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Journal of Statistical Physics

, Volume 26, Issue 2, pp 347–364 | Cite as

States of one-dimensional Coulomb systems as simple examples of θ vacua and confinement

  • Michael Aizenman
  • Jürg Fröhlich
Articles

Abstract

The one-dimensional Coulomb system is known to have equilibrium states with nonvanishing electric field. These states are shown here to be analogous, and related, to theθ vacua which have been discussed for gauge theories in two or more space-time dimensions. The system exhibits confinement of fractional charges, which we dicuss with the purpose of offering a simple example of theθ-vacua phenomenology. Precise relations and connections between one-dimensional Coulomb gases and two-dimensional Abelian gauge theories, and quantum-mechanical matter systems, are discussed.

Key words

Coulomb systems θ vacua confinement one-dimensional model 

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References

  1. 1.
    M. Aizenman and Ph. A. Martin,Commun. Math. Phys.,78:99 (1980).Google Scholar
  2. 2.
    H. Kunz,Ann. Phys. (N.Y.) 85:303 (1974).Google Scholar
  3. 3.
    H. J. Brascamp and E. H. Lieb, inFunctional Integration and its Applications, A. M. Arthurs, ed. (Clarendon Press, Oxford, 1979).Google Scholar
  4. 4.
    S. Coleman, R. Jackiw, and L. Susskind,Ann. Phys. (N.Y.) 93:267 (1975); S. Callan, R. Dashen, and D. Gross,Phys. Lett. 63B:334 (1976); R. Jackiw and C. Rebbi,Phys. Rev. Lett. 37:172 (1976).Google Scholar
  5. 5.
    A. Lenard,J. Math. Phys. 4:533 (1963).Google Scholar
  6. 6.
    A. J. F. Siegert,Physica 26:530 (1960); S. Edwards and A. Lenard,J. Math. Phys. 3:778 (1962).Google Scholar
  7. 7.
    A. Lenard,J. Math. Phys. 2:682 (1961).Google Scholar
  8. 8.
    S. Prager, inAdvances in Chemical Physics IV, (Wiley-Interscience, New York, 1962), p. 201.Google Scholar
  9. 9.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. IV, (Academic Press, New York, 1978), Sec. XIII.16.Google Scholar
  10. 10.
    J. Fröhlich and Y. M. Park,Commun. Math. Phys.,59:235 (1978).Google Scholar
  11. 11.
    J. Fröhlich and T. Spencer,J. Stat. Phys., in press.Google Scholar
  12. 12.
    D. Brydges, J. Fröhlich, and E. Seiler,Nucl. Phys. B 152:521 (1979).Google Scholar
  13. 13.
    J. Fröhlich and B. Simon,Ann Math. 105:493 (1977).Google Scholar
  14. 14.
    J. Fröhlich, R. Israel, E. H. Lieb, and B. Simon,Commun. Math. Phys. 62:1 (1978).Google Scholar
  15. 15.
    B. Simon:The-P(φ)2 Euclidean (Quantum) Field Theory, (Princeton University Press, Princeton, New Jersey 1974).Google Scholar
  16. 16.
    D. Brydges,Commun. Math. Phys. 58:313 (1978).Google Scholar
  17. 17.
    K. Osterwalder and R. Schrader,Commun. Math. Phys. 31:83 (1973);42:281 (1975).Google Scholar
  18. 18.
    S. Coleman,Phys. Rev. D 11:2088 (1975).Google Scholar
  19. 19.
    J. Fröhlich and E. Seiler,Helv. Phys. Acta 49:889 (1976).Google Scholar
  20. 20.
    D. H. Weingarten and J. L. Challifour,Ann. Phys. (N.Y.) 123:61 (1979).Google Scholar
  21. 21.
    D. Brydges, J. Fröhlich, and E. Seiler,Ann. Phys. (N. Y.) 121:227 (1979);Commun. Math. Phys. 71:159 (1980);Commun Math. Phys. 79:353 (1981).Google Scholar
  22. 22.
    S. Colemanet. al, see Ref. 4; S. Coleman,Ann Phys. (N.Y.) 101:239 (1976).Google Scholar
  23. 23.
    R. Israel and C. Nappi,Commun. Math. Phys. 68:29 (1979).Google Scholar
  24. 24.
    E. Witten,Phys. Lett. 86B:283 (1979); B. Durhuus and J. Fröhlich,Commun. Math. Phys. 75:103 (1980).Google Scholar
  25. 25.
    J. Fröhlich,Acta Phys. Austr., Suppl. XV:133 (1976).Google Scholar
  26. 26.
    C. Callan, R. Dashen, and D. Gross, inQuantum Chromodynamics (La Jolla Institute, 1978), W. Frazer and F. Henyey, eds., AIP Conference Proceeding 55 (American Institute of Physics, New York, 1979); S. Coleman, in Proc. International School of Subnuclear Physics (Ettore Majorana, 1977) A. Zichichi, ed.; M. Lüscher,Phys. Lett. 78B:456 (1978).Google Scholar
  27. 27.
    H. P. McKean and E. Trubowitz,Commun. Pure Appl. Math. 29:143 (1976).Google Scholar
  28. 28.
    E. I. Dinaburg and Y. G. Sinai,Funct. Anal. Appl. 9:279 (1975).Google Scholar
  29. 29.
    H. Rüssmann,Commun. Pure Appl. Math. 29:755 (1976).Google Scholar
  30. 30.
    S. Aubrey, Saclay preprint (1979).Google Scholar

Copyright information

© Plenum Publishing Corporation 1981

Authors and Affiliations

  • Michael Aizenman
    • 1
  • Jürg Fröhlich
    • 2
  1. 1.Department of PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance

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