Journal of High Energy Physics

, 2019:57 | Cite as

The Schwinger point

  • Howard GeorgiEmail author
Open Access
Regular Article - Theoretical Physics


The Sommerfield model with a massive vector field coupled to a massless fermion in 1+1 dimensions is an exactly solvable analog of a Bank-Zaks model. The “physics” of the model comprises a massive boson and an unparticle sector that survives at low energy as a conformal field theory (Thirring model). I discuss the “Schwinger point” of the Sommerfield model in which the vector boson mass goes to zero. The limit is singular but gauge invariant quantities should be well-defined. I give a number of examples, both (trivially) with local operators and with nonlocal products connected by Wilson lines (the primary technical accomplishment in this note is the explicit and very pedestrian calculation of correlators involving straight Wilson lines). I hope that this may give some insight into the nature of bosonization in the Schwinger model and its connection with unparticle physics which in this simple case may be thought of as “incomplete bosonization.”


Field Theories in Lower Dimensions Gauge Symmetry 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited


  1. [1]
    H. Georgi and Y. Kats, Unparticle self-interactions, JHEP02 (2010) 065 [arXiv:0904.1962] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    C.M. Sommerfield, On the definition of currents and the action principle in field theories of one spatial dimension, Annals Phys.26 (1964) 1.ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    L.S. Brown, Gauge invariance and mass in a two-dimensional model, Nuovo Cim.29 (1963) 617.ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    W.E. Thirring and J.E. Wess, Solution of a field theoretical model in one space-one time dimension, Annals Phys.27 (1964) 331.ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    D.A. Dubin and J. Tarski, Interactions of massless spinors in two dimensions, Annals Phys.43 (1967) 263.ADSCrossRefGoogle Scholar
  6. [6]
    C.R. Hagen, Current definition and mass renormalization in a model field theory, Nuovo Cim.A 51 (1967) 1033.ADSCrossRefGoogle Scholar
  7. [7]
    H. Georgi and Y. Kats, An Unparticle Example in 2D, Phys. Rev. Lett.101 (2008) 131603 [arXiv:0805.3953] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    R. Roskies and F. Schaposnik, Comment on Fujikawa’s Analysis Applied to the Schwinger Model, Phys. Rev.D 23 (1981) 558 [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    H. Georgi and J.M. Rawls, Anomalies of the axial-vector current in two dimensions, Phys. Rev.D 3 (1971) 874 [INSPIRE].ADSGoogle Scholar
  10. [10]
    J.S. Schwinger, Gauge Invariance and Mass. 2., Phys. Rev.128 (1962) 2425 [INSPIRE].
  11. [11]
    K.G. Wilson, Confinement of Quarks, Phys. Rev.D 10 (1974) 2445 [INSPIRE].ADSGoogle Scholar
  12. [12]
    S.R. Coleman, The Quantum sine-Gordon Equation as the Massive Thirring Model, Phys. Rev.D 11 (1975) 2088 [INSPIRE].ADSGoogle Scholar
  13. [13]
    A.V. Smilga, On the fermion condensate in Schwinger model, Phys. Lett.B 278 (1992) 371 [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    C. Jayewardena, Schwinger model on S2 , Helv. Phys. Acta61 (1988) 636 [INSPIRE].MathSciNetGoogle Scholar
  15. [15]
    J.E. Hetrick and Y. Hosotani, QED on a circle, Phys. Rev.D 38 (1988) 2621 [INSPIRE].ADSMathSciNetGoogle Scholar
  16. [16]
    H. Georgi and B. Warner, Generalizations of the Sommerfield and Schwinger models, arXiv:1907.12705 [INSPIRE].
  17. [17]
    H.A. Falomir, R.E. Gamboa Saravi and F.A. Schaposnik, Wilson Loop Dependence on the Contour Shape, Phys. Rev.D 25 (1982) 547 [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Center for the Fundamental Laws of Nature, Jefferson Physical LaboratoryHarvard UniversityCambridgeU.S.A.

Personalised recommendations