[FOM] Question about theoretical physics

[Jay Sulzberger]

On Sun, 24 Feb 2013, Joe Shipman <JoeShipman at aol.com> wrote:

> What does that mean, precisely?
>
> Are you interpreting "asymptotic" to mean that a value is
> assumed to exist, but the relationship of the series to the
> value is not convergence, but is rather "the sum of the first N
> terms differs from the value by less than the size of the Nth
> term"?
>
> Under this interpretation, the series provides an interval of
> positive width within which the value lies, and that's all, and
> the physical theory is inherently incomplete and imprecise.
>
> -- JS

Joe, I was not directly answering your question.  Rather I was
attempting to suggest that the issue of "What is the radius of
convergence of this one series, or do we have an asymptotic
sequence of series?" is two things:

1. The question is ill defined: which series, which sequence
under what, perhaps not entirely formalized to the usual
standard, theory?  I think also in your question as posed there
is a suggestion that we are dealing with a sequence with only one
series in it.  I just looked at the Wikipedia article

   http://en.wikipedia.org/wiki/Asymptotic_series
   [page was last modified on 25 May 2012 at 07:10]

and though, for helping with this discussion, a few sentences of
the Lower Predicate Calculus, interpreted in some calculus 101
structure, would be useful, I think the article is OK.

2. The range of techniques by which we get predictions from our
theory is wider than "The theory, uncontroversially and with full
clarity, gives us this series/sequence of series and we plug in
input values.".  In particular one path to predictions starts
with a theory which one can show leads to no finite real numbers
in its predictions.  One then "corrects" the theory by a
"subtraction scheme".  I looked at this, and up until I found
Manoukian could make neither head nor tail of it.  When I read
Manoukian, I saw that the theory used crucially, in order to
figure out which diagram-integrals to add or subtract, a Moebius
inversion.  This is not a negligible matter.  Serious physicists
sometimes disagreed as to the sign which should be given to a
particular diagram-integral.  Later Kreimer et al clarified all
this.  Today some people think that one version of QED has been
laid out in such a way that the theory is "consistent" by
(some|the) usual standard of mathematicians.  Other competent
people think not.  If the subtraction scheme is part of a well
defined and persuasively consistent theory, then I think that
indeed, your criterion is met.  But only at the last step in one
line of exposition do we get a series about which it makes
sense to ask "Where does this series converge?".

The more interesting question, which I now think you are asking,
is:

   Is there a consistent theory of QED, consistent to the usual
   standard of professors of mathematics?

If I before misunderstood your question to be a less interesting
question, please accept my apology!

Finally, in some expositions of QED something like this is said:

"Our theory is an effective theory and we leave undetermined some
constants.  We do assume that their values are set so that the
predictions of our theory match well established experimental
results, and we think certain further predictions made by our
theory will be approximately right."

I have seen suchlike statements made in expositions which suggest
that electro-weak unification, or something like it, is required
to get a consistent theory.

Your question is close in feel, I think, to questions like this
one: "What set theory is needed for Grothendieck et al's attack
on arithmetic algebraic geometry?".

To repeat: Experts disagree as to whether there has been
published a consistent theory of QED.  There is further
disgareement as to whether such a theory is possible without some
stuff most would consider not part of QED.

oo--JS.


>
> Sent from my iPhone
>
> On Feb 24, 2013, at 4:37 AM, Jaykov Foukzon <jaykovfoukzon at list.ru> wrote:
>
> Joe Shipman wrote:
>
>> If my account is correct, then even the supposedly well-established
>> theory of Quantum Electrodynamics does not have a valid mathematical > foundation, because it unjustifiably assumes convergence of the relevant
>> series.
>
>
> No. Convergence of the relevant series is not supposed.There on many weaker assumption of that is made: this series is gives asymptotically decomposition on degrees of small parametre alpha
>
>
>
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