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\sectionLink{slac-pub-7205-0-0-3}{slac-pub-7205-0-0-3}{Above: 3. Lessons from the rejection of the Fine Structure paper}%
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\subsection{\usemenu{slac-pub-7205::context::slac-pub-7205-0-0-3-1}{Background}}\label{subsection::slac-pub-7205-0-0-3-1}
In preparation for ANPA 9, Christoffer Gefwert\cite{32},
David McGoveran\cite{49} and I\cite{61} prepared
three papers intended to present a common philosophical and methodological
approach to discrete physics. Unfortunately, in order to get the
first two papers typed and processed by slac, I had to put my name on
them, but I want it in the record that my share in Gefwert's and
McGoveran's papers amounted mainly to criticism; I made no substantial
contribution to their work.
We started the report on ANPA 9\cite{60}
with these three papers, followed by a paper on Combinatorial Physics
by Ted Bastin, John Amson's
Parker Rhodes Memorial Lecture (the first in this series), a second
paper by John, and a number of first rate contributed papers. Clive
Kilmister's concluding remarks closed the volume. I went to
considerable trouble to get the whole thing into camera ready format
and tried to get the volume
into the Springer-Verlag lecture note series, but they were unwilling
to accept such a mixed bag. They were interested in the first three
papers and were willing to discuss what else to include, but I was
unwilling to abandon my comrades at ANPA by dropping any of their
contributions to ANPA 9. We ended up publishing the proceedings
ourselves, with some much needed help on the physical production
from Herb Doughty, which we gratefully acknowledge.
David and I did considerably more work on my paper, and I tried to get
it into the mainstream literature, but to no avail. Our joint version
ended up in {\it Physics Essays}\cite{73}. In the
interim David had seen how to calculate the fine structure
of hydrogen using the discrete and combinatorial approach, and
presented a preliminary version at ANPA 10\cite{46}. I was so
impressed by this result (see below) that I tried to get it
published in {\it Physical Review Letters}. It was rejected even after
we rewrote it in a vain attempt to meet the referee's objections.
In order for the reader to form his own opinion about this rejection,
I review the paper\cite{51} here and quote extensively from it.
The first three pages of the paper reviewed the arguments leading to CH
and the essential results already achieved. These will already
be familiar to the careful reader of the material given above. With this as
background we turned to the critical argument:
\begin{quote}
We consider a system composed of two masses, $m_p$ and $m_e$ |
which we claim to have computed from first principles\cite{62}
in terms of $\hbar, c$ and $G_{[Newton]}$ | and identified by their labels using
our quantum number mapping onto the combinatorial hierarchy
\cite{73}.
In this framework, their mass ratio
(to order $\alpha ^3$ and $(m_e/m_p)^2$)
has also been computed using only $\hbar, c$ and 137.
However, to put us in a situation more analagous to that of Bohr,
we can take $m_p$ and $m_e$ from experiment, and treat $1/137$
as a counting number representing the coulomb interaction;
we recognize that corrections of the order of the square of this number
{\it may} become important one we have to include degrees
of freedom involving electron-positron pairs.
We attribute the binding of $m_e$ to $m_p$ in the hydrogen atom
to coulomb events, i.e. only to those events which involve a specific
one of the 137 labels at level 3 and hence
occur with probability $1/137$;
the changes due to other events average out (are {\it indistinguishable}
in the absence of additional information).
We can have
any periodicity of the form $137 j$ where $j$ is any positive
integer.
So long as this is the only periodicity,
we can write this restriction as $137 j$ {\it steps}
$= 1$ {\it coulomb event}.
Since the internal frequency $1/137j $ is generated independently from
the {\it zitterbewegung} frequency which specifies the mass scale,
the normalization condition
combining the two must be in quadrature. We meet the bound
state requirement that the energy E be less than the system
rest energy $m_{ep} c^2$ (
where $m_{ep}= m_em_p/(m_e +m_p)$ is used to take account of 3-momentum
conservation) by requiring
that $(E/\mu c^2)^2[1 + (1/137N_B)^2]=1$.
If we take $e^2/\hbar c= 1/137$, this is just the relativistic
Bohr formula\cite{14} with $N_B$ the principle quantum number.
\end{quote}
[Here I inserted into McGoveran's argument a discussion of the
Bohr formula and how it might
be derived from dispersion theory. This insertion was motivated
by the vain hope that any referee would see that our reasoning was in fact
closely related to standard physics. We will look at this result,
called the handy-dandy formula, in a new way
in the section of this paper carrying that title.]
\begin{quotation}
The Sommerfeld model for the hydrogen atom
(and, for superficially different but profoundly similar reasons
\cite{12}, the Dirac model as well)
requires two {\it independent} periodicities.
If we take our reference period $j$ to be integer
and the second period $s$ to differ from an integer
by some rational fraction $\Delta$, there will be two minimum
values $s_0^{\pm} = 1 \pm \Delta$, and other values of $s$ will
differ from one or the other of these values by integers: $s_n=n+s_0$.
This means that we can relate (``synchronize'') the fundamental
period $j$ to this second period in two different ways, namely to
\begin{equation}
137 j {steps \over (coulomb \ event)}
+137 s_0 {steps \over (coulomb \ event)}
= 1+e=b_+\nonumber
\end{equation}
or to
\begin{equation}
137 j {steps \over (coulomb \ event)}
-137 s_0{steps \over (coulomb \ event)}
= 1-e=b_-\nonumber
\end{equation}
where $e$ is an event probability.
Hence we can form
\begin{equation}
a^2=j^2-s_0^2=(b_+/137)(b_-/137)=(1-e^2)/137^2
\end{equation}
Note that if we want a finite numerical value for
$a$, we cannot simply take a square root,
but must determine from context which of the symmetric factors
[i.e. $(1-e)$ or $(1+e)$]
we should
take (c.f. the discussion about factoring a quadratic above).
With this understood, we write $s_n=n+\sqrt{j^2 -a^2}$.
\par
We must now compute the probability $e$ that $j$ and $s$ are
mapped to the same label, using a
single basis representation constructed within the combinatorial
hierarchy. We can consider the quantity $a$ as an event
probability corresponding to an event {\bf A} generated by a global
ordering operator which ultimately generates the entire structure
under consideration. Each of the two events $j$ and $s$ can be
thought of as derived by sampling from the same population. That
population consists of 127 strings defined at level three of the
hierarchy. In order that $j$ and $s$ be independent, at least the
last of the 127 strings generated in the construction of $s$ (thus
completing level three for $s$) must not coincide with any string
generated in the construction of $j$. There are 127 ways in which
this can happen.
There is an additional constraint. Prior to the completion
of level three for $s$, we have available the $m_2 = 16$ possible
strings constructed as a level two representation
basis to map (i.e. represent)
level three.
One of these is the null string and cannot be used, so there are
15 possibilities from which the actual construction of the label for
$s$ that
completes level 3 are drawn.
The level can be completed
just before or just after some $j$ cycle is completed.
So, employing the usual frequency theory of probability,
the expectation $e$
that $j$ and $s$ as constructed
will be indistinguishable is $e=1/(30\times 127)$.
\par
In accordance with the symmetric factors $(1-e)$ or $(1+e)$
the value $e$ can either subtract from or add to the probability of
a coulomb event.
These two cases
correspond to two different combinatorial paths by which the
independently generated sequences of events may close
(the ``relative phase'' may be either positive or negative).
However we require only the probability that all $s_0$ events
be generated within one period of $j$, which is $1-e$.
Hence the difference between $j^2$ and $s^2$ is to be computed as
the ``square'' of this ``root'', $j^2-s_0^2=(1-e)^2$.
Thus, for a system dynamically bound by the coulomb interaction
with two internal periodicities, as in the
Sommerfeld or Dirac
models for the hydrogen atom,
we conclude that the value of the fine structure constant to be used
should be
\begin{equation}
{1 \over a} = {137\over 1- {1 \over 30 \times 127}} =
137.0359 \ 674...\nonumber
\end{equation}
in comparison to the accepted empirical value of\cite{1}
\begin{equation}
{1 \over \alpha} \simeq 137.0359 \ 895(61)\nonumber
\end{equation}
Now that we have the relationship between $s,j$ and $a$, we consider
a quantity $H'$ interpreted as the energy attribute expressed in
dynamical variables at the $137j$ value of the system containing
two periods.
We represent $H' $ in units of the invariant system
energy $\mu c^2$.
The independent additional energy due to the shift of
$s_n$ relative to $j$
for a period can then be given as a fraction of this energy by
$(a/s_n)H'$, and can be added or subtracted, giving us the two
factors $(1-(a/s_n)H')$ and $(1+(a/s_n)H')$.
These are to be multiplied
just as we multiplied the factors of $a$ above,
giving the (elliptic)
equation $(H')^2/(\mu¢2c^4) +(a^2/s_n^2)(H')^2/\mu ^2c^4=1$,
Thanks to the previously derived expression of $s=n+s_0$ this
can be rearranged to give us the Sommerfeld formula\cite{87}
\begin{equation}
H'/\mu c^2=[1 + {a^2 \over (n +\sqrt{j^2-a^2})^2}]^{{-1/2}}\nonumber
\end{equation}
Several corrections to our calculated value for $\alpha$
can be anticipated,....
\end{quotation}