Skip to main content

Small distance behaviour in field theory and power counting

Abstract

For infinitesimal changes of vertex functions under infinitesimal variation of all renormalized parameters, linear combinations are found such that the net infinitesimal changes of all vertex functions are negligible relative to those functions themselves at large momenta in all orders of renormalized perturbation theory. The resulting linear first order partial differential equations for the asymptotic forms of the vertex functions are, in quantum electrodynamics, solved in terms of one universal function of one variable and one function of one variable for each vertex function whereby, in contrast to the renormalization group treatment of this problem, the universal function is obtained from nonasymptotic considerations. A relation to the breaking of scale invariance in renormalizable theories is described.

This is a preview of subscription content, access via your institution.

References

  1. Gell-Mann, M., Low, F. E.: Phys. Rev.95, 1300 (1954).

    Google Scholar 

  2. Bogoliubov, N. N., Shirkov, D. V.: Introduction to the Theory of Quantized Fields. New York: Interscience Publ. 1959.

    Google Scholar 

  3. Wilson, K.: Phys. Rev.179, 1499 (1969).

    Google Scholar 

  4. Gell-Mann, M., Zachariasen, F.: Phys. Rev.123, 1065 (1961).

    Google Scholar 

  5. Symanzik, K.: Commun. Math. Phys.16, 48 (1970).

    Google Scholar 

  6. —— Coral Gables Conference on Fundamental Interactions at High Energy II, p. 263, Eds. A. Perlmutter, G. J. Iverson, R. M. Williams, New York: Gordon and Breach, 1970.

    Google Scholar 

  7. Stora, R., Symanzik, K.: (in preparation).

  8. Landau, L. D., Abrikosov, A., Halatnikov, L.: Suppl. al Nuovo Cimento3, 80 (1956).

    Google Scholar 

  9. Appelquist, T., Primack, J. R.: Phys. Rev.1D, 1144 (1970).

    Google Scholar 

  10. Pais, A., Uhlenbeck, G. E.: Phys. Rev.79, 145 (1950).

    Google Scholar 

  11. Gupta, S. N.: Proc. Phys. Soc. (London)A 66, 129 (1953).

    Google Scholar 

  12. Johnson, R. W.: J. Math. Phys.10, (to appear).

  13. Symanzik, K.: In: Lectures on High Energy Physics, Ed. B. Jakšić, Zagreb: 1961. Reprinted. New-York: Gordon and Breach 1966.

    Google Scholar 

  14. Taylor, J. G.: Suppl. al Nuovo Cimento1, 857 (1963).

    Google Scholar 

  15. Schwinger, J.: Proc. Natl. Acad. Sci. U.S.37, 452, 455 (1951).

    Google Scholar 

  16. —— Proc. Natl. Acad. Sci. U.S.48, 603 (1962).

    Google Scholar 

  17. Brandt, R. A.: Ann. Phys. (N. Y.)52, 122 (1969).

    Google Scholar 

  18. Jauch, J. M., Rohrlich, F.: The Theory of Photons and Electrons. Cambridge (Mass.): Addison-Wesley 1955.

    Google Scholar 

  19. Kazes, E.: Nuovo Cimento13, 1226 (1959).

    Google Scholar 

  20. Bjorken, J. D., Drell, S. D.: Relativistic Quantum Fields. New York: McGraw-Hill 1965.

    Google Scholar 

  21. Eriksson, K. E.: Nuovo Cimento27, 178 (1963).

    Google Scholar 

  22. Jackiw, R.: Nucl. Phys.B 5, 158 (1968).

    Google Scholar 

  23. Ward, J. C.: Phys. Rev.84, 897 (1951).

    Google Scholar 

  24. Hepp, K.: Commun. Math. Phys.2, 301 (1966).

    Google Scholar 

  25. Gross, D. J., Wess, J.: Phys. Rev., (to appear).

  26. Callan Jr., C. G., Coleman, S., Jackiw, R.: Ann. Phys. (N. Y.), (to appear).

  27. Lam, C. S.: Nuovo Cimento38, 1755 (1965).

    Google Scholar 

  28. Weinberg, S.: Phys. Rev.118, 838 (1960).

    Google Scholar 

  29. Fink, J. P.: J. Math. Phys.9, 1389 (1968).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Symanzik, K. Small distance behaviour in field theory and power counting. Commun.Math. Phys. 18, 227–246 (1970). https://doi.org/10.1007/BF01649434

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01649434

Keywords

  • Neural Network
  • Partial Differential Equation
  • Perturbation Theory
  • Nonlinear Dynamics
  • Renormalization Group