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%% subsection 1.2 From ``NON-LOCALITY'' to ``PITCH'': 1974-1979 [slac-pub-7205-0-0-1-2 in slac-pub-7205-0-0-1: slac-pub-7205-0-0-1-3]
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\subsection{\usemenu{slac-pub-7205::context::slac-pub-7205-0-0-1-2}{From ``NON-LOCALITY'' to ``PITCH'': 1974-1979}}\label{subsection::slac-pub-7205-0-0-1-2}
My interest in how to resolve this puzzle has obviously continued to this
day. I was already impressed by the quality of Fredrick's results in
1973, and made a point of keeping in contact with Ted Bastin. This led to my
meeting with Fredrick Parker-Rhodes, Clive Kilmister and several of the other Epiphany
Philosophers at a retreat in the windmill at Kings Lynn, followed by discussions in
Cambridge. At that point the group were trying to put together a volume
on {\it Revisionary Philosophy and Science}. I agreed to contribute a
chapter, and finished about half of the first draft of what was to
become ``Non-Locality in Particle Physics'' \cite{55,56} on the
plane going back to Stanford. In the course of finishing that article,
I noted for the first time that the Dyson route to the
hierarchy number places an energy cutoff on the validity of QED at
$E_{max}=2\times 137 m_ec^2$, which is approximately equal to the pion (Yukawa
particle) mass. I have subsequently realized that this explains a
puzzle I had been carrying with me since I was a graduate student.
This puzzle, which I have sometimes called the {\it Joe Weinberg memnonic},
came from quite another direction\cite{91}. An easy way to
remember the hierarchy of nuclear, QED, and atomic dimensions expressed
in terms of fundamental constants is the fact that
\begin{equation}
1.4 \ Fermi \approx {e^2 \over 2m_ec^2} =\left[{e^2\over \hbar c}\right]{\hbar \over 2m_ec}
=\left[{e^2\over \hbar c}\right]^2{\hbar^2 \over 2m_ee^2} \approx 0.265
\ Angstrom
\end{equation}
Why nuclear dimensions should be approximately half the ``classical
electron radius'' (i.e. ${e^2 \over 2m_ec^2}\approx 1.4\times 10^{-15}
meter$) and hence $[1/137]^2$
smaller than than the radius of the positronium atom (i.e. ${\hbar^2
\over 2m_ee^2}\approx 2.65\times 10^{-10} meter $) was almost completely mysterious
in 1947. It was known that the mass of the electron attributed to it's
electrostatic ``self-energy'' as due to its charge distributed over a spherical shell
fixed at this radius would have the mass $m_e$, but the success of Einstein's relativity
had shown that this electron model made no sense\cite{78}. The square of this
parameter was also known to be proportional to the cross section for scattering a low energy
electromagnetic wave from this model electron (Thompson cross section
$[8\pi/3](e^2/m_ec^2)^2$),
but again why this should have anything to do with nuclear forces was
completely mysterious.
As we have already seen, it {\it was} known that the Wick-Yukawa principle
\cite{96} accounted roughly for the range of nuclear forces if
those forces were
attributed to a strongly interacting particle intermediate in mass between
proton and electron. However, the only known particle in that mass range
(the muon) had been shown experimentally to interact with
nuclei with an energy $10^{13}$ times smaller than the Yukawa theory of nuclear forces
demanded\cite{18}.
The Yukawa particle (the pion) was indeed discovered later that year,
but there was still no reason to connect it with the ``classical
electron radius''. Joseph Weinberg left his students to ponder this puzzle.
The trail to the solution of this conundrum starts with
a 1952 paper by Dyson\cite{19}, despite the fact that
neither he nor I realized it at the time.
Two decades later, when I first heard a detailed account of the {\it
combinatorial hierarchy}\cite{8}, and was puzzled by the problem
of how a counting number (i.e. 137) could approximate a combination of
empirical constants (i.e. $\hbar c/e^2$), I realized that this
number is both the number of terms in the perturbation series
and the number of virtual electron-positron pairs where QED ceases
to be self-contained. But, empirically, $m_{\pi}\approx 2\times 137
m_e$. Of course, if neutral this system is highly unstable due
to $2\gamma$ decay, but if we add an electron-antineutrino or a
positron-neutrino pair to the system, and identify the system with
$\pi^-$ or $\pi^+$ respectively, the system {\it is} stable until
we include weak decays in the model. This suggests that the QED theory
of electrons, positrons and $\gamma$-rays breaks down at an energy
of $[2(\hbar c/e^2) +1]m_ec^2$ due to the formation of charged pions,
finally providing me with a tentative explanation for the Joe Weinberg memnonic.
As noted above, I first presented this speculative idea some time ago
\cite{55,56}.
By the time I wrote ``NON-LOCALITY'', I was obviously committed to
engaging in serious research on the combinatorial hierarchy as part of my
professional activity. Ted was able to get a research contract to spend
a month with me at Stanford. I had hoped that this extended period
of interaction would give me a better
understanding of what was going on; in the event little progress was made on my
side. By 1978 I had met Irving Stein, and was also struggling to
understand how he could get both special relativity and the quantum
mechanical uncertainty principle from an elementary random walk. His
work, after much subsequent development, is now available in final form
\cite{88}.
Meanwhile Ted had attended the 1976 Tutzing Conference organized by
Carl Friedrick von Weizsacker and presented a paper on the
combinatorial hierarchy by John Amson. I agreed to accompany
Ted to the 1978 meeting and present a joint paper. I arrived in England to
learn of the startlingly successful successful calculation of the
proton-electron mass ratio, which Ted and I had to discuss and
digest in order to present Fredrick's result \cite{11,81}
at the Tutzing meeting,
which followed almost immediately
thereafter. This formula has been extensively discussed at
ANPA meetings. It was originally arrived at by assuming that the
electron's charge could come apart, as a statistical fluctuation,
in three steps with three degrees of freedom
corresponding to the three dimensions of space and that the
electrostatic energy corresponding to these pieces could be computed
by taking the appropriate statistical average cut off at the proton
Compton radius $\hbar/m_pc$. The only additional physical input
is the CH value for the electronic charge $e^2=\hbar c/137$. Take $0
\leq x \leq 1$ to be the fractional charge in these units and
$x(1-x)$ the charge factor in Coulomb's law. Take $0\leq y \leq 1$
to be the inverse distance between the charge fractions in that law in
units of the proton Compton radius. Then, averaging between these
limits with the appropriate weighting factors of $x^2(1-x)^2$ and
$1/y^3$ respectively, Fredrick's straightforward statistical calculation
gives
\begin{equation}
{m_p\over m_e} ={137 \pi\over <{1\over y}>}=
{137 \pi\over ({3\over 14})[1 + {2\over 7} + {4\over 49}]({4\over 5})}
\end{equation}
At that time the result was within a tenth of a standard deviation of
the accepted value. I knew this was much too good because, for example,
the calculation does not include the effect of the weak interactions.
I was therefore greatly relieved when a revision of the fit to the
fundamental constants changed the empirical value by 20 standard
deviations, giving us something to aim at when we know how to
include additional effects.
I also learned from the group during those few days before Tutzing
that up to that point no one had proved the {\it existence}
of the combinatorial hierarchy in
a mathematical sense! Subsequent to the Tutzing meeting, thanks to the kind
hospitality of K.V.Laurikainen in Finland, I was able to devote considerable time
to an empirical attack on that problem and
get a start on actually {\it constructing} specific representations of
both the $level \ 2 \longrightarrow level \ 3$ and the
$level \ 3 \longrightarrow level \ 4$ mappings.
It turned out that neither John Amson's nor our contributions to the Tutzing
conferences, despite promises, appeared in the conference proceedings.
Fortunately we had had an inkling at the meeting that this contingency
might arise. In the event we were able to turn to David Finkelstein
and write a more careful presentation of the developments up to that
point for publication in the {\it International Journal of Theoretical
Physics}\cite{11}. The first version, called ``Physical Interpretation
of the Combinatorial Hierarchy'' (or PICH for short) still lacked a formal existence
proof, but Clive came up with one; further, he and John Amson (whose
unpublished 1976 Tutzing contribution had been extended and completed to serve as an
Appendix) were able to say precisely in what sense the CH is {\it
unique}. The final title was therefore changed to ``Physical Interpretation
and mathematical structure of The Combinatorial Hierarchy''
affectionately known as PITCH. The finishing touches on this paper were
completed at the first meeting of ANPA. This brings my informal
history to the point at which Clive ended his historical sketch in
his first lecture.