------------------------
Summing divergent series
------------------------

There is a second kind of divergences, different from those cured 
by renormalization.

Most perturbation series in QFT are believed to be asymptotic only, 
hence divergent. Strong arguments (which haven't lost in half a 
century their persuasive power) supporting the view that one should 
expect the divergence of the QED (and other relatvistic QFTs) 
power series for S-matrix elements, for all values of 
alpha>0 (and independent of energy) are given in
    F.J. Dyson,
    Divergence of perturbation theory in quantum electrodynamics,
    Phys. Rev. 85 (1952), 613--632. 

The remarkable fact is that QED is very accurate in spite of this.
It produces verifiable predictions by restricting attention to the 
first few terms of a (most probably divergent) asymptotic series, 
but it has no way to make sense of the whole series.
This is what Dirac found deficient in the foundations. 

An asymptotic series is a series expansion of the form 
    f(x) = sum_{k=0:inf} a_k x^k.      (*)
A simple example is the power series for the integral 
    integral_0^inf dt e^{-t}/(1-tx) = sum_{k=0:inf} k! x^k
(easy to check by expansion of the integrand and integration of the 
terms), which has radius of convergence zero, hence diverges for every 
nonzero value of x. The mathematical meaning of an asymptotic expansion 
(*) is that, for all n, 
    |f(x)-sum_{k=0:n} a_k x^k|<=C_n(x)x^{n+1},
with a function C_n(x) that is bounded near x=0. Therefore, for small 
enough x, the first few terms give good approximations, but if one
includes - for any fixed nonzero x - enough terms, the series diverges
since the convergence radius is zero.
Thus, as Dirac asserted, one neglects arbitrarily large terms to get 
the approximations which work so well in QED.

There are infinitely many different ways to assign to an 
asymptotic series a function with this series as Taylor expansion. 
The problem is to have a way to choose the right one. Borel summation 
is often taken as default, but seems to be no cure for QFT in view 
of the so-called renormalon problem.

At present, there is no sound mathematical foundation of relativistic 
quantum field theory in 4 dimensions. Who finds one will be awarded one 
of the 1 Million Dollar Clay Millenium prizes...



If we have a well-defined Hamiltonian H(g) depending infinitely 
differentiably on a parameter g, it typically has a well-defined 
S-matrix S(g), also depending infinitely differentiably on g. 
Perturbation theory computes a power series expansion 
    S(g)=S_0 + S_1 g + ...
which often diverges for all g although each S_k is finite. 
This happens already for the anharmonic oscillator with 
    H(g)= 1/2 (p^2+q^2) +g q^4.
Thus a correct Hamiltonian with a convergent (in the harmonic 
oscillator case even finite, hence trivially convergent) expansion 
is quite consistent with a divergent expansion of the S-matrix.

However, one can one still extract information by so-called
resumming techniques. One can study these things quite well with 
functions which have known asymptotic expansions 
(e.g., improper integrals, using Watson's lemma). 

In many cases (and under well-defined conditions), the resulting 
infinite series is Borel summable in the following sense: To sum 
   f(x) = sum a_k x^k        (1)
if it is divergent or very slowly convergent, one can sum instead
its Borel transform
   Bf(z) = sum a_k/k! z^k    (2)
which obviously converges much faster (if not yet, one could probably 
repeat the procedure). In many cases, f can be reconstructed from Bf 
by means of 
   Sf(x) = integral_0^inf dz/x exp(-z/x) Bf(z)
        = integral_0^inf dt exp(-t) Bf(tx).
Sf is called the Borel summation of the asymptotic series (1), 
and is defined whenever Bf is convergent. 
It is easy to show that BSf=Bf and that Sf has the same asymptotic 
expansion as f. Moreover, the identity Sf(x)=f(x) can be easily 
verified if (1) has a positive radius of convergence, but also under
other natural assumptions (but stronger than simply asserting that (1) 
is an asymptotic expansion for f). If Bf has singularities, 
the integral over t may have to be done along a contour in the complex 
plane; see, e.g., physics/0010038. A good account of Borel summation
in the context of eigenvalue calculations for perturbed Hamiltonians is
   B. Simon
   Large orders and summability of eigenvalue perturbation theory: 
   A mathematical overview
   Int. J. Quantum Chemistry 21 (1982), 3-25
   http://onlinelibrary.wiley.com/doi/10.1002/qua.560210103/abstract

The book
    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
claims to prove on p. 112ff that the coefficients in the loop 
expansion of the QED S-matrix are bounded by const*(N!)^{1/2)/R^n for 
some R>0, which would imply that it is locally Borel summable.
But hep-ph/9701418 seems to make oppsite claims.
See also hep-ph/9807443.

Of course, since there are many functions with the same asymptotic 
expansion (e.g., one can add arbitrary multiples of terms like 
e^{-a/x}, e^{-a/x) log x, etc.),
one has to show that the Borel summed Sf actually has the
properties that the original f was supposed to have (and from which 
the asymptotic series was derived). If, in addition, f is
uniquely determined by these properties, we know that f=Sf.

Unfortunately, a proof for such a statement is missing in QED.
In some 2D cases, where nonperturbative QM applies, one can show that
the nonperturbative result satisfies the properties needed to show
that Borel summation of the perturbative expansion reproduces the
nonperturbative result. See also the thread 
   Re: unsolved problems in QED
starting with
   http://www.lns.cornell.edu/spr/2003-03/msg0049669.html

With experimental results one just has numbers, and not infinite
series, so questions of convergence do not occur.
On the other hand, if one knows of an infinite series a finite number 
of terms only, the result can be, strictly speaking, anything.
But usually one applies some extrapolation algorithm
(e.g., the epsilon or eta algorithm) to get a meaningful guess for the
limit, and estimates the error by doing the same several times,
keeping a variable number of terms. The difference between
consecutive results can count as a reasonable (though not foolproof)
error estimate of these results. Similarly, given a finite number
of coefficients of a power series, one can use Pade approximation to
find an often excellent approximation of the 'intended' function,
although of course, a finite series says, strictly speaking, nothing
about the limit of the sequence.

But to have reliable bounds one needs to know an exact definition of
what one is approximating, and work from there. Such an exact
defintion is, at present, missing for quantum electrodynamics.


More about asymptotic series in quantum field theory can be found in
   J. Fischer 
   On the role of power expansions in quantum field theory
   Int.J.Mod.Phys. A12 (1997) 3625-3663
   arXiv:hep-ph/9704351