Re: Is general relativity incompatible with the Newtonian limit?

Igor Khavkine wrote:

Juan R. wrote:

In one of his last works Mathematical Foundations of
Quantum Theory. (Academic Press, Inc., 1978) Dirac claimed:

Most physicists are very satisfied with this situation [refer to
divergences of QFT]. They argue that if one has rules for doing
calculations and the results agree with observation, that is all that
one requires. But it is not all that one requires. One requires a
single comprehensive theory applying to all physical phenomena. Not one
theory for dealing with non-relativistic effects and a separate
disjoint theory for dealing with certain relativistic effects.
Furthermore, the theory has to be based on sound mathematics, in which
one neglects only quantities that are small. One is not allowed to
neglect infinitely large quantities [...]


However great a theorist was Dirac, he was wrong about this assessment
of renormalization. Perturbative renormalization is based on sound
mathematics *and* is capable of produce correct veriable (and verified)
predictions.

True and not true. It produces verifiable predictions if you restrict
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 such as
    f(x) = sum_{k=0:inf} k! x^k
with radius of convergence zero. For small enough x, the first few
terms give seemingly good approximations, but if one includes for
any fixed nonzero x enough terms, the series diverges. Thus, as Dirac
asserts, 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.

For more details, see the entry ''Summing divergent series'' in my
theoretical physics FAQ at
        http://www.mat.univie.ac.at/~neum/physics-faq.txt


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


Arnold Neumaier


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