Abstract
We develop the quantization of topological solitons (vortices) in three-dimensional quantum field theory, in terms of the Euclidean region functional integral. We analyze in some detail the vortices of the abelian Higgs model. If a Chern-Simons term is added to the action, the vortices turn out to be “anyons,” i.e. particles with arbitrary real spin and intermediate (Θ) statistics. Localization properties of the interpolating field, scattering theory and spin-statistics connection of anyons are discussed. Such analysis might be relevant in connection with the fractional quantum Hall effect and two-dimensional models of HighT csuperconductors.
Similar content being viewed by others
References
Fröhlich, J., Marchetti, P. A.: Commun. Math. Phys.112, 343 (1987)
Marchetti, P. A.: Europhys. Lett.4, 663 (1987); Fröhlich, J., Marchetti, P.A.: Commun. Math. Phys.116, 127 (1988)
Buchholz, K. Fredenhagen: Commun. Math. Phys.84, 1 (1982)
Wilczek, F.: Phys. Rev. Lett.48, 1144 (1982);49, 957 (1982)
Abrikosov, A. A.: Sov. Phys. JETP5, 1174 (1957)
Laughlin, R. B.: Phys. Rev. Lett.50, 1395 (1983); Halperin, B. I.: Phys. Rev. Lett.52, 1583 (1984); Arovas, D.A., Schrieffer, R., Wilczek, F.: Phys. Rev. Lett.53, 772 (1984); Tao, R., Wu, Y. S.: Rev.B31, 6859 (1985); Thouless, D. J., Wu, Y.S.: Phys. Rev.B31, 1191 (1985); Arovas, D. A., Schrieffer, R., Zee, A.: Nucl. Phys.B251, 117 (1985) see also [7]
Girvin, S. M.: The quantum hall effect, R. E. Prange, S. M. Girvin (eds.) Berlin, Heidelberg, New York: Springer 1987
Wiegman, P. B.: Phys. Rev. Lett.60, 821 (1988); Dzialoshinsky, I. E., Polyakov, A. M., Weigman, P. B.: Phys. Lett.127A, 112 (1988); Polyakov, A. M.: Mod. Phys. Lett.A3, 325 (1988)
Laughlin, R. B.: Superconducting ground state of noninteracting particles obeying fractional statistics. preprint 1988
Jaffe, A. Taubes, C.: Vortices and monopoles. Structure of static gauge theories. Progress in Phys.2. A. Jaffe, D. Ruelle (eds.). Boston Basel, Stuttgart: Birkhäuser 1980
Julia, B., Zee, A.: Phys. Rev.D11, 2227 (1975)
Paul, S. K., Khare A.: Phys. Lett.B174, 420 (1986)
de Vega, H. J.: Phys. Rev.D18, 2932 (1978); Hasenfratz, P.: Phys. Lett.B85, 338 (1979); Schwartz, A. S., Tyupkin, Y. S.: Phys. Lett.B90, 135 (1980)
de Vega, H. J., Schaposnik, F. A.: Phys. Rev. Lett.56, 2564 (1986)
Fröhlich, J.: In: Recent developments in gauge theories. (Cargése 1979). G. t'Hooft et al. (eds.). New York: Plenum Press 1980; Trautman, A.: Prep. Math. Phys.1, 29 (1970); Percacci, R.: Geometry of nonlinear field theories, Singapore: World Scientific 1986
See e.g. N. Steenrod: The topology of fibre bundles, Princeton, NJ: Princeton Math. Series, 1951; Hirzebruch, F.: Topological methods in algebraic geometry, Berlin, Heidelberg, New York: Springer 1978
de Rham, G., Kodaira, K.: Harmonic integrals, Princeton, NJ: Princeton 1953
Allendoerfer, C. B., Eells, J. Jr.: Comment Math. Helv.32, 165 (1958)
Singer, I.M.: Commun. Math. Phys.60, 7 (1978); Narasimhan, N. S., Ramadas, T. R.: Commun. Math. Phys.67, 121 (1979); Asorey, M., Mitter, P. K.: Commun. Math. Phys.80, 43 (1981)
King, C.: Commun. Math. Phys.102, 649 (1986),103, 323 (1986)
Balaban, T. B., Imbrie, J. Z., Jaffe, A.: Commun. Math. Phys.97, 299 (1985);114, 257 (1988); Imbrie, J. Z.: in Les Houches, 1984. In: Critical phenomena, Random systems, Gauge theories, K. Osterwalder, R. Stora (eds.). Amsterdam, New York, Oxford, Tokyo: North Holland 1986
Guth, A.: Phys. Rev.D21, 2291 (1980); Fröhlich, J., Spencer, T.: Commun. Math. Phys.93, 411 (1982)
Lüscher, M.: Commun. Math. Phys.85, 39 (1982); Marchetti, P. A.: unpublished
Balaban, T.: Renormalization group methods in non-abelian gauge theories, preprint HUTMP B134
Seiler, E.: Gauge theories as a problem in constructive quantum field theory and statistical mechanics, Lecture Notes in Physics, vol.159. Berlin, Heidelberg, New York: Springer 1982
Fröhlich, J., Oesterwalder, K., Seiler, E.: Ann. Math.18, 461 (1981)
See e.g. Fröhlich, J.: In: Invariant wave equations. Lecture Notes in Physics vol.73, Berlin, Heidelberg, New York: Springer 1978; Fröhlich, J., Morchio, G., Strocchi, F.: Ann. Phys.119, 241 (1979)
Marchetti, P. A.: Commun. Math. Phys.117, 501 (1988)
Balaban, T., Brydges, D., Imbrie, J., Jaffe, A.: Ann Phys.158, 281 (1984)
Deser, S., Jackiw, R., Templeton, S.: Ann. Phys.140, 372 (1982); Siegel, W.: Nucl. Phys.B156, 135 (1981)
Morchio, G., Strocchi, F.: In: Erice 1985, Fundamental Problems of Gauge Field Theory, G. Velo, A. S. Wightman (eds.). New York, London: Plenum Press 1986
Eguchi, T., Gilkey, P. G., Hanson, A.J.: Prep. Rep.66, 215 (1980)
Doplicher, S., Haag, R., Roberts, J.E.: Commun. Math. Phys.13, 1 (1969);15, 173 (1969);23, 199 (1971);35, 49 (1974)
Fröhlich, J., Morchio, G., Strocchi, F.: Ann. Phys.119, 241 (1979); Fröhlich, J.: Commun. Math. Phys.66, 223 (1979)
Wu, Y. S.: Fractional quantum statistics in two-dimensional systems, In: Proc. 2nd Int. Symp. Foundations of Quantum Mechanics, Tokyo 1986, pp. 171 and refs given there
Bargmann, V.: Ann. Math.48, 568 (1986)
Mackey, G. W.: Ann. Math.55, 101 (1952); see also Wigner, E. P.: Ann. Math.40, 149 (1959); Simms, D. J.: Lie groups and quantum mechanics. Lecture Notes in Mathematics, vol.52. Berlin, Heidelberg, New York: Springer 1968
See e.g. Isham, C. J.: Phys. Lett.B106, 188 (1981)
Becher, P., Joos, H.: Z. Phys.C15, 343 (1982)
Fröhlich, J.: In: Progress in gauge theory. Cargése 1979, (eds.) G. t'Hooft et al. New York: Plenum Press 1980
Marchetti, P. A., Percacci, R.: Lett. Math. Phys.6, 405 (1982)
Buchholz, D., Epstein, H.: Fisika17, 329 (1986)
Author information
Authors and Affiliations
Additional information
Communicated by K. Gawedzki
Rights and permissions
About this article
Cite this article
Fröhlich, J., Marchetti, P.A. Quantum field theories of vortices and anyons. Commun.Math. Phys. 121, 177–223 (1989). https://doi.org/10.1007/BF01217803
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01217803