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Spin-Statistics Transmutation in Quantum Field Theory

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Abstract

Spin-statistics transmutation is the phenomenon occurring when a “dressing” transformation introduced for physical reasons (e.g. gauge invariance) modifies the “bare” spin and statistics of particles or fields. Historically, it first appeared in Quantum Mechanics and in semiclassical approximation to Quantum Field Theory. After a brief historical introduction, we sketch how to describe such phenomenon in Quantum Field Theory beyond the semiclassical approximation, using a path-integral formulation of euclidean correlation functions, exemplifying with anyons, dyons and skyrmions.

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References

  1. Tamm, I.: Die verallgemeinerten Kugelfunktionen und die Wellenfunktionen eines Elektrons im Felde eines Magnetpoles. Z. Phys. 71, 141–150 (1931)

    Article  MATH  ADS  Google Scholar 

  2. Dirac, P.A.M.: Quantized Singularities in the Electromagnetic Field. Proc. R. Soc. A 133, 60–72 (1931)

    Article  MATH  ADS  Google Scholar 

  3. Hurst, C.A.: Charge quantization and nonintegrable Lie algebras. Ann. Phys. 50, 51–75 (1968)

    Article  MATH  ADS  Google Scholar 

  4. Jackiw, R., Rebbi, C.: Solitons with Fermion number 1/2. Phys. Rev. D 13, 3398–3409 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  5. Jackiw, R., Rebbi, C.: Spin from isospin in a gauge theory. Phys. Rev. Lett. 36, 1116–1119 (1976)

    Article  ADS  Google Scholar 

  6. Hasenfratz, P., ’t Hooft, G.: A Fermion-Boson puzzle in a gauge theory. Phys. Rev. Lett. 36, 1119–1122 (1976)

    Article  ADS  Google Scholar 

  7. Goldhaber, A.: Connection of spin and statistics for charge-monopole composites. Phys. Rev. Lett. 36, 1122–1125 (1976)

    Article  ADS  Google Scholar 

  8. Friedman, J.L., Sorkin, R.D.: Dyon spin and statistics: a fiber bundle theory of interacting magnetic and electric charges. Phys. Rev. D 20, 2511–2525 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  9. Friedman, J.L., Sorkin, R.D.: A spin statistics theorem for composites containing both electric and magnetic charges. Commun. Math. Phys. 73, 161–196 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  10. Wilczek, F.: Magnetic flux, angular momentum, and statistics. Phys. Rev. Lett. 48, 1144–1146 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  11. Wilczek, F.: Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957–959 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  12. Polyakov, A.M.: Fermi-Bose transmutations induced by gauge fields. Mod. Phys. Lett. A 3, 325–328 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  13. Wilczek, F., Zee, A.: Linking numbers, spin, and statistics of solitons. Phys. Rev. Lett. 51, 2250–2252 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  14. Finkelstein, D., Rubinstein, J.: Connection between spin, statistics, and kinks. J. Math. Phys. 9, 1762–1779 (1968)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Kauffman, L.H.: On Knots. Ann. Math. Studies, vol. 115. Princeton University Press, Princeton (1987)

    MATH  Google Scholar 

  16. Ehrenberg, W., Siday, R.E.: The refractive index in electron optics and the principles of dynamics. Proc. Phys. Soc. B 62, 8–21 (1949) ;

    Article  ADS  Google Scholar 

  17. Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in quantum theory. Phys. Rev. 115, 485–491 (1959)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Bargmann, V.: Irreducible unitary representations of the Lorentz group. Ann. Math. 48, 568–640 (1947)

    Article  MathSciNet  Google Scholar 

  19. Birman, J.: Braids, Links and Mapping Class Group. Ann. Math. Studies, vol. 82. Princeton University Press, Princeton (1974)

    Google Scholar 

  20. Leinaas, J.M., Myrheim, J.: On the theory of identical particles. Nuovo Cimento B 37, 1–23 (1977)

    Article  ADS  Google Scholar 

  21. Goldin, G.A., Menikoff, R., Sharp, D.H.: Representations of a local current algebra in nonsimply connected space and the Aharonov-Bohm effect. J. Math. Phys. 22, 1664–1668 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  22. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  23. Skyrme, T.H.R.: A unified field theory of mesons and baryons. Nucl. Phys. 31, 556–569 (1962)

    Article  MathSciNet  Google Scholar 

  24. Witten, E.: Global aspects of current algebra. Nucl. Phys. B 223, 422–432 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  25. Witten, E.: Current algebra, baryons, and quark confinement. Nucl. Phys. B 223, 433–444 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  26. Feynman, R.P.: Space-time approach to nonrelativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948)

    Article  MathSciNet  ADS  Google Scholar 

  27. Schwinger, J.: On gauge invariance and vacuum polarization. Phys. Rev. 82, 664–679 (1951)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Dirac, P.A.M.: Gauge invariant formulation of quantum electrodynamics. Can. J. Phys. 33, 650–660 (1955)

    MATH  MathSciNet  Google Scholar 

  29. Strocchi, F.: Spontaneous symmetry breaking in local gauge quantum field theory: the Higgs mechanism. Commun. Math. Phys. 56, 57–78 (1977)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. Morchio, G., Strocchi, F.: Infrared problem, Higgs phenomenon and long range interactions. In: Velo, G., et al. (eds.) Fundamental Problems of Gauge Field Theory (Erice 1985), pp. 301–344. Plenum, New York (1986)

    Google Scholar 

  31. Becchi, C., Rouet, A., Stora, R.: The Abelian Higgs-Kibble model. Unitarity of the S operator. Phys. Lett. B 52, 344–346 (1974)

    Article  ADS  Google Scholar 

  32. Becchi, C., Rouet, A., Stora, R.: Renormalization of gauge theories. Ann. Phys. 98, 287–321 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  33. Tyutin, I.V.: Gauge invariance in field theory and statistical mechanics. Lebedev preprint FIAN39 (1975)

  34. Kugo, T., Ojima, I.: Local covariant operator formalism of nonAbelian gauge theories and quark confinement problem. Prog. Theor. Phys. Suppl. 66, 1–130 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  35. Froehlich, J., Marchetti, P.A.: Superselection sectors in quantum field models: Kinks in Phi4 in two-dimensions and charged states in lattice (Q.E.D.) in four-dimensions. In: Kastler, D. (ed.) Algebraic Theory of Superselection Sectors. Introduction and Recent Results, pp. 428–484. World Scientific, Singapore (1990)

    Google Scholar 

  36. Candlin, D.J.: On sums over trajactories for systems with Fermi statistics. Nuovo Cimento 4, 231–239 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  37. Berezin, A.: The Method of Second Quantization. Academic Press, New York (1966)

    MATH  Google Scholar 

  38. Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions. Commun. Math. Phys. 31, 83–112 (1973)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  39. Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions. 2. Commun. Math. Phys. 42, 281–305 (1975)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  40. Seiler, E.: Gauge Theories as a Problem in Constructive Quantum Field Theory and Statistical Mechanics. Lecture Notes in Physics, vol. 159. Springer, Berlin (1982)

    Google Scholar 

  41. Froehlich, J., Marchetti, P.A.: Quantum field theory of anyons. Lett. Math. Phys. 16, 347–358 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  42. Froehlich, J., Marchetti, P.A.: Quantum field theories of vortices and anyons. Commun. Math. Phys. 121, 177–223 (1989)

    Article  MATH  ADS  Google Scholar 

  43. Lechner, K., Marchetti, P.A.: Spin statistics transmutation in relativistic quantum field theories of dyons. J. High Energy Phys. 0012, 028 (2000). doi:10.1088/1126-6708/2000/12/028

    Article  MathSciNet  ADS  Google Scholar 

  44. Froehlich, J., Marchetti, P.A.: Quantum skyrmions. Nucl. Phys. B 335, 1–22 (1990)

    Article  ADS  Google Scholar 

  45. Kruczenski, M., Oxman, L.: Covariant quantization of the skyrmion. Nucl. Phys. B 488, 513–538 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  46. Laughlin, R.B.: Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983)

    Article  ADS  Google Scholar 

  47. Chakraborty, T., Pietilainen, P.: The Fractional Quantum Hall Effect. Springer, Berlin (1988)

    Google Scholar 

  48. Chern, S.-S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math. 99, 48–69 (1974)

    Article  MathSciNet  Google Scholar 

  49. Marchetti, P.A.: Bosonization and duality in condensed matter systems. In: Barone, A., Devoto, A. (eds.) Proceedings of “Common Trends in Condensed Matter and High Energy Physics”, Chia Laguna 1995, pp. 178–188. Istituto Italiano per gli studi Filosofici, Napoli (2000) (Also in arXiv:hep-th/9511100)

    Google Scholar 

  50. Schonfeld, J.F.: A mass term for three-dimensional gauge fields. Nucl. Phys. B 185, 157–171 (1981)

    Article  ADS  Google Scholar 

  51. Deser, S., Jackiw, R., Templeton, S.: Topologically massive gauge theories. Ann. Phys. 140, 372–411 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  52. Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  53. Seiberg, N., Witten, E.: Electric—magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19–52 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  54. Dirac, P.A.M.: The theory of magnetic poles. Phys. Rev. 74, 817–830 (1948)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  55. Schwinger, J.: Magnetic charge and quantum field theory. Phys. Rev. 144, 1087–1093 (1966)

    Article  MathSciNet  ADS  Google Scholar 

  56. Schwinger, J.: Magnetic charge and the charge quantization condition. Phys. Rev. D 12, 3105–3111 (1975)

    Article  ADS  Google Scholar 

  57. de Rham, G.: Differentiable Manifolds. Forms, Currents, Harmonic Forms. Springer, Berlin (1984)

    MATH  Google Scholar 

  58. Mandelstam, S.: Quantum electrodynamics without potentials. Ann. Phys. 19, 1–24 (1962)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  59. Froehlich, J., Marchetti, P.A.: Gauge invariant charged, monopole and dyon fields in gauge theories. Nucl. Phys. B 511, 770–812 (1999)

    Article  ADS  Google Scholar 

  60. Belavin, V.A., Chernodub, M.N., Polikarpov, M.I.: Numerical study of Froehlich and Marchetti monopole creation operator. Nucl. Phys. Proc. Suppl. 106, 610–612 (2002)

    Article  ADS  Google Scholar 

  61. Froehlich, J., Marchetti, P.A.: Soliton quantization in lattice field theories. Commun. Math. Phys. 112, 343–383 (1987)

    Article  MATH  ADS  Google Scholar 

  62. Froehlich, J., Marchetti, P.A.: Bosonization, topological solitons and fractional charges in two-dimensional quantum field theory. Commun. Math. Phys. 116, 127–173 (1988)

    Article  MATH  ADS  Google Scholar 

  63. Marino, E.C., Schroer, B., Swieca, J.A.: Euclidean functional integral approach for disorder variables and kinks. Nucl. Phys. B 200, 473–497 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  64. Belavin, A.A., Polyakov, A.M., Schwartz, A.S., Tyupkin, Y.S.: Pseudoparticle solution of the Yang-Mills equations. Phys. Lett. B 59, 85–87 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  65. Wess, J., Zumino, B.: Consequences of anomalous Ward identities. Phys. Lett. B 37, 95–97 (1971)

    Article  MathSciNet  ADS  Google Scholar 

  66. Chou, K., Guo, H., Wu, K.: On the gauge invariance and anomaly free condition of Wess-Zumino-Witten effective action. Phys. Lett. B 134, 67–69 (1984)

    Article  MathSciNet  Google Scholar 

  67. Balachandran, A.P., Lizzi, F., Rodgers, V.E.J., Stern, A.: Dibaryons as chiral solitons. Nucl. Phys. B 256, 525–556 (1985)

    Article  ADS  Google Scholar 

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Marchetti, P.A. Spin-Statistics Transmutation in Quantum Field Theory. Found Phys 40, 746–764 (2010). https://doi.org/10.1007/s10701-009-9345-2

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