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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Weinberg, S.: Phys. Rev.118, 838 (1960).
Fink, J. P.: J. Math. Phys.9, 1389 (1968).
Westwater, M.J.: Fortschr. d. Phys.17, 1 (1969).
Gell-Mann, M., Low, F.E.: Phys. Rev.95, 1300 (1954).
Bogoliubov, N. N., Shirkov, D. V.: Introduction to the Theory of Quantized Fields. New York: Interscience Publ. 1959.
Landau, L. D., Abrikosov, A., Khalatnikov, I.: Nuovo Cimento Suppl.3, 80 (1956).
Appelquist, T., Primack, J.R.: Phys. Rev.1D, 1144 (1970).
Symanzik, K.: Commun. math. Phys.18, 227 (1970).
Wilson, K.G.: Phys. Rev.179, 1499 (1969).
Symanzik, K.: In: Springer Tracts in Modern Physics Vol.57, (1971), p. 222.
Wilson, K. G.: preprint, Cornell U., 1964.
Brandt, R.A.: Ann. Phys.44, 221 (1967).
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.
Coleman, R., Jackiw, R.: Ann. Phys. (to appear).
Wilson, K.G.: Phys. Rev. D3, 1818 (1971).
Baker, M., Johnson, K.: Phys. Rev. D3, 2516, 2541 (1971).
Symanzik, K.: Commun. math. Phys.16, 48 (1970).
—— In: Cargèse Lectures in Physics, Vol. 6. Ed. J. D. Bessis. New York: Gordon and Breach 1971.
Wu, T. T.: Phys. Rev.125, 1436 (1962).
Johnson, R. W.: J. Math. Phys.11, 2161 (1970).
Callan Jr., C. G.: Phys. Rev. D2, 1541 (1970).
Kinoshita, T.: J. Math. Phys.3, 650 (1962).
Ruelle, D.: Nuovo Cimento19, 356 (1961).
Schroer, B.: Fortschr. d. Phys.11, 1 (1963).
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.
Pohlmeyer, K.: Commun. math. Phys.12, 204 (1969).
e.g., Fritzsch, H., Gell-Mann, M., CALT-68-297.
Lehmann, H.: Nuovo Cimento11, 342 (1954).
Symanzik, K.: In: Lectures on High Energy Physics, ed. B. Jakśič, Zagreb: 1961. New York: Gordon and Breach 1965.
Taylor, J. G.: Suppl. al Nuovo Cimento1, 857 (1963).
Ferretti, B.: Nuovo Cimento (L)12, 457 (1954).
Hahn, Y., Zimmermann, W.: Commun. math. Phys.10, 330 (1968); Zimmermann, W.: Ibid.11, 1 (1968).
Hepp, K.: Théorie de la renormalisation. Berlin-Heidelberg-New York: Springer 1969.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Symanzik, K. Small-distance-behaviour analysis and Wilson expansions. Commun.Math. Phys. 23, 49–86 (1971). https://doi.org/10.1007/BF01877596
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01877596