Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
Abdalla E., Abdalla M.C.B., Rothe K.D.: Nonperturbative Methods in Two-Dimensional Quantum Field Theory. World Scientific, Singapore (1991)
DiFrancesco P., Mathieu P., Senechal D.: Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, Berlin (1997)
Schroer B.: Infrateilchen in quantenfeldtheorie. Fort. der Phys. 11, 1–31 (1963)
Thirring, W.: A soluble relativistic field theory? Ann. Phys. 3, 91–112 (1958)
Thirring W., Wess J.: Solution of a field theory model in one time and one space dimension. Ann. Phys. 27, 331–337 (1964)
Federbush K.: A two-dimensional relativistic field theory. Phys. Rev. 121, 1247–1249 (1961)
Schwinger J.: Gauge invariance and mass II. Phys. Rev. 128, 2425–2429 (1962)
Dirac P.A.M.: Forms od relativistic dynamics. Rev. Mod. Phys. 21, 392–399 (1949)
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)
McCartor G.: Schwinger model in the light cone representation. Z. Phys. C 64, 349–354 (1994)
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)
Martinovic L., Grangé P.: Hamiltonian formulation of exactly solvable models and their physical vacuum states. Phys. Lett. B 724, 310–315 (2013)
Martinovic L.: Solvable models in the conventional and light-front field thory: recent progress. Few Body Syst. 55, 527–534 (2014)
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)
Salmons S., Grangé P., Werner E.: Field dynamics on the light cone: compact versus continuum quantization. Phys. Rev. D 60, 067701–067709 (1998)
Leutwyler H., Klauder J.R., Streit L.: Quantum field theory on light like slabs. Nuovo Cim. A 66, 536–554 (1970)
Bergknoff T.: Physical particles of the massive Schwinger model. Nucl. Phys. B 122, 215–229 (1977)
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)
Johnson K.: Solution of the equations for the Green’s functions of a two-dimensional relativistic field theory. Nuovo Cim. 20, 773–790 (1961)
Klaiber, B.: The Thirring model. In: Lectures in Theoretical Physics, vol. XA, pp. 141–176. New York (1968)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Martinovic̆, L., Grangé, P. Solvable Models with Massless Light-Front Fermions. Few-Body Syst 56, 607–613 (2015). https://doi.org/10.1007/s00601-015-0983-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00601-015-0983-y