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Small-distance-behaviour analysis and Wilson expansions

Communications in Mathematical Physics volume 23, pages 49–86 (1971)Cite this article

Abstract

A previously described method to obtain the asymptotic forms of vertex functions at large momenta is, with the help of Wilson operator product expansion formulas, extended to momenta where the vertex functions of the zero-mass theory underlying the asymptotic forms are infrared singular. To obtain from asymptotic forms information on asymptotic behaviour requires assumptions on the behaviour of the zero-mass theory in the limit of infinite dilatation. One particular set of assumptions is discussed and found to pass a simple consistency test; this set of assumptions leads to power laws, or slight modifications thereof, with coupling-constant-independent exponents. The detailed discussion is given for the ф4 model.

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References

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

    Google Scholar 

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

    Google Scholar 

  3. Westwater, M.J.: Fortschr. d. Phys.17, 1 (1969).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  6. Landau, L. D., Abrikosov, A., Khalatnikov, I.: Nuovo Cimento Suppl.3, 80 (1956).

    Google Scholar 

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

    Google Scholar 

  8. Symanzik, K.: Commun. math. Phys.18, 227 (1970).

    Google Scholar 

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

    Google Scholar 

  10. Symanzik, K.: In: Springer Tracts in Modern Physics Vol.57, (1971), p. 222.

  11. Wilson, K. G.: preprint, Cornell U., 1964.

  12. Brandt, R.A.: Ann. Phys.44, 221 (1967).

    Google Scholar 

  13. Zimmermann, W.: In: Lectures on Elementary Particles and Fields (1970 Brandeis University Summer Institute), Vol. 2. Eds.: Deser, S., Grisarv, M., Pendleton, H. Cambridge, Mass.: MIT Press 1971.

    Google Scholar 

  14. Coleman, R., Jackiw, R.: Ann. Phys. (to appear).

  15. Wilson, K.G.: Phys. Rev. D3, 1818 (1971).

    Google Scholar 

  16. Baker, M., Johnson, K.: Phys. Rev. D3, 2516, 2541 (1971).

    Google Scholar 

  17. Symanzik, K.: Commun. math. Phys.16, 48 (1970).

    Google Scholar 

  18. —— In: Cargèse Lectures in Physics, Vol. 6. Ed. J. D. Bessis. New York: Gordon and Breach 1971.

    Google Scholar 

  19. Wu, T. T.: Phys. Rev.125, 1436 (1962).

    Google Scholar 

  20. Johnson, R. W.: J. Math. Phys.11, 2161 (1970).

    Google Scholar 

  21. Callan Jr., C. G.: Phys. Rev. D2, 1541 (1970).

    Google Scholar 

  22. Kinoshita, T.: J. Math. Phys.3, 650 (1962).

    Google Scholar 

  23. Ruelle, D.: Nuovo Cimento19, 356 (1961).

    Google Scholar 

  24. Schroer, B.: Fortschr. d. Phys.11, 1 (1963).

    Google Scholar 

  25. Landau, L. D.: In: Niels Bohr and the developments of Physics, ed. W. Pauli, New York: McGraw Hill Book Co. 1955; and In: Physics in the Twentieth Century, eds. R. Jost, V. Weisskopf.

    Google Scholar 

  26. Pohlmeyer, K.: Commun. math. Phys.12, 204 (1969).

    Google Scholar 

  27. e.g., Fritzsch, H., Gell-Mann, M., CALT-68-297.

  28. Lehmann, H.: Nuovo Cimento11, 342 (1954).

    Google Scholar 

  29. Symanzik, K.: In: Lectures on High Energy Physics, ed. B. Jakśič, Zagreb: 1961. New York: Gordon and Breach 1965.

    Google Scholar 

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

    Google Scholar 

  31. Ferretti, B.: Nuovo Cimento (L)12, 457 (1954).

    Google Scholar 

  32. Hahn, Y., Zimmermann, W.: Commun. math. Phys.10, 330 (1968); Zimmermann, W.: Ibid.11, 1 (1968).

    Google Scholar 

  33. Hepp, K.: Théorie de la renormalisation. Berlin-Heidelberg-New York: Springer 1969.

    Google Scholar 

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  1. Deutsches Elektronen-Synchrotron DESY, Hamburg

    K. Symanzik

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  1. K. Symanzik
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Symanzik, K. Small-distance-behaviour analysis and Wilson expansions. Commun.Math. Phys. 23, 49–86 (1971). https://doi.org/10.1007/BF01877596

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  • DOI: https://doi.org/10.1007/BF01877596

Keywords

  • Neural Network
  • Statistical Physic
  • Complex System
  • Asymptotic Behaviour
  • Nonlinear Dynamics