[FOM] Question about theoretical physics
Jay Sulzberger
jays at panix.com
Sun Feb 24 20:04:46 EST 2013
On Sun, 24 Feb 2013, Joe Shipman <JoeShipman at aol.com> wrote:
>
> Neumaier wrote:
> ***
>> My understanding is that it is the sum of a power series in the
>> fine-structure constant "alpha", where the coefficients of the
>> power series are computable numbers, but no computable modulus
>> of convergence is involved and no proof of convergence is known.
>> This makes the predicted value Pi^0_3 in the parameter alpha.
>
> No. The sum of a power series with convergence radius zero is logically
> meaningless.
No 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:
Hopf algebraic Renormalization of Kreimer's toy model
by Erik Panzer
http://arxiv.org/abs/1202.3552
[[v1] Thu, 16 Feb 2012 10:07:05 GMT]
http://en.wikipedia.org/wiki/Combinatorics_and_physics
[page was last modified on 21 January 2013 at 03:10]
http://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula
[page was last modified on 15 January 2013 at 19:32]
The best introduction I know for the subtractive scheme of
renormalization is Edward B. Manoukian's Renormalization,
published in 1983:
@book{manoukian1983renormalization,
title={Renormalization},
author={Manoukian, E.B.},
isbn={9780080874258},
series={Pure and Applied Mathematics},
url={http://books.google.com/books?id=XENkSYN-dT8C},
year={1983},
publisher={Elsevier Science}
}
And there is more than one puzzle about what QED as she is spoke
really is, and about whether and if so where, and with what force,
"ex contradictione sequitur quodlibet" comes in.
oo--JS.
PS. I just found page 993 of E. Zeidler's Quantum Field Theory,
which was shown to me by Google's Partial Book Publication System,
@book{zeidler2009quantum,
title={Quantum field theory},
author={Zeidler, E.},
number={v. 2},
isbn={9783540853770},
lccn={2006929535},
series={Quantum Field Theory},
url={http://books.google.com/books?id=Pk9yyC239scC},
year={2009},
publisher={Springer London, Limited}
}
The page has a quote about Kreimer et al's work, which quote
deals in part with this nexus of puzzlements.
> ***
>
> My reply:
>
> It probably diverges, but there is no proof yet that it diverges, and my point was that if it doesn't diverge than it is Pi^0_3, and that IN GENERAL the sum of a power series with computable coefficients, or simply the sum of a sequence of computable reals, is, if it exists, Pi^0_3.
>
>
> Neumaier wrote:
> ***
>> Furthermore, alpha is measured by the same experimental technique
>> that is used to measure g; there is a range of values of alpha
>> which is consistent with the measured value for g.
>> QED would be considered falsified if it were shown that for all
>> alpha, the predicted value of g would lie outside the experimentally
>> measured range of possible values for g.
>
> No. By convention, falsification in high energy physics requires a
> discrepancy of 5 times the standard deviation of the measurement errors
> (which are realizations of a random variable).
> But since the predicted value is itself an uncontrolled approximation,
> it is not 100% clear when precisely QED would count as being falsified.
> ***
>
> I reply:
>
> But it is clear if you are using the number the calculation actually gives in practice (since as you point out we are far from the point where the terms will be expected to stop decreasing): what I meant in that case was precisely that even for the value of alpha that allows the best fit, the corresponding calculated value of g is outside the 5-sigma error bars of the best measurements of g, so that it's not necessary to measure alpha or g too closely to falsify QED.
>
> -- JS
>
>
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