Feynman, R. P. (1949) Spacetime approach to quantum electrodynamics. Physical Review, 76 (6). pp. 769789. ISSN 0031899X. http://resolver.caltech.edu/CaltechAUTHORS:FEYpr49c

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Abstract
In this paper two things are done. (1) It is shown that a considerable simplification can be attained in writing down matrix elements for complex processes in electrodynamics. Further, a physical point of view is available which permits them to be written down directly for any specific problem. Being simply a restatement of conventional electrodynamics, however, the matrix elements diverge for complex processes. (2) Electrodynamics is modified by altering the interaction of electrons at short distances. All matrix elements are now finite, with the exception of those relating to problems of vacuum polarization. The latter are evaluated in a manner suggested by Pauli and Bethe, which gives finite results for these matrices also. The only effects sensitive to the modification are changes in mass and charge of the electrons. Such changes could not be directly observed. Phenomena directly observable, are insensitive to the details of the modification used (except at extreme energies). For such phenomena, a limit can be taken as the range of the modification goes to zero. The results then agree with those of Schwinger. A complete, unambiguous, and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons. The simplification in writing the expressions results from an emphasis on the overall spacetime view resulting from a study of the solution of the equations of electrodynamics. The relation of this to the more conventional Hamiltonian point of view is discussed. It would be very difficult to make the modification which is proposed if one insisted on having the equations in Hamiltonian form. The methods apply as well to charges obeying the KleinGordon equation, and to the various meson theories of nuclear forces. Illustrative examples are given. Although a modification like that used in electrodynamics can make all matrices finite for all of the meson theories, for some of the theories it is no longer true that all directly observable phenomena are insensitive to the details of the modification used. The actual evaluation of integrals appearing in the matrix elements may be facilitated, in the simpler cases, by methods described in the appendix.
Item Type:  Article 

Additional Information:  ©1949 The American Physical Society. Received 9 May 1949. The author is particularly indebted to Professor H. A. Bethe for his explanation of a method of obtaining finite and gauge invariant results for the problem of vacuum polarization. He is also grateful for Professor Bethe’s criticisms of the manuscript, and for innumerable discussions during the development of this work. He wishes to thank Professor J. Ashkin for his careful reading of the manuscript. 
Issue or Number:  6 
Record Number:  CaltechAUTHORS:FEYpr49c 
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:FEYpr49c 
Alternative URL:  http://dx.doi.org/10.1103/PhysRev.76.769 
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  3523 
Collection:  CaltechAUTHORS 
Deposited By:  Tony Diaz 
Deposited On:  12 Jun 2006 
Last Modified:  26 Dec 2012 08:54 
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