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.
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References
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.
Wilson, K.: Phys. Rev.179, 1499 (1969).
Gell-Mann, M., Zachariasen, F.: Phys. Rev.123, 1065 (1961).
Symanzik, K.: Commun. Math. Phys.16, 48 (1970).
—— 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.
Stora, R., Symanzik, K.: (in preparation).
Landau, L. D., Abrikosov, A., Halatnikov, L.: Suppl. al Nuovo Cimento3, 80 (1956).
Appelquist, T., Primack, J. R.: Phys. Rev.1D, 1144 (1970).
Pais, A., Uhlenbeck, G. E.: Phys. Rev.79, 145 (1950).
Gupta, S. N.: Proc. Phys. Soc. (London)A 66, 129 (1953).
Johnson, R. W.: J. Math. Phys.10, (to appear).
Symanzik, K.: In: Lectures on High Energy Physics, Ed. B. Jakšić, Zagreb: 1961. Reprinted. New-York: Gordon and Breach 1966.
Taylor, J. G.: Suppl. al Nuovo Cimento1, 857 (1963).
Schwinger, J.: Proc. Natl. Acad. Sci. U.S.37, 452, 455 (1951).
—— Proc. Natl. Acad. Sci. U.S.48, 603 (1962).
Brandt, R. A.: Ann. Phys. (N. Y.)52, 122 (1969).
Jauch, J. M., Rohrlich, F.: The Theory of Photons and Electrons. Cambridge (Mass.): Addison-Wesley 1955.
Kazes, E.: Nuovo Cimento13, 1226 (1959).
Bjorken, J. D., Drell, S. D.: Relativistic Quantum Fields. New York: McGraw-Hill 1965.
Eriksson, K. E.: Nuovo Cimento27, 178 (1963).
Jackiw, R.: Nucl. Phys.B 5, 158 (1968).
Ward, J. C.: Phys. Rev.84, 897 (1951).
Hepp, K.: Commun. Math. Phys.2, 301 (1966).
Gross, D. J., Wess, J.: Phys. Rev., (to appear).
Callan Jr., C. G., Coleman, S., Jackiw, R.: Ann. Phys. (N. Y.), (to appear).
Lam, C. S.: Nuovo Cimento38, 1755 (1965).
Weinberg, S.: Phys. Rev.118, 838 (1960).
Fink, J. P.: J. Math. Phys.9, 1389 (1968).
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Symanzik, K. Small distance behaviour in field theory and power counting. Commun.Math. Phys. 18, 227–246 (1970). https://doi.org/10.1007/BF01649434
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DOI: https://doi.org/10.1007/BF01649434