[FOM] Question about theoretical physics

[Jay Sulzberger]

On Tue, 26 Feb 2013, Arnold Neumaier <Arnold.Neumaier at univie.ac.at> wrote:

> On 02/25/2013 02:04 AM, Jay Sulzberger wrote:
>> 
>> [The sum of a power series with convergence radius zero is logically]
>> Not necessarily meaningless.  One can define other "summation
>> methods".  Work of the past about fifteen years connects certain
>> classical summation methods with old puzzlements about what
>> "renormalization by iterated subtractive over and under
>> corrections" using Feynman diagrams is:
>
> The problem is that any fixed summation methods give the same results
> for the asymptotic expansion of g(alpha)+c*e^{-1/alpha^2} for any c (as these 
> have the same asymptotic expansion). Hence there is no guarantee at all that 
> the result of the summation method corresponds to the
> ''true'' value predicted by a future nonperturbative formulation of QED.
>
> For example, to use Borel summation one needs to know (or pretend to know) 
> analytic properties in a cone with vertex 0, properties that are most likely 
> invalid for QED.
>
> This is completely independent of the various renormalization tools.

I should have stated with greater generality my main point, which is:

   There are several competing "foundations" for QED, such as the
   various subtraction schemes, the "causal" QED as set forth in
   G. Scharf's "Finite quantum electrodynamics", the "net of
   algebras" QED as set forth in R. Haag's "Local Quantum Physics"
   and more.

Thank you for a clear statement of the state of play today!  I
did not know that the recent study of summability methods has so
far not improved the "theory" (say under still uncertain
assumptions of "consistency") beyond what you state above.

oo--JS.