-----------------------
S9b. 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).
    http://arxiv.org/abs/hep-ph/0606174

    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)
    http://hussle.harvard.edu/~gabrielse/gabrielse/papers/2006/NewElectronMagneticMoment.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
    http://arxiv.org/abs/hep-ph/0509010
(For proposals to derive the Lamb shift without using QED, see work by 
Barut and collegues in the report 
    http://streaming.ictp.trieste.it/preprints/P/87/248.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: 
    http://en.wikipedia.org/wiki/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 
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';
which can be seen 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 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 '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<alpha<1/137 
(rather than only for infinitesimal alpha, as currently established)
as a unitary operator S(alpha) in the Hilbert space H(Emax) of all 
in-states of energy <= E_max=E_limit(alpha), for some reasonable
Emax.


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.