:quant-ph/0306189
  16. 16.
    Peshkin, M.: Spin and statistics in nonrelativistic quantum mechanics: The spin-zero case. Phys. Rev. A 67, 042102 (2003) CrossRefADSGoogle Scholar
  17. 17.
    Peshkin, M.: On spin and statistics in quantum mechanics. arXiv:quant-ph/0207017
  18. 18.
    Morgan, J.A.: Demonstration of the spin-statistics connection in elementary quantum mechanics. arXiv:physics/0702058
  19. 19.
    Kuckert, B.: Spin and statistics in nonrelativistic quantum mechanics, I. Phys. Lett. A 322, 47–53 (2004) zbMATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Donth, E.: Ein einfacher nichtrelativistischer Beweis des Spin-Statistik-Theorems und das Verhältnis von Geometrie und Physik in der Quantenmechanik. Wissenschaftl. Z. Tech. Hochsch. “Carl Schorlemmer” Leuna-Merseburg 19, 602–606 (1977) Google Scholar
  21. 21.
    Donth, E.: Non-relativistic proof of the spin statistics theorem. Phys. Lett. A 32, 209–210 (1970) CrossRefADSGoogle Scholar
  22. 22.
    Kuckert, B., Mund, J.: Spin & statistics in nonrelativistic quantum mechanics, II. Ann. Phys. (Leipz.) 14, 309–311 (2005). arXiv:quant-ph/0411197 zbMATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Bacry, H.: Answer to Question #7 [“The spin-statistics theorem,” Dwight E. Neuenschwander, Am. J. Phys. 62 (11), 972 (1994)]. Am. J. Phys. 63, 297–298 (1995) CrossRefADSGoogle Scholar
  24. 24.
    Bacry, H.: Introduction aux concepts de la physique statistique, pp. 198–200. Ellipses, Paris (1991) Google Scholar
  25. 25.
    Piron, C.: Mécanique quantique, Bases et applications, pp. 166–167. Presses polytechniques et universitaires romandes, Lausanne (1990) zbMATHGoogle Scholar
  26. 26.
    Balachandran, A.P., Daughton, A., Gu, Z.-C., Sorkin, R.D., Marmo, G., Srivastava, A.M.: Spin-statistics theorems without relativity or field theory. Int. J. Modern Phys. A 8, 2993–3044 (1993) zbMATHCrossRefMathSciNetADSGoogle Scholar
  27. 27.
    Feynman, R.P.: The reason for antiparticles. In: Feynman, R.P., Weinberg, S. (eds.) Elementary Particles and the Laws of Physics, pp. 1–59, especially pp. 56–59. Cambridge University Press, Cambridge (1987) Google Scholar
  28. 28.
    York, M.: Symmetrizing the symmetrization postulate. In: Hilborn, R.C., Tino, G.M. (eds.) Spin-Statistics Connection and Commutation Relations, pp. 104–110. American Institute of Physics, Melville (2000). arXiv:quant-ph/0006101 Google Scholar
  29. 29.
    York, M.: Identity, geometry, permutation, and the spin-statistics theorem. arXiv:quant-ph/9908078
  30. 30.
    Jabs, A.: Quantum mechanics in terms of realism. Phys. Essays 9, 36–95 (1996). arXiv:quant-ph/9606017 CrossRefMathSciNetGoogle Scholar
  31. 31.
    Jabs, A.: An interpretation of the formalism of quantum mechanics in terms of epistemological realism. Br. J. Philos. Sci. 43, 405–421 (1992) CrossRefMathSciNetGoogle Scholar
  32. 32.
    Cohen-Tannoudji, C., Diu, B., Laloë, F.: Quantum Mechanics, vols. I, II. Wiley, New York (1977) Google Scholar
  33. 33.
    Bjorken, J.D., Drell, S.D.: Relativistic Quantum Mechanics, p. 136. McGraw-Hill, New York (1964) Google Scholar
  34. 34.
    Feynman, R.P.: Quantum Electrodynamics, pp. 124–125. Benjamin, Elmsford (1961). The argumentation formulated here in quantum electrodynamics carries over to quantum mechanics. This holds also for equivalent formulations in many books on quantum field theory Google Scholar
  35. 35.
    Knopp, K.: Funktionentheorie, second part, pp. 90–91. de Gruyter, Berlin (1955). English translation by Bagemihl, F.: Theory of Functions, part II, pp. 101–103. Dover, New York (1996) Google Scholar
  36. 36.
    Weyl, H.: The Theory of Groups and Quantum Mechanics, p. 184. Dover, New York (1950) Google Scholar
  37. 37.
    von Foerster, T.: Answer to Question #7 [“The spin-statistics theorem,” Dwight E. Neuenschwander, Am. J. Phys. 62 (11), 972 (1994)]. Am. J. Phys. 64(5), 526 (1996) CrossRefADSGoogle Scholar
  38. 38.
    Gould, R.R.: Answer to Question #7 [“The spin-statistics theorem,” Dwight E. Neuenschwander, Am. J. Phys. 62 (11), 972 (1994)]. Am. J. Phys. 63(2), 109 (1995) CrossRefADSGoogle Scholar
  39. 39.
    Penrose, R., Rindler, W.: Spinors and Space-Time, vol. 1, p. 43. Cambridge University Press, Cambridge (1984) zbMATHGoogle Scholar
  40. 40.
    Biedenharn, L.C., Louck, J.D.: Angular Momentum in Quantum Physics. Addison-Wesley, Reading (1981). Chapter 2 zbMATHGoogle Scholar
  41. 41.
    Hartung, R.W.: Pauli principle in Euclidean geometry. Am. J. Phys. 47(10), 900–910 (1979) CrossRefMathSciNetADSGoogle Scholar
  42. 42.
    Rieflin, E.: Some mechanisms related to Dirac’s strings. Am. J. Phys. 47(4), 378–381 (1979) CrossRefADSGoogle Scholar
  43. 43.
    Schrödinger, E.: Die Mehrdeutigkeit der Wellenfunktion. Ann. Phys. (Leipz.) 32(5), 49–55 (1938). Reprinted in: Schrödinger, E.: Collected Papers, vol. 3, pp. 583–589. Verlag der Österreichischen Akademie der Wissenschaften, Wien (1984) CrossRefGoogle Scholar
  44. 44.
    van Winter, C.: Orbital angular momentum and group representations. Ann. Phys. (New York) 47, 232–274 (1968) zbMATHCrossRefADSGoogle Scholar
  45. 45.
    Altmann, S.L.: Rotations, Quaternions, and Double Groups. Dover, New York (2005). Chapter 10 Google Scholar
  46. 46.
    Pauli, W.: Die allgemeinen Prinzipien der Wellenmechanik. In: Geiger, H., Scheel, K. (eds.) Handbuch der Physik, vol. 24, part 1, pp. 189–193. Springer, Berlin (1933). English translation of a 1958 reprint: Achutan, P., Venkatesan, K.: General Principles of Quantum Mechanics, pp. 116–121. Springer, Berlin (1980) Google Scholar
  47. 47.
    Jacob, M., Wick, G.C.: On the general theory of collisions for particles with spin. Ann. Phys. (New York) 7, 404–428 (1959) zbMATHCrossRefMathSciNetADSGoogle Scholar

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Authors and Affiliations

  1. 1.AlumnusTechnical University BerlinBerlinGermany

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