From: pusch@mcs.anl.gov (Gordon D. Pusch)
Newsgroups: sci.physics.particle
Subject: Re: Divergence of QED
Date: 07 Aug 1996 14:50:25 -0500

In article <4uaedl$7es@news.nyu.edu> mallahj@acf2.nyu.edu 
(Jacques Maurice Mallah) writes:

> Douglas A. Singleton (das3y@faraday.clas.Virginia.EDU) wrote:
> : The basic idea is this. Many physical quantities (such as the magnetic 
> : moment of the electron) can be calculated as a power series in e^2
> : (e=electric coupling) so that
> : 
> : F(e^2) = a_0 + a_1 e^2  + a_2 e^4 + ....
> : 
> : where the coeffs. a_i are gotten via the renormalization program. Now
> : if this series converges then so should the series if one lets
> : e ---> -e. However Dyson argues that this transformation will give
> : you a theory where like charges attract each other.
>
>   Thanks, but why is that?  If the charge of the electron is -e and
> the charge of the proton is +e, then changing e to -e doesn't change
> anything.  That's why I thought it was the sign of e^2, actually the
> sign of Coulomb's constant, that was to be changed, and then it
> would matter whether the a_n are positive definate.

Jacques is correct. Note that the expansion only contains powers of
'e^2', not 'e'; this is because each term in the series corresponds 
physically to a set of diagrams with 0,1,2,3,... virtual photons,
and you get one power of 'e' for each fermion/photon interaction
vertex. Since 'virtual' photons BY DEFINITION must be both _emitted_
AND _absorbed_ internal to the diagram (if they could escape, they'd 
be REAL photons, not virtual!), one always gets factors of 'e^2' from
virtual photons, not 'e'. Hence it is only the sign of 'e^2' that matters.
(A.O. Barut described this procedure as ``...begining by assuming an
electron has no coulomb field; then, putting the coulomb field back in, 
one [virtual] photon at a time...'')

[Actually, the expansion is not in terms of 'e' itself, since 'e' is 
a dimensional quantity, but rather the ``fine structure constant''
'\alpha := e^2/(\hbar c)' which is a small dimensionless number = ~1/137. 
It doesn't make  sense to expand =ANYTHING= in powers of a dimensional
parameter, since different powers have different units, nor can said 
parameter be meaningfully said to be 'large' or 'small' --- except in
comparison to another quantity having the same units...]


Dyson's argument, while not entirely rigorous, strongly suggests that
the physical quantities being expanded in the perturbation series are
=NOT= analytic functions of the fine structure constant --- that is,
their singularities are not just simple poles. It is now generally
accepted that =ALL= perturbative expansions are neither convergent 
nor even conditionally convergent --- at best, they are =ASYMPTOTIC
SERIES=. That means that, while these series are formally divergent,
(have =ZERO= radii of convergence), partial summations of their
leading terms APPEAR to converge up through some finite order, 
then begin to diverge rapidly. (In the case of QED, Dyson's argument
suggests that the series will appear to converge for about about the
first ~137 terms; this is of little ``practical'' significance, 
since the combinatoric complexity of computing each term increase
super-exponentially...)

In fact, later workers have shown that the so-called ``S matrix'' of 
QED, which in some sense contains ALL the physical information that 
CAN be observed in a quantum-mechanical theory, must NECESSARILY have
``branch cut'' singularities. This is actually a quite profound result, 
because quantum mechanically, only simple poles can be interpreted as
asymptotically observable states --- that is, as ``physical particles''
that may be detected experimentally. However in reality, it appears 
that what we call ``an electron'' is =NOT= the manifestation of a 
simple pole --- it is not a ``particle'' in the sense of quantum field
theory, but rather an ``infraparticle'' (See, e.g., Jauch and Rohrlich's 
``Theory of Photons and Electrons''). In fact, the basis of the 
renormalization procedure is to =FORCE= poles to exist at the
``physically observed'' locations --- even though these ``poles'' 
(particles) are in fact not poles at all, but branch cuts !!!

In other words: perturbative quantum field theory begins by assuming
something =KNOWN= to be untrue, yet blindly proceeds ahead to expand
quantities =AS IF= this falsehood were true, regardless !!!  

Personally, I've long felt the whole perturbative QFT approach is
therefore wrong-headed: we should be searching for ways to handle
infraparticles =AS= infraparticles, rather than than blindly
pretending they are merely simple poles. (Most of my colleagues
disagreed, of course; eventually, I stopped beating my head 
against the establishment wall, and left high-energy physics 
for more tractable problems --- like weather-simulation... :-T)


Gordon D. Pusch                     |  Internet: <pusch@mcs.anl.gov>
Math and C.S. Div., Bldg.203/C254   |  FAX:      (708) 252-5986
Argonne National Laboratory         |  Phone:    (708) 252-3843
9700 South Cass Ave.                |  
Argonne, IL  USA  60439-4844        |  http://www.mcs.anl.gov/people/pusch/

But I don't speak for ANL or the DOE, and they *sure* don't speak for =ME=...

From: pusch@mcs.anl.gov (Gordon D. Pusch)
Newsgroups: sci.physics,sci.physics.particle
Subject: Re: Doubt about photons
Date: 09 Apr 1997 00:22:34 -0500
In-reply-to: Patrick Van Esch's message of Tue, 08 Apr 1997 23:54:10 +0000

It might interest all of you to know that, in fact, what Martin Green
is proposing is not new --- Ed Jaynes and his collaborators worked out
this ``neoclassical electrodynamics'' approach to matter/radiation
interactions in considerable detail back in the early seventies,
and it caused about as much controversy and consternation back then
as it is here and now... :-T

It's quite remarkable just how far one can take this approach ---
they get a reasonable spontaneous lifetime (albeit with the ``wrong''
lineshape --- however this particular prediction vs. QED's *still*
hasn't been tested experimentally, yet, to the best of my knowledge),
a Lamb shift of the correct order of magnitude (if Barut et.al.'s
related calculations are correct, it is in fact =exact=!), and claims
to reproduce both the photoelectric effect and the correct Einstein
'A' and 'B' coefficients. Jaynes also argues plausibly that the
ultraviolet catastrophe may be evaded via the same nonlinear effects
that prevent equipartion from occuring in, e.g., the Toda lattice and
other nonlinear systems that exhibit Fermi-Pasta-Ulam recurrance...

For those interested, I strongly recommend reading Jaynes'
review-paper on the status of neoclassical electrodynamics in
``Coherence and Quantum Optics'' [eds. Mandel and Wolf; 1977].
It's worth reading, if only for Jaynes' critical comments on the
experimental and theoretical foundations of quantum mechanics, and
the chapter near the end on the ``reality'' of vacuum zero-point
energy, and how most of the ``evidence'' for vacuum zero-point
fluctuations can more easily be understood as simple consequences
of radiation damping and the Fluctuation-Dissipation Theorem...


--  Gordon D. Pusch   <pusch@mcs.anl.gov>

Disclaimer:  I'm a consultant ---  I don't speak for ANL or the DOE,
and they *certainly* don't speak for =ME=  !!!