Generalized Feynman amplitudes PDF
Preview Generalized Feynman amplitudes
Annals of Mathematics Studies Number 62 GENERALIZED FEYNMAN AMPLITUDES BY Eugene R. Speer PRINCETON UNIVERSITY PRESS AND THE UNIVERSITY OF TOKYO PRESS PRINCETON, NEW JERSEY 1969 Copyright © 1969, by Princeton University Press ALL RIGHTS RESERVED LC. Card: 72-77595 Published in Japan exclusively by the University of Tokyo Press; in other parts of the world by Princeton University Press Printed in the United States of America Acknowledgements I would like to thank my advisor, Prof. Arthur Wightman, for giving generously of his time during the past two years. His help and advice were invaluable in the preparation of this thesis. I would also like to thank Prof. Edward Nelson for reading the manuscript and for various suggestions, and Prof. Tullio Regge for several helpful discussions. I am grateful to the National Science Foundation for support during four years of grad uate school, to the Princeton University Mathematics Department for support during the summer of 1968, and to Dr. Carl Kaysen for his hospitality at the Institute for Advanced Study. Part of this work was sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant 68-1365. ABSTRACT Renormalization in the context of Lagrangian quantum field theory is reviewed, with emphasis on two points: (a) the Bogoliubov-Parasiuk definition of the renormalized amplitude of an arbitrary Feynman graph, including some generalizations of the rigorous work of Hepp, and (b) a discussion of the implementation of this renormalization by counter terms in an arbitrary interaction Lagrangian. A new quantity called a generalized Feynman amplitude is then defined. It depends analytically on complex parameters A1, ... , ,\L, and these analytic properties may be used to define renormalized Feynman amplitudes in a new way; the method is shown to be equivalent to that of Bogoliubov, Parasiuk, and Hepp. The generalized Feynman amplitude depends on other parameters also; when these take on certain values, it is equal to the Feynman amplitudes for various graphs (aside from problems of renormalization, which are handled via the ,\ dependence). The generalized amplitude thus interpolates the Feynman amplitude between different graphs. Some partial results are ob tained which exploit this interpolation to give an integral representation for a sum of Feynman amplitudes.
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