Skip to main content
Log in

Summing up the perturbation series in the Schwinger Model

  • Published:
The European Physical Journal C - Particles and Fields Aims and scope Submit manuscript

Abstract

Perturbation series for the electron propagator in the Schwinger Model is summed up in a direct way by adding contributions coming from individual Feynman diagrams. The calculation, performed entirely in momentum space, shows the complete agreement between nonperturbative and perturbative approaches.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. F.J. Dyson, Phys. Rev. 85, 631 (1952)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. F. Calogero, N. Cim. 30, 916 (1963)

    Article  MathSciNet  Google Scholar 

  3. J.S. Langer, Ann. Phys. 41, 108 (1967)

    Article  ADS  Google Scholar 

  4. S. Graffi, V. Grecchi, B. Simon, Phys. Lett. 32B, 631 (1970)

    ADS  MathSciNet  Google Scholar 

  5. C.M. Bender, T.T. Wu, Phys. Rev. Lett. 27, 461 (1971)

    Article  ADS  Google Scholar 

  6. D.V. Shirkov, Lett. N. Cim. 18, 452 (1977)

    Article  Google Scholar 

  7. L.N. Lipatov, Sov. Phys. JETP 72, 411 (1977)

    MathSciNet  Google Scholar 

  8. E. Brézin, J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. D15, 1544 and 1558 (1977)

    ADS  Google Scholar 

  9. W.-C. Ng, W.-B. Yeung, Lett. N. Cim. 23, 413 (1978)

    Article  MathSciNet  Google Scholar 

  10. J.C. Le Guillou, J. Zinn-Justin (eds.), Large-Order Behaviour of Perturbation Theory (North Holland, 1990)

  11. C. Bachas, Theor. Math. Phys. 95, 491 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. C. Itzykson, B. Parisi, J.-B. Zuber, Phys. Rev. D16, 996 (1977); R. Balian et al., Phys. Rev. D17, 1041 (1978)

    ADS  Google Scholar 

  13. E.B. Bogomolny, V.A. Fateyev, Phys. Lett. 76B, 210 (1978)

    ADS  MathSciNet  Google Scholar 

  14. J. Fischer, Int. J. Mod. Phys. A12, 3625 (1997)

    ADS  Google Scholar 

  15. D.I. Kazakov, D.V. Shirkov, Fortschr. Phys. 28, 465 (1980)

    Article  MathSciNet  Google Scholar 

  16. J. Zinn-Justin, Phys. Rep. 70, 109 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  17. I.W. Herbst, B. Simon, Phys. Lett. 78 B, 304 (1978); 80B, 433 (1979)

    ADS  Google Scholar 

  18. F. Calogero, Lett. N. Cim 25, 533 (1979)

    Article  MathSciNet  Google Scholar 

  19. S.N. Behera, A. Khare, Phys. Lett. 2, 169 (1981)

    Google Scholar 

  20. J.M. Jauch, F. Rohrlich, The theory of photons and electrons (Springer, New York1971)

    Google Scholar 

  21. T. Radozycki, I. Bialynicki-Birula, Phys. Rev. D52, 2439 (1995)

    ADS  Google Scholar 

  22. A.L. Fetter, J.D. Walecka, Quantum theory of manyparticle system (McGraw-Hill, New York 1971)

    Google Scholar 

  23. K. Johnson, M. Baker, R. Willey, Phys. Rev. 136, B111 (1964)

    MathSciNet  Google Scholar 

  24. K. Johnson, R. Willey, M. Baker, Phys. Rev. 183, 1292 (1969)

    Article  ADS  Google Scholar 

  25. M. Baker, K. Johnson, Phys. Rev. D3, 2516 and 2541 (1971)

    ADS  Google Scholar 

  26. S.L. Adler, Phys. Rev. D5, 3021 (1972)

    ADS  Google Scholar 

  27. K. Johnson, M. Baker, Phys. Rev. D8, 1110 (1973)

    ADS  Google Scholar 

  28. I. Bialynicki-Birula, Phys. Rev. Lett. 5, 584 (1961)

    Article  ADS  Google Scholar 

  29. I. Bialynicki-Birula, Phys. Rev. 122, 1942 (1961)

    Article  ADS  Google Scholar 

  30. I. Bialynicki-Birula, in Functional Integration, Geometry and Strings, ed. by Z. Haba, J. Sobczyk (Birkhäuser Verlag, Basel-Boston-Berlin1989)

    Google Scholar 

  31. J. Schwinger, in Theoretical Physics, Trieste Lectures 1962 (I.A.E.A., Vienna 1963), p. 89; Phys. Rev. 128, 2425 (1962)

    Google Scholar 

  32. J.-P. Eckmann, J. Magnen, R. Sénéor, Commun. Math. Phys. 39, 251 (1975)

    Article  ADS  Google Scholar 

  33. M. Fry, Phys. Lett. B80, 65 (1978)

    ADS  Google Scholar 

  34. A.R. Zhitnitsky, Phys. Rev.D53, 5821 (1996)

    ADS  Google Scholar 

  35. J. Spanier, K.B. Oldham, An atlas of functions (HPC-Springer, Washington, Berlin 1987)

    MATH  Google Scholar 

  36. I.O. Stamatescu, T.T. Wu, Nucl. Phys. B143, 503 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  37. O. Schnetz, Ph. D. Thesis, Nurnberg 1995

  38. It is worthy noting that the iteration equation (4.7) of [36] is a kind of Dyson-Schwinger equation which follows from the very construction of the diagram on Fig. 4, where 2n-th vertex is fixed. It is also reflected by the presence of p + on the left hand side of (4.8). This gives in fact the expansion of the derivative of S (in coordinate space), as also seen in (4.10), whereas our equation (15) gives the expansion of S itself. Of course both approaches are equivalent.

  39. I.S. Gradshteyn’s, I.M. Ryzhik’s, Table of Integrals Series and Products (Academic Press, New York 1980)

    Google Scholar 

  40. K. Stam, J. Phys. G: Nucl. Phys. 9, L229 (1983). Please note the slight disagreement in coefficient

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tomasz Radozycki.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Radozycki, T. Summing up the perturbation series in the Schwinger Model. Eur. Phys. J. C 6, 549–553 (1999). https://doi.org/10.1007/s100529800933

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s100529800933

Keywords

Navigation