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\sectionLink{slac-pub-7205-0-0-1}{slac-pub-7205-0-0-1}{Above: 1. Pre-ANPA IDEAS: A personal memoir}%
\subsectionLink{slac-pub-7205-0-0-1}{slac-pub-7205-0-0-1-2}{Next: 1.2. From ``NON-LOCALITY'' to ``PITCH'': 1974-1979}%
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\subsection{\usemenu{slac-pub-7205::context::slac-pub-7205-0-0-1-1}{First Encounters}}\label{subsection::slac-pub-7205-0-0-1-1}
When I first met Ted Bastin in 1972 and heard of the Combinatorial Hierarchy
(hereinafter CH),
my immediate reaction was that it must be dangerous nonsense. Nonsense,
because the two numbers computed to reasonable accuracy ---
$137 \approx \hbar c/e^2$ and $2^{127} + 136 \approx \hbar c/Gm_p^2$ ---
are {\it empirically determined}, according to
conventional wisdom. Dangerous, because the idea that one can gain
insight into the physical world by ``pure thought'' without empirical
input struck me then (and still strikes me) as subversive of the
fundamental Enlightenment rationality which was so hard won, and which
is proving to be all too fragile in the ``new age'' environment
that the approach to the end of the millennium seems to encourage
\cite{84,86}.
Consequently when Ted came back to Stanford the next year
(1973)\cite{8}, I made sure
to be at his seminar so as to raise the point about empirical input with as much force
as I could. Despite my bias, I was struck from the start of his talk by his obvious
sanity, and by a remark he made early on (but has since forgotten) to the
effect that {\it the basic quantization is the quantization of mass}.
When his presentation came around to the two ``empirical'' numbers,
I was struck by the thought that some time ago Dyson\cite{19}
had proved that if one calculates perturbative QED up to the
approximation in which 137 electron-positron pairs can be present,
the perturbation series in powers of $\alpha =e^2/\hbar c \approx
1/137$ is no longer uniformly convergent. Hence, the number 137 {\it as
a counting number} already had a respectable place in the paradigm
for relativistic quantum field theory known as renormalized quantum
electrodynamics (QED). The problem for me became {\it why} should the
arguments leading to CH produce a number which {\it also}
supports this particular physical interpretation.
As to the CH itself, I refer you to Clive Kilmister's introductory talk
in these proceedings\cite{39}, where he discusses an
early version of the bit-string construction of the sequence of
discriminately closed subsets with cardinals $2^2-1=3\rightarrow 2^3-1=7\rightarrow
2^7-1=127\rightarrow 2^{127}-1\approx 1.7\times 10^{38}$ based on
bit-strings of length 2,4,16,256 respectively..
The first three terms can be mapped by square matrices of
dimension $2^2=4\rightarrow 4^2=16 \rightarrow 16^2=256$.
The $256^2$ discriminately independent matrices made available by
squaring the dimension needed to map the third level
are many two few to map the $2^{127}-1$ discriminately closed subsets
in the fourth level, terminating
the construction. In the historical spirit of this memoir, I add that thanks to some
archeological work John Amson and I did in John's attic in St. Andrews,
the original paper on the hierarchy by Fredrick Parker-Rhodes, drafted
late in 1961, is now available\cite{79}.
I now ask you to join with me here in my continuing investigation of how the CH
can be connected to conventional physics. As you will see in due course,
this research objective differs considerably from the aims of Ted
Bastin and Clive Kilmister. They, in my view, are unnecessarily dismissive
of the results obtained in particle physics and physical cosmology
using the conventional (if mathematically inconsistent) relativistic
quantum field theory, in particular quantum electrodynamics (QED),
quantum chromodynamics (QCD) and weak-electromagnetic unification
(WEU).
Before we embark on that
journey, I think it useful to understand some of the physics
background. Dyson's argument itself rests on one of the most profound
and important papers in twentieth century physics. In 1937 Carl
Anderson discovered in the cosmic radiation a charged particle he
could show to be intermediate in mass between the proton and electron.
This was the first to be discovered of the host of particles now called collectively
``mesons'' . {\it One} such particle had already been postulated by
Yukawa, a sort of ``heavy photon'' which he showed, using a ``massive
QED'', gave rise to an exponentially bounded force of finite range.
If the mass of the Yukawa particle was taken to be a few hundred
electron masses, this could be the ``nuclear force quantum''.
Anderson's discovery prompted Gian Carlo Wick to try
to see if the existence of such a particle could be accounted for
simply by invoking the basic principles of quantum mechanics and
special relativity. He succeeded brilliantly, using only one column
in {\it Nature}\cite{96}. We summarize his argument here.
Consider two massive particles which are
within a distance $R$ of each other during a time $\Delta t$. If they are to act coherently,
we must require $R \leq c\Delta t$. [Note that this postulate, in the
context of my {\it neo-operationalist} approach\cite{68}
based on measurement accuracy\cite{76}, opens the
door to {\it supraluminal} effects at short distance, which I am now
starting to explore\cite{69}]. Because of the uncertainty
principle this short-range coherence tells us that the energy
is uncertain by an amount $\Delta E\approx \hbar/\Delta t$.
But then mass-energy equivalence allows a
particle of mass $\mu$ or rest-energy $\mu c^2\geq \Delta E$ to
be present in the space time-volume of linear dimension $\approx
R\Delta t$. Putting this together, we have the {\it Wick-Yukawa
Principle}:
\begin{equation}
R\leq c\Delta t\approx {c\hbar \over \Delta E} \leq {\hbar/\mu c}
\end{equation}
Put succinctly, if we try to localize two massive particles within
a distance $R \leq \hbar/\mu c$, then the uncertainty principle allows a
particle of mass $\mu$ to be present. If this meson has the proper
quantum numbers to allow it to transfer momentum between the two
massive particles we brought together in this region, they will
experience a force, and will emerge moving in different directions
than those with which they entered the {\it scattering region}.
Using estimates of the range of nuclear forces obtained from
deviations from Rutherford scattering in the 1930's one can then estimate the mass
of the ``Yukawa particle'' to be $\approx 200-300$ electron masses.
We are now ready to try to follow Dyson's argument. By 1952, one was
used to picturing the result of Wick-Yukawa uncertainty at short
distance as due to ``vacuum fluctuations'' which would allow
$N_e$ electron-positron pairs to be present at distances $r\leq
\hbar/2Nm_ec$. This corresponds to taking $\mu =2Nm_e$ in Eq. 1.
Although you will not find it in the reference\cite{19}, in a seminar
Dyson gave on this paper he presented what he called a crude way to understand
his calculation making use of the non-relativistic coulomb potential.
I construct here my own version of the argument.
Consider the case where there are $N_e$ positive charges in one clump
and $N_e$ negative charges in the other, the two clumps being a
distance $r=\hbar/m_ec$ apart. Then a single charge from one clump will have
an electrostatic energy $N_ee^2/r= N_e[e^2/\hbar c]m_ec^2$
due to the other clump and visa
versa. I do recall that Dyson said that the system is dilute enough
so that non-relativistic electrostatic estimates of this type are
a reasonable approximation. If under the force of this
attraction, these two charges we are considering come together
and scatter producing a Dalitz pair ($e^+ +e^-\rightarrow 2e^+ +2e^-$)
the energy from the fluctuation will add another pair to the system.
Of course this process doesn't happen physically because like charges
repel and the clumps never form in this way. However, in a theory
in which like charges attract [which is equivalent to renormalized QED
with $\alpha_e= [e^2/\hbar c] \rightarrow -\alpha_e$ in the renormalized
perturbation series], once one goes beyond 137 terms such a process
will result in the system {\it gaining} energy by producing another
pair and the system collapses to negatively infinite energy.
Dyson concluded that the renormalized perturbation
series cannot be uniformly convergent, and hence that QED cannot be a
fundamental theory, as I have subsequently learned from Schweber's
history of those heroic years\cite{85}.
Returning to 1973, once I had understood that, thanks to
Dyson's argument, 137 can be interpreted as
a {\it counting number}, I saw immediately that $2^{127}+136\approx
1.7\times 10^{38}\approx \hbar c/Gm_p^2$ could {\it also} be interpreted
as a counting number, namely the number of baryons of protonic mass
which, if found within the Compton wavelength of any one of them, would
form a black hole.
These two observations removed my objection to the
calculation of two pure numbers that, conventionally interpreted,
depend on laboratory measurements using arbitrary units of mass, length
and time. I could hardly restrain my enthusiasm long enough
to allow Ted to finish his seminar before bursting out with this
insight. If this cusp turns out to be the point
at which a new fundamental theory takes off --- as I had hoped to make
plausible at ANPA 18 --- then we can tie it firmly into the
history of ``normal science'' as the point where a ``paradigm shift'',
in Kuhn's sense of the word\cite{40}, became possible.
However, my problem with {\it why} the calculation made by Fredrick
Parker-Rhodes\cite{79,80} lead to these
numbers remained unresolved. Indeed, I do not find a satisfactory
answer to that question even in Ted and Clive's book published last
year\cite{10}. [I had hoped to get further with that quest
at the meeting (ANPA 18, Sept.5-8, 1996), but discussions during
and subsequent to the meeting still leave many of my questions
unanswered. I intend to review these discussions and draw my own
conclusions at ANPA 19 (August 14-17,1997).]