Few-Body Systems

, Volume 56, Issue 6–9, pp 607–613 | Cite as

Solvable Models with Massless Light-Front Fermions

  • L’ubomír Martinovic̆Email author
  • Pierre Grangé


Two-dimensional models with massless fermions (Thirring model, Thirring–Wess and Schwinger model, among others) have been solved exactly a long time ago in the conventional (space-like) form of field theory and in some cases also in the conformal field theoretical approach. However, solutions in the light-front form of the theory have not been obtained so far. The primary obstacle is the apparent difficulty with light-front quantization of free massless fermions, where one half of the fermionic degrees of freedom seems to “disappear” due to the structure of a non-dynamical constraint equation. We shall show a simple way how the missing degree of freedom can be recovered as the massless limit of the massive solution of the constraint. This opens the door to the genuine light front solution of the above models since their solvability is related to free Heisenberg fields, which are the true dynamical variables in these models. In the present contribution, we give an operator solution of the light front Thirring model, including the correct form of the interacting quantum currents and of the Hamiltonian. A few remarks on the light-front Thirring–Wess models are also added. Simplifications and clarity of the light-front formalism turn out to be quite remarkable.


Dirac Equation Solvable Model Operator Solution Massive Solution Massless Limit 
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.
    Wightman, A.S.: Introduction to Some Aspects of the Relativistic Dynamics of Quantized Fields. In: Cargése Lectures in Theoretical Physics, Gordon and Breach, pp. 171–291. New York (1967)Google Scholar
  2. 2.
    Abdalla E., Abdalla M.C.B., Rothe K.D.: Nonperturbative Methods in Two-Dimensional Quantum Field Theory. World Scientific, Singapore (1991)CrossRefGoogle Scholar
  3. 3.
    DiFrancesco P., Mathieu P., Senechal D.: Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, Berlin (1997)Google Scholar
  4. 4.
    Schroer B.: Infrateilchen in quantenfeldtheorie. Fort. der Phys. 11, 1–31 (1963)MathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Thirring, W.: A soluble relativistic field theory? Ann. Phys. 3, 91–112 (1958)Google Scholar
  6. 6.
    Thirring W., Wess J.: Solution of a field theory model in one time and one space dimension. Ann. Phys. 27, 331–337 (1964)MathSciNetCrossRefADSzbMATHGoogle Scholar
  7. 7.
    Federbush K.: A two-dimensional relativistic field theory. Phys. Rev. 121, 1247–1249 (1961)MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    Schwinger J.: Gauge invariance and mass II. Phys. Rev. 128, 2425–2429 (1962)MathSciNetCrossRefADSzbMATHGoogle Scholar
  9. 9.
    Dirac P.A.M.: Forms od relativistic dynamics. Rev. Mod. Phys. 21, 392–399 (1949)MathSciNetCrossRefADSzbMATHGoogle Scholar
  10. 10.
    Chang S.-J., Root R.G., Yan T.-M.: Quantum field theories in the infinite momentum frame. 1. Quantization of scalar and Dirac fields. Phys. Rev. D 7, 1133–1148 (1973)CrossRefADSGoogle Scholar
  11. 11.
    McCartor G.: Schwinger model in the light cone representation. Z. Phys. C 64, 349–354 (1994)MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    McCartor G., Pinsky S.S., Robertson D.G.: Vacuum structure of two-dimensional gauge theories on the light front. Phys. Rev. D 56, 1035–1049 (1997)MathSciNetCrossRefADSGoogle Scholar
  13. 13.
    Martinovic L., Grangé P.: Hamiltonian formulation of exactly solvable models and their physical vacuum states. Phys. Lett. B 724, 310–315 (2013)MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Martinovic L.: Solvable models in the conventional and light-front field thory: recent progress. Few Body Syst. 55, 527–534 (2014)CrossRefADSGoogle Scholar
  15. 15.
    Grangé P., Ullrich P., Werner E.: The continuum version of the ϕ 4(1 + 1) theory in light front quantization. Phys. Rev. D 57, 4981–4989 (1998)CrossRefADSGoogle Scholar
  16. 16.
    Salmons S., Grangé P., Werner E.: Field dynamics on the light cone: compact versus continuum quantization. Phys. Rev. D 60, 067701–067709 (1998)CrossRefADSGoogle Scholar
  17. 17.
    Leutwyler H., Klauder J.R., Streit L.: Quantum field theory on light like slabs. Nuovo Cim. A 66, 536–554 (1970)MathSciNetCrossRefADSzbMATHGoogle Scholar
  18. 18.
    Bergknoff T.: Physical particles of the massive Schwinger model. Nucl. Phys. B 122, 215–229 (1977)CrossRefADSGoogle Scholar
  19. 19.
    Dell-Antonio G.F., Frishman Y., Zwanziger D.: Thirring model in terms of currents: solution and light cone expansions. Phys. Rev. D 6, 988–1007 (1972)CrossRefADSGoogle Scholar
  20. 20.
    Johnson K.: Solution of the equations for the Green’s functions of a two-dimensional relativistic field theory. Nuovo Cim. 20, 773–790 (1961)CrossRefGoogle Scholar
  21. 21.
    Klaiber, B.: The Thirring model. In: Lectures in Theoretical Physics, vol. XA, pp. 141–176. New York (1968)Google Scholar

Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  1. 1.Institute of PhysicsSlovak Academy of SciencesBratislavaSlovakia
  2. 2.Bogoliubov Laboratory of Theoretical PhysicsJINRDubnaRussia
  3. 3.LUPMUniversité Montpellier IIMontpellier Cedex 05France

Personalised recommendations