Natural Science Forum / Physics / Research / July 2009

On Jun 22, 2:21 pm, Arnold Neumaier <Arnold.Neuma...@univie.ac.at>
wrote:

Dear Arnold,

There is no "my technique" or "my method". There is a general
methodological
approach to search better initial approximations. I do not even call
it "mine".
Yes, it is based on my own experience and I made it available to
everybody via
my journal and arXiv publications, for example:
http://arxiv.org/abs/0806.2635,
http://arxiv.org/abs/0811.4416, and
http://arxiv.org/abs/0906.3504.

Most of your questions are clearly posed and answered in them.

Yes, it would be nice if I could solve all "renormalization" problems
myself but my working possibilities are currently limited (you know
that from
our private correspondence).

Concerning cited by you publication with the short-range potential in
atom,
according to the author the matrix elements are finite, not divergent.
It is a
spectral sum which diverges. It has nothing in common with the
"potential
singularity" (or deepness of the short-range potential well) - the
spectral sum
diverges at any infinitely shallow potential well. His cut-off means
excluding
a part of the spectrum from consideration.
The author tries to calculate the eigenvalues by the perturbation
theory. It is
known that the exact solutions are finite and can be found
numerically. They
can also be found perturbatively but not in the frame of his approach.
Do you
want me to develop another PT approach to this problem? I repeat, I do
not
have a "special technique" or a universal formula for that. Each
problem should
be solved creatively. So even though I resolved a particular problem,
it would
not mean, generally speaking, creating a universal "technique" or
"method".

I provided sufficient number of examples in my publications. They are
simple
and more or less relevant. In particular, I "discovered" (first of all
for myself)
the positive charge atomic form-factor and generalized this physical
and
mathematical solution to an electronium. This explains clearly all
wrongness
of "vacuum polarization" physics around a "point-like charge".
I made a poll and it turned out that nobody even suspected the
positive charge
form-factor existence in atoms. Meanwhile it is an exact QM result. It
helped
me to advance the notion of electronium where the charge and the
quantized
electromagnetic field form a compound system. In other words, it
helped me to
build (or to guess) the exact solution at least for the ground
electronium state.
There is no "potential singularity" at short distances in such a
system. The divergence
appears when one tries to expand the naturally smeared potential in
powers of a
formally small parameter and integrates the resulting series over the
region where
this parameter is big (infinite). I propose not to do such expansions.

In my recent arXiv publication on the perturbation theory I provided
at least four
examples of importance of better choice of the initial approximation.
In Appendix-4
I considered also a "renormalization" approach - perturbative
treatment (resummation)
of divergent terms into new finite eigenfunctions and eigenvalues.
This also might give
finite expansions but with a lot of difficulties and complications.
That is why I insist
on better choice of the initial approximations from the very
beginning.

So I repeat my question:

Do you feel it necessary, desirable, or preferable to have a short-cut
to finite series
with the right physics or is it a way to be banned? Experts, express
yourself on this
matter, please. It is important to me to potentially extend my working
possibilities.

Regards,

Vladimir Kalitvianski.