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On the Structure of the Finite Parts of the Generating Functional of Propagators in Quantumelectrodynamics
1.H. Lehman, K. Symanzik, and W. Zimmerman, Nuovo Cimento 1, 1425 (1955);
1.K. Symanzik, Z. Naturforsch. 9, 809 (1954);
1.H. Umezawa and A. Visconti, Nuovo Cimento 1, 1079 (1955).
2.A. Visconti and Y. Le Gaillard, Nuovo Cimento 34, 914 (1964).
2.For a more rigorous foundation of this formalism (using formal series, regularization, etc.) and for application to the renormalization of two point propagators in general and for simple models, see J. Soffer and A. Visconti, Nuovo Cimento 38, 917 (1965)., Nuovo Cimento
3.For the definition of functional derivatives considered as distributions, see M. B. Donsker and J. L. Lions, Acta Math. 108, 148 (1962).
4.J. Schwinger, Proc. Natl. Acad. Sci. U.S. 37, 452, 455 (1951);
4.K. Symanzik, J. Math. Phys. 1, 249 (1960);
4.A. Visconti and J. Carmona, Nuovo Cimento 29, 742 (1963).
5.For a general definition of formal series, see N. Bourbaki, Algèbre (Hermann & Cie, Paris, 1962), Vol. 2, Chap. IV, Sec. 5.
5.The less mathematically minded reader may consult H. Cartan, Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes (Hermann & Cie., Paris, 1961), p. 9, where all rules are clearly stated.
5.Formal series have been already considered by S. Wightman, Proc. Int. Congr. Math., Amsterdam 1962, 587.
6.L. Schwartz, Theorie des Distributions (Hermann & Cie., Paris, 1957) Vol. I, p. 38.
7.E. Caianiello, Nuovo Cimento 13, 637 (1959).
8.Note that the order of fermions sources in the series (3.11) is important: any modification introduces sign changes due to the anticommuting character of this kind of sources.
9.Note that the differentiation is a well defined operation in the theory of formal series.
10.A. Salam, Phys. Rev. 82, 226 (1951).
11.In the published expressions of there is an infrared divergency which should be disregarded.
12.This solution, in the frame of formal series, is indeed uniquely determined by and the recurrence formulas in expressed by (3.7).
13.The assumption is certainly valid for corresponding to renormalizable theories.
14.See Y. Le Gaillard, these de 3ème Cycle Faculté des Sciences de Marseille.
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