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\section{\usemenu{slacpub7115::context::slacpub7115005}{Epilogue}}\label{section::slacpub7115005}
If we had started by saying (in the RI case) we had a simple
solution for the Dirac equation (discretized) using nothing but {\it
bitstrings} (L,R choice sequences) and appropriate signs, then it
would have been natural to ask: How are these signs justified on the
basis of a philosophy of bitstrings? In retrospect we can answer:
This pattern of signs is very simple, but not (yet) to be deduced
from the notion of a distinction alone. Nevertheless, it does arise
naturally from the simple structures that are available at that
primitive level. The $i$ operator ($i[a,b] = [b,a]$) does not
involve anything more sophisticated that the idea of exchanging the
labels on the two sides of a distinction followed by the flipping of
a label on a given side:
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\noindent
as is discussed elsewhere \cite{11,12}. A
choice sequence such as
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has ``corners'' wherever R meets L or L meets R. We have
characterized these corners into two types RL and LR:
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We then enumerate the choice sequences in terms of lattice paths in
Minkowski space and the solutions to the Dirac equation emerge,
along with a precursor to spin and the role of $i=\sqrt{1}$ in
quantum mechanics. We have shown exactly how this point of view
interfaces with Feynman's Checkerboard.
Corners in the bitstring sequence alternate from RL to LR and from
LR to RL. The moral of Feynman's $(i)^c$ where $c$ is the number of
corners is that this alteration should be regarded as an elementary
rotation:
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\noindent
One may wonder, why does this simple combinatorics occur in a level
so close to the making of one distinction, and yet implicate fully
the solutions to the Dirac equation in continuum 1+1 physics?! We
cannot begin to answer such a question except with another question:
If you believe that simple combinatorial principles underlie not
only physics and physical law, but the generation of spacetime
herself, then these principles remain to be discovered. What are
they? What are these principles? It is no surprise to the
mathematician that $i$ ends up as central to the quest. For $i$ is a
strange amphibian not only neither 1 nor $1$, $i$ is neither discrete
nor continuous, not algebra, not geometry, but a communicator of
both. In this essay we have seen the beginning of a true connection
of discrete and continuum physics.
The continuum version of our theory merges the paths on the lattice
to a sum over all possible paths on an infinitely divided rectangle
in Minkowski spacetime. The individual paths disappear into the
values of the series $ \psi_0 = \Sigma_{k=0}^{\infty} (1)^k
\frac{r^{k}}{k!}\,\frac{\ell^{k}}{k!} $, $
\psi_L=\Sigma_{k=0}^{\infty}(1)^k
\frac{r^{k}}{k!}\,\frac{\ell^{k+1}}{(k+1)!}$ , $
\psi_R=\Sigma_{k=0}^{\infty}(1)^k
\frac{r^{k+1}}{(k+1)!}\,\frac{\ell^{k}}{k!} $. Here we have a glimpse
of the possibilities inherent in a complete story of discrete
physics {\it and} its continuum limit. The continuum limit will be
seen as a {\it summary} of the real physics. It is a way to view,
through the glass darkly, the crystalline reality of simple quantum
choice.
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