Review of Consistency Problems in Quantum Electrodynamics

  • Gunnar Källén
Conference paper
Part of the Supplementa book series (FEWBODY, volume 2/1965)


The question of the consistency of quantized field theories in general and quantum electrodynamics in particular has been the subject of several papers during the last 15 years. In the literature one can find controversial statements ranging from the claim that any renormalized field theory is completely meaningless1 to the assertion that even unrenormalized quantum electrodynamics has an exact solution with well behaved physical properties.2 In view of this rather controversial situation, we find it of some interest to review the actual situation in this field. As can perhaps be expected, our conclusion is that the truth lies somewhere in between the two extreme opinions quoted above. One of the corner stones in our discussion is concerned with an argument published more than 10 years ago with the result that at least one of the renormalization constants in ordinary quantum electrodynamics is infinite.3 For some time this result was accepted by most research workers in the field. However, later it has been criticized besause its lack of rigour.4


Matrix Element Perturbation Theory Vacuum Expectation Ward Identity Current Operator 
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  1. 1.
    This statement was published in the middle 1950’s by L. D. Landau and col aborators. The original papers here are in Russian, but a convenient summary can be found in the article by L. D. Landau “On the quantum theory of fields” in “Niels Bohr and the development of physics”. Pergamon Press, London 1955.Google Scholar
  2. 2.
    K. Johnson, M. Baker, R. S. Willey, Phys. Rev. Lett. 11, 518 (1963). At the end of this paper one finds the statement: “In conclusion, we find that quantum electrodynamics may be regarded as a perfectly consistent theory. The usual divergences from our point of view arise from an unjustified use of perturbation theory. . . “ In a later paper by the same authors, Phys. Rev. 136 B, 1111 (1964) this somewhat aggressive formulation is considerably modified. To be impartial, we quote also the summary of this last reference. It reads: “A perturbation theory is developed within the usual formalism of quantum electrodynamics which yields a finite unrenormalized electron Green’s function and a finite value for the electron electromagnetic self mass in each order. This is subject only to the qualifications in this paper, that the vacuum polarization is also obtained without divergences. Furthermore, the bare mass of the electron vanishes; the electron mass must be totally dynamical in origin.”MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    G. Källén, Dan. Mat. Fys. Medd. 27, Nr. 12 (1953). A short summary of the argument is included in G. Källén. Quantenelektrodynamik, Handbuch der Physik V1 (1958).Google Scholar
  4. 4.
    S. G. Gasiorowicz, D. R. Yennie, H. Suura, Phys. Rev. Lett. 2, 153 (1959). The remarks of these authors are uncritically repeated in the textbook by S. S. Schweber “An introduction to quantum field theory” (1961) esp. p. 683.ADSCrossRefGoogle Scholar
  5. 4a.
    Other objections of a slightty different nature have been raised by K. Johnson, Phys. Rev. 112, 1367 (1958). Cf. footnote 26 below.MathSciNetADSMATHCrossRefGoogle Scholar
  6. 5.
    Cf. esp. the concluding remarks in the first paper quoted in ref. 3.Google Scholar
  7. 6.
    Cf. esp. pp. 11 and 12 in the first paper in ref. 3.Google Scholar
  8. 7.
    CERN/T/GK/3 Nov. 1955.Google Scholar
  9. 8.
    Cf. Proc. of the CERN Symposion on High Energy Accelerators and Pion Physics, June 1956, p. 187.Google Scholar
  10. 9.
    To be published in Ergebn. d. exakten Naturwissenschaften.Google Scholar
  11. 10.
    Historically, spectral representations of the kind used here and for the particular case of quantum electrodynamics were first published by H. Umezawa and S. Kamefuchi, Progr. of Theor. Phys. 6, 543 (1951).MathSciNetADSMATHCrossRefGoogle Scholar
  12. 10a.
    The particular notation which we are using was introduced in G. Källén, Helv. Phys. Acta 25, 417 (1952).MATHGoogle Scholar
  13. 10b.
    A few years later (and without any claim of priority) a related discussion was also given by M. Gell-Mann and F. Low in Phys. Rev. 95, 1300 (1954).MathSciNetADSMATHCrossRefGoogle Scholar
  14. 10c.
    About simultaneously with this last paper appeared a paper by H. Lehmann in Nuovo Cim. 11, 342 (1954). This last paper is widely missquoted as being the first place where spectral representations were introduced.MathSciNetMATHCrossRefGoogle Scholar
  15. 11.
    G. Källén, Helv. Phys. Acta 25, 417 (1952) and Handbuch der Physik V1 (1958).MATHGoogle Scholar
  16. 12.
    To make this statement into an exact mathematical result the argument above has to be considerably amplified. We do not insist on the details as the reader should be able to work them out for himself along the lines indicated here, assuming the convergence of the integral (2.10) but not of the integral (2.9). Further, it can perhaps be remarked that, in the language of perturbation theory, this introduction of the function II* corresponds to the transition from “self energy graphs” to “proper self energy graphs”.Google Scholar
  17. 13.
    Cf. the well-known situation in the theory of the electromagnetic form factors of the nucleon. We have here adopted the same notational convention which is used in form factor discussions. The expression F 1 is sometimes referred to as the “Dirac form factor’ and F 2 is normally referred to as the “Pauli form factor”. In the papers mentioned in ref. 3, linear combinations of these two functions were denoted by R(p 2 ) and S(p2). Google Scholar
  18. 14.
    This constant is actually a combination of the conventional renormalization constants. Note also the similarity of the algebraic structure of Eq. (3.3) and Eq. (2.4).Google Scholar
  19. 15.
    Cf. R. Jost, Helv. Phys. Acta 31, 263 (1958).MathSciNetMATHGoogle Scholar
  20. 16.
    This, simplifying assumption is also made, e.g., in the first paper of ref. 4.Google Scholar
  21. 17.
    In obtaining the estimate (3.9) we have used that not only the function F 2 (p 2 ) goes to zero in the high energy limit but that also the term — p 2 F 2 2 can be neglected compared to the right-hand-side of Eq. (3.8 a). A more careful investigation which we do not give here shows that the high energy behaviour of the function — p 2 F 2 (p 2 ) is not worse than some power of log (— p 2 ). Consequently, the estimate (3.9) is correct. To obtain this result one has to assume certain well-behaved properties of the weight functions f 2 (p 2 ) but these smoothness properties appear very reasonable from the point of view of physics. Some of the details of this argument are given in the appendix of the first paper in ref. 3, but we do not want to give them here. However, it can be mentioned that the explicit form of the weight functions f 1 (p 2 ) computed in perturbation theory fulfill these smoothness conditions.Google Scholar
  22. 18.
    Eq. (3.13) is most easily obtained by considering the conventional Hilbert transformation relation between the real and imaginary parts of the analytic functionGoogle Scholar
  23. 19.
    A discussion similar to the argument presented here but related to the behaviour of the vertex part for a scalar meson theory has been published by H. Lehmann, K. Symanzik, B. Zimmermann, Nuovo Cim. 2, 425 (1955). This paper appeared at rather exactly the time when the manuscript mentioned in ref. 7 was issued. However, the authors just mentioned did not discuss the relation between their result and the conventional renormalization constants.MathSciNetMATHCrossRefGoogle Scholar
  24. 19a.
    Cf. in this connection also the paper by K. W. Ford, Phys. Rev. 105, 320 (1957).MathSciNetADSMATHCrossRefGoogle Scholar
  25. 20.
    Actually, Eq. (4.1 b) is oversimplified as the relation between the renormalized and the unrenormalized electromagnetic potentials is more involved and different for different values of the index µ. For our present discussion, these complications are irrelevant. The notation L comes from the German expression for charge renormalization, viz. “Ladungsrenormierung”.Google Scholar
  26. 21.
    Note that we get a factor 1 — L in the denominator in Eq. (4.4) and not the square root of this number. This happens because we both have to express the unrenormalized charge in terms of the renormalized charge e and divide the whole current operator by the renormalization factor in Eq. (4.1 b). For µ = 4 complications similar to those indicated in footnote 20 also occur, but we do not consider them here. For details we refer to the papers in ref. 11.Google Scholar
  27. 22.
    J. C. Ward, Phys. Rev. 78, 182 (1950).ADSMATHCrossRefGoogle Scholar
  28. 22a.
    The first order perturbation theory expression for this identity was used already by J. Schwinger, Phys. Rev. 76, 790 (1949).MathSciNetADSMATHCrossRefGoogle Scholar
  29. 22b.
    A proof of the Ward identity without reference to perturbation theory was given by G. Källen, Helv. Phys. Acta 26, 755 (1953) and, later, by T. Taka-hashi, Nuovo Cim. 6, 371 (1957).MATHGoogle Scholar
  30. 23.
    The actual calculation is given, e.g., in ref. 3.Google Scholar
  31. 24.
    As before, we disregard some formal complications appearing for the special case µ = 4. These complications make the algebra somewhat more involved than what is indicated here, but do not in any way influence the final result. They are described in ref. 3, esp. the last paper, Section 46.Google Scholar
  32. 25.
    S. Gupta, Proc. Phys. Soc. London A 63, 681 (1950).ADSCrossRefGoogle Scholar
  33. 25a.
    S. Gupta, Proc. Phys. Soc. London A 64, 850 (1951).ADSMATHCrossRefGoogle Scholar
  34. 25b.
    K. Bleuler, Helv. Phys. Acta 23, 567 (1950).MathSciNetMATHGoogle Scholar
  35. 26.
    We mention this elementary point explicitly because it appears to have caused some confusion in the literature. K. Johnson, Phys. Rev. 112, 1367 (1958).MathSciNetADSMATHCrossRefGoogle Scholar
  36. 27.
    Cf. esp. Eqs. (42.a), (42.b) and (42.c) and the remarks immediately after these equations in the original paper.Google Scholar
  37. 28.
    A short note to this effect was communicated to the Phys. Rev. Lett, in the summer of 1959. However, the note has never appeared in print.Google Scholar
  38. 29.
    Some of the relevant papers in this connection are G. Källén, A. Wightman, Dan. Mat. Fys. Skr. 1, Nr. 6 (1958).Google Scholar
  39. 29a.
    R. Oehme, Phys. Rev. 111, 1430 (1951).MathSciNetADSCrossRefGoogle Scholar
  40. 29b.
    R. Oehme, Phys. Rev. 117, 1151 (1960).MathSciNetADSMATHCrossRefGoogle Scholar
  41. 29c.
    G. Kallén, J. Toll, Helv. Phys. Acta 33, 753 (1960).MathSciNetGoogle Scholar
  42. 30.
    For an illustration of these remarks it is of interest to follow the historical development of the theory as reflected, e.g., in the following papers: N. M. Kroll, W.E. Lamb, Phys. Rev. 75, 388 (1949) .ADSMATHCrossRefGoogle Scholar
  43. 30a.
    J. B. French, V. F. Weisskopf, Phys. Rev. 75, 1240 (1949)ADSMATHCrossRefGoogle Scholar
  44. 30b.
    and G. Wentzel, Phys. Rev. 74, 1070 (1948).ADSMATHCrossRefGoogle Scholar
  45. 31.
    B. Zumino, Nuovo Cim. 27, 547 (1960).MathSciNetCrossRefGoogle Scholar
  46. 32.
    This logical situation appears to have been overlooked in the paper by B. Zumino, referred to in the previous footnote as well as in the paper by O. Fleischman, Nuovo Cim. 29, 1098 (1963).MathSciNetCrossRefGoogle Scholar
  47. 33.
    Although no explicit mentioning of these facts can be found in the published papers referred to in footnote 2, Dr. Johnson has informed the author in a private letter that he is aware of these circumstances. Dr. Johnson also suggests that the physical vacuum state should be degenerate.Google Scholar
  48. 34.
    Apart from the particular matrix element considered here, similar properties can be shown to hold exactly for the operators of the Lee model and for the operators in the Wentzel pair theory. These are the only four-dimensional models of field theories where the exact solution is known, at least to some extent. For the two-dimensional Thirring model, where the exact solution is also known, the situation is slightly ambiguous but the explicit formulae derived do not violate the intuitive idea suggested here.Google Scholar
  49. 35.
    Cf. further the paper by S. Kamefuchi, Dan. Mat. Fys. Medd., 31, Nr. 6 (1957).MathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 1965

Authors and Affiliations

  • Gunnar Källén
    • 1
  1. 1.Department of Theoretical PhysicsUniversity of LundLundSweden

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