Quantum Electrodynamics pp 133-161 | Cite as

# Review of Consistency Problems in Quantum Electrodynamics

## Abstract

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 meaningless^{1} 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}

## Keywords

Matrix Element Perturbation Theory Vacuum Expectation Ward Identity Current Operator## Preview

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## Literatur

- 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.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.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 V_{1}(1958).Google Scholar - 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 - 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 - 5.Cf. esp. the concluding remarks in the first paper quoted in ref.
**3**.Google Scholar - 6.Cf. esp. pp. 11 and 12 in the first paper in ref. 3.Google Scholar
- 7.CERN/T/GK/3 Nov. 1955.Google Scholar
- 8.Cf. Proc. of the CERN Symposion on High Energy Accelerators and Pion Physics, June 1956, p. 187.Google Scholar
- 9.To be published in Ergebn. d. exakten Naturwissenschaften.Google Scholar
- 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 - 10a.The particular notation which we are using was introduced in G. Källén, Helv. Phys. Acta
**25**, 417 (1952).MATHGoogle Scholar - 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 - 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 - 11.
- 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 - 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(p^{2}). Google Scholar - 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
- 15.
- 16.This, simplifying assumption is also made, e.g., in the first paper of ref. 4.Google Scholar
- 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 - 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
- 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 - 19a.Cf. in this connection also the paper by K. W. Ford, Phys. Rev.
**105**, 320 (1957).MathSciNetADSMATHCrossRefGoogle Scholar - 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 - 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 - 22.
- 22a.The first order perturbation theory expression for this identity was used already by J. Schwinger, Phys. Rev.
**76**, 790 (1949).MathSciNetADSMATHCrossRefGoogle Scholar - 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 - 23.The actual calculation is given, e.g., in ref. 3.Google Scholar
- 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 - 25.
- 25a.
- 25b.
- 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 - 27.Cf. esp. Eqs. (42.a), (42.b) and (42.c) and the remarks immediately after these equations in the original paper.Google Scholar
- 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
- 29.Some of the relevant papers in this connection are G. Källén, A. Wightman, Dan. Mat. Fys. Skr.
**1**, Nr. 6 (1958).Google Scholar - 29a.
- 29b.
- 29c.
- 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 - 30a.
- 30b.
- 31.
- 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 - 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
- 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
- 35.Cf. further the paper by S. Kamefuchi, Dan. Mat. Fys. Medd.,
**31**, Nr. 6 (1957).MathSciNetGoogle Scholar