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= !!!

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