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No Gell-Mann and Low eigenvalue in quantum electrodynamics

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Il Nuovo Cimento A (1965-1970)

An Erratum to this article was published on 01 April 1991

Summary

It is argued that the Gell-Mann and Low eigenvalue equation,ψ(x)=0 in quantum electrodynamics doesnot possess a nontrivial solution,x 0≠0 and hence that quantum electrodynamics cannot be afinite field theory. This result reinforces the Baker-Johnson conjecture of 1979.

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References

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  8. The Baker-Johnson argument has been examined in great detail inP. Langacker andH. Pagels:Phys. Rev. D,9, 3413 (1974) and references therein. The latter analysis employs the results ofR. J. Crewther, S. Shei andT. Yan:Phys. Rev. D,8, 3396 (1973). See, however,R. Crewther andN. Nielsen:Nucl. Phys. B,87, 52 (1975). It should be pointed here that Adler and Bardeen (ref.[4]) have established that\(m_0 (\psi \gamma 5\psi )0\) is finite order-by-order in perturbation theory. If this result remains valid in the “summed” up theory (as would be expected in the renormalization group philosophy), then γ5 symmetry would be broken explicitly.

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  13. Whenm=0, eq. (2.9) is, of course, trivially stisfied in ordinary QED, order by order, since then{S \( ^{ - 1} ,\gamma 5\} \) vanishes in each order of perturbation theory. We are, however, interested in Gell-Mann-Low limit of the exact theory: on grounds of Lorentz invariance and parity conservation, we can writeS \( ^{ - 1} (p) = A(p^2 ,a)\gamma \cdot p + B(p^2 ,a) \). In the Gell-Mann-Low limit ofZ3/−1<∞ andm=0, eq. (2.9) givesB(p 2, α)=0 andA(p 2, α)=A(x 0) independent ofp 2 since scale invariance becomesexact. (It is worth recalling that, for finiteZ3/−1, one can always find a gauge in whichZ 2 is finite (i.e. cut-off independent) and therefore the «anomalous dimension”\(\gamma _e (a) = - \Lambda (\partial /\partial \Lambda )\ln Z_2 = 0\). In QED, there is the further exceptional simplification that the Callan-Symanzik function β(α)=0 and the anomalous dimension\(\gamma _e (a) = - \Lambda (\partial /\partial \Lambda )\ln Z_2 = 0\) are proportional to each other. Now, sinceZ3/−1 is finite,\(\gamma _e (a) = - \Lambda (\partial /\partial \Lambda )\ln Z_2 = 0\) and hence, β(α)=0. Consequently, in the finite gauge,β(α)=γ e(α)=0. Thus scale invariance becomes exact in the zero physical mass limit and eq. (2.11b) follows.) See.C. Callan andD. Gross:Phys. Rev. D,8, 4383 (1973), eq. (4.1).

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  15. In arriving at eq. (2.15), we have not madeexplicit use of the short-distance expansions which are known to be beset with «anomalies»; seeH. Schnitzer:Phys. Rev. D,8, 385 (1973).

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  16. We call attention to the work ofG. Eilam andM. Gluck:Phys. Rev. D,13, 279 (1976), who have shown earlier that the vertex function in massless, finite QED is «bare»,i.e. Z 1=1. Their result depends, however, on the assumption of theexistence of anS-matrix form=0 theory and hence the analysis could be questioned.

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Authors and Affiliations

  1. Physics Department, Arizona State University, 85287, Tempe, AZ

    R. Acharya

  2. Physics Department, Southern Illinois University, 62026, Edwardsville, IL

    P. Narayana Swamy

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  1. R. Acharya
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This work is dedicated to the memory of Heinz Pagels.

The authors of this paper have agreed to not receive the proofs for correction.

An erratum to this article is available at http://dx.doi.org/10.1007/BF02813597.

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Acharya, R., Narayana Swamy, P. No Gell-Mann and Low eigenvalue in quantum electrodynamics. Nuov Cim A 103, 1131–1138 (1990). https://doi.org/10.1007/BF02820540

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