Communications in Mathematical Physics

, Volume 116, Issue 2, pp 309–342 | Cite as

Confinement, deconfinement and freezing in lattice Yang-Mills theories with continuous time

  • Christian Borgs


In this paper I analyse lattice Yang-Mills theories with continuous time. After a short discussion of more conceptual questions, such as the existence of a Hamilton operator in the infinite volume limit, I study the phase diagram. The existence of a strong coupling/low temperature confinement phase (which was not proven up to now) is established for arbitrary compact groups, continuous or discrete. For discrete compact groups the deconfinement region decomposes into (at least) two phases, which are distinguished by the behaviour of spatial Wilson loops: a deconfinement phase where spatial Wilson loops still show area law behaviour, and a “freezing” phase with perimeter law behaviour for spatial Wilson loops. The methods to prove these results rely on cluster expansion methods, combined with renormalisation ideas.


Continuous Time Compact Group Expansion Method Volume Limit Hamilton Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Borgs, C., Seiler, E.: Lattice Yang-Mills theories at nonzero temperature and the confinement problem. Commun. Math. Phys.91, 329 (1983)Google Scholar
  2. 2.
    Tomboulis, E.T., Yaffe, L.G.: Finite temperatureSU(2) lattice gauge theory. Commun. Math. Phys.100, 313 (1985)Google Scholar
  3. 3.
    Borgs, C.: Area law for spatial Wilson loops in high-temperature lattice gauge theories. Nucl. Phys.B 261, 455 (1985)Google Scholar
  4. 4.
    Tomboulis, E.T., Yaffe, L.G.: Chiral symmetrie restoration at finite temperature. Phys. Rev. Lett.52, 2115 (1984)Google Scholar
  5. 5.
    Kogut, J., Susskind, L.: Hamiltonian formulation of Wilson's lattice gauge theories. Phys. Rev. D11, 395 (1975)Google Scholar
  6. 6.
    Osterwalder, K., Seiler, E.: Gauge field theories on a lattice. Ann. Phys.110, 440 (1978)Google Scholar
  7. 7.
    Seiler, E.: Gauge theories as a problem of constructive quantum field theory and statistical mechanics. Lecture Notes in Physics, Vol. 159. Berlin, Heidelberg, New York: Springer 1982Google Scholar
  8. 8.
    Svetitski, B., Yaffe, L.G.: Critical behaviour at finite temperature confinement transitions. Nucl. Phys. B210 [FS6], 423 (1982)Google Scholar
  9. 9.
    Polyakov, A.: Thermal properties of gauge fields and quark liberation. Phys. Lett.72, 477 (1978)Google Scholar
  10. 10.
    Ginibre, J.: Existence of phase transitions for quantum lattice systems. Commun. Math. Phys.14, 205 (1969)Google Scholar
  11. 11.
    Gruber, C., Kunz, H.: General properties of polymer systems. Commun. Math. Phys.22, 133 (1971)Google Scholar
  12. 12.
    Hugenhotz, N.M.: States and representations in statistical mechanics. In: Mathematics of contempory physics. Streater, R.F. (ed.). New York, London: Academic Press 1972Google Scholar
  13. 13.
    Glimm, J., Jaffe, A.: The λ(φ4)3 quantum field theory without cutoffs. III. The physical vacuum. Acta Math.125, 203 (1970)Google Scholar
  14. 14.
    Simon, B., Yaffe, L.G.: Rigorous perimeter law upper bound on Wilson loops. Phys. Lett.115 B, 145 (1982)Google Scholar
  15. 15.
    Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. 2. New York, London: Academic Press 1975Google Scholar
  16. 16.
    Seiler, E., Simon, B.: Nelson's symmetry and all that in the Yukawa and (φ4)3 theories. Ann. Phys.97, 470 (1976)Google Scholar
  17. 17.
    Brydges, D.: A short course on cluster expansions. In: Critical phenomena, random systems, gauge theories. Osterwalder, K., Stora, R. (eds.). Les Houches 1984. Amsterdam: North-Holland 1986Google Scholar
  18. 18.
    Borgs, C.: Zufallsflächen und Clusterentwicklungen in Gitter-Yang-Mills Theorien. Thesis, University of Munich 1986Google Scholar
  19. 19.
    Mack, G.: Nonperturbative methods. In: Gauge theories of the eighties. Raitio, R., Lindfors, J. (eds.). Lecture Notes in Physics, Vol. 181. Berlin, Heidelberg, New York: Springer 1983Google Scholar
  20. 20.
    Nill, F.: Untersuchungen zur Phasenstruktur abel'scher Higgsmodelle in der Gittereich-theorie. Thesis, University of Munich 1987Google Scholar
  21. 21.
    Behnke, H., Sommer, F.: Theorie der analytischen Funktionen einer Veränderlichen. Grundlehren der math. Wissenschaften, Bd. 77. Berlin, Heidelberg, New York: Springer 1965Google Scholar
  22. 22.
    Borgs, C.: Charged surfaces and the analyticity properties of the string tension in lattice gauge theories. Trebon 1986. J. Stat. Phys.47, 867 (1987)Google Scholar
  23. 23.
    Bricmont, J., Fröhlich, J.: Statistical mechanical methods in particle structure analysis of lattice gauge field theories I–III. Nucl. Phys. B251 [FS13], 517 (1985); Commun. Math. Phys.98, 553 (1985); Nucl. Phys. B280 [FS 18], 385 (1987)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Christian Borgs
    • 1
  1. 1.Theoretische Physik, ETH-HönggerbergZürichSwitzerland

Personalised recommendations