Small distance behaviour in field theory and power counting
- 649 Downloads
- 748 Citations
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.
Keywords
Neural Network Partial Differential Equation Perturbation Theory Nonlinear Dynamics Renormalization GroupPreview
Unable to display preview. Download preview PDF.
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).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 26.Callan Jr., C. G., Coleman, S., Jackiw, R.: Ann. Phys. (N. Y.), (to appear).Google Scholar
- 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