Is QED consistent?

Quantum electrodynamics (QED) gives the most accurate predictions quantum physics currently has to offer.
The anomalous magnetic dipole moment matches the experimental data to 12 significant digits:

  • M. Passera, Precise mass-dependent QED contributions to leptonic g-2 at order alpha^2 and alpha^3, Phys. Rev. D 75, 013002 (2007). arxiv paper
  • B. Odom, D. Hanneke, B. D'Urso, and G. Gabrielse, New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron, Phys. Rev. Lett. 97, 030801 (2006) pdf
    The Lamb shift, whose prediction made QED and renormalization respectable, is much more difficult to measure with high precision, hence offers no such phenomenal test of accuracy:
  • S.G. Karshenboim, Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants, Phys. Rep. 422 (2005), 1-63 arxiv paper
    (For proposals to derive the Lamb shift without using QED, see work by Barut and collegues in the report pdf and the papers
  • Phys. Rev. A 34 (1986), 3500-3501 Phys. Rev. A 34, (1986) 3502-3503 Int. J. Theor. Phys. 32 (1993), 961-968.
    They don't seem to have convinced the mainstream.)

    In spite of these successes of QED, many physicists think that QED cannot be a consistent theory. There is a phenomenon called the Landau pole: It indicates that at extremely large energies (far beyond the range of physical validity of QED, even far beyond the Planck energy) something might go wrong with QED. (QED loses its validity already at energies of about 10^11 eV, where the weak interaction becomes essential. The Planck energy at about 10^28 eV is the limit where some current theories try to make predictions. But the Landau pole, if it exists, has an energy far larger than the latter.) This is probably why Yang-Mills and not quantum electrodynamics was chosen as the model theory for the millenium prize.

    Since the existence of the Landau pole is confirmed only in low order perturbation theory and in lattice calculations,

  • hep-lat/9801004 and hep-th/9712244
    the question whether the alleged Landau pole implies limits to the consistency of QED has currently no rigorous mathematical substance.
    The observations about the Landau pole in perturbation theory can be recast in mathematically rigorous terms using so-called renormalons, obstructions to Borel summability; see
  • V Rivasseau, From Peturbative to Constructive Renormalization, Princeton 1991
    But the resulting analysis is inconclusive as regards the existence of the theory.

    The quality of the computed approximations to QED are a strong indication that there should be a consistent mathematical foundation (for not too high energies), although it hasn't been found yet. There is no indication at all that at the energies where QED suffices to describe our world (with electrons and nuclei considered as elementary particles), it should be inconsistent. To show this rigorously, or to disprove therefore remains another unsolved (and for physics more important) problem.
    Perturbative QED is only a rudimentary version of the 'real QED'. This can be seen from the fact that Scharf's results on the external field case

  • G. Scharf, Finite Quantum Electrodynamics: The Causal Approach, 2nd ed., New York: Springer-Verlag, 1995.
    are much stronger (he constructs in his book the S-matrix) than those for QED proper (where he only shows the existence of the power series in alpha, but not their convergence).
  • J.S. Feldman, T.R. Hurd, L. Rosen and J.D. Wright, QED: A proof of renormalizability, Lecture Notes in Physics 312, Springer, Berlin 1988
    gives a rigorous proof of perturbative existence of QED at all orders. This means that a formal power series for the S-matrix is shown to exist rigorously. This includes renormalization and is sufficient for actual computations of single electron effects in weak fields (such as the anomalous magnetic moment and the Lamb shift in light atoms) since a few terms in the power series give very high accuracy.
    However, the power series is believed to diverge if enough (i.e., infinitely many) terms are added, and a consistent nonperturbative treatment of full QED is presently missing.

    The quest for the 'existence' of QED is the quest for a framework where the formulas make sense nonperturbatively, and where the power series in alpha is a Taylor expansion of a (presumably nonanalytic) function of alpha that is mathematically well-defined for alpha around 1/137 and not too high energy. This is still open.
    More precisely: Probably QED (and thus the QED S-matrix exists nonperturbatively as a 2-parameter theory depending on the fine structure constant alpha and the electron mass m_e; these parameters are the zero energy limits of the corresponding renormalized running coupling constants, and is defined for alpha <= 1/137 and input energies <= some number E_limit(alpha,m_e) larger than the physical validity of pure QED. What is needed is a mathematical proof that the QED S-matrix exists for 0

    We know from perturbation theory how to compute in such a range the coefficients of an asymptotic series in alpha for S(alpha). We also have a number of nonperturbative approximation schemes that give certain nonperturbative results (such as the Lamb shift).
    But we currently do not have a way to ascertain that some well-defined object S(alpha) exists that has this asymptotic series. The quest for proving that QED exists is that of finding a construction for S(alpha) that makes rigorous sense and has the known asymptotic expansion.

    QED is renormalizable at all loops, which means that the power series expansion of the S-matrix is mathematically well-defined at ordinary energies. The _only_ thing that is missing is to give its limit a mathematically well-defined meaning. Note that the S-matrix S commutes with the Hamiltonian; hence if P is the orthogonal projector to the space H_limit of states involving only energies < E_limit(alpha) then PSP is unitary on H_limit, and my conjecture is that PSP has some (yet unknown but) rigorous nonperturbative construction.
    The Landau pole (if it exists) just gives an upper bound to the allowed energies. E_limit(alpha) is a function of alpha, which according to perturbation theory has to satisfy

  • E_limit(0) < (Landau-pole in lowest order)
    (and possibly decreases with increasing alpha); apart from that, the known approximate results do not restrict the likely mathematical validity of pure quantum electrodynamics.
    A cautious evaluation of the situation is given in Weinberg's QFT book, Vol. 2, pp.136-138 - all options are left open. On the other hand,
  • D. Espriu and R. Tarrach, The case for triviality, Phys. Lett. B383 (1996) 482,
    argue that, because of the Landau pole, quantum electrodynamics is only an effective field theory.

    To summarize:
    QED is renormalizable at all loops, which means that the power series expansion of the S-matrix is mathematically well-defined at ordinary energies. The _only_ thing that is missing is to give its limit a mathematically well-defined meaning derived from a formulation of QED that makes sense also at finite times and not only as a transition from t=-infinity to t=+infinity.

    Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
    A theoretical physics FAQ