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\section{\usemenu{slacpub7115::context::slacpub7115001}{Introduction}}\label{section::slacpub7115001}
In this paper we give explicit solutions to the Dirac equation for
1+1 spacetime. These solutions are valid for discrete physics
\cite{1} using the calculus of finite differences, and they
have as limiting values solutions to the Dirac equation using
infinitesimal calculus. We find that the discrete solutions can be
directly interpreted in terms of sums over lattice paths in discrete
spacetime. We document the relationship of this latticepath with
the checkerboard model of Richard Feynman \cite{2}. Here we
see how his model leads directly to an {\it exact} solution to the
Dirac equation in discrete physics and thence to an exact continuum
solution by taking a limit. This simplifies previous approaches to
the Feynman checkerboard \cite{3,4}.
We also interpret these solutions in terms of choice sequences
(bitstrings) and we show how the elementary combinatorics of
$i=\sqrt{1}$ as an operator on ordered pairs ($i[a,b] =[b,a]$)
informs the discrete physics. In this way we see how solutions to
the Dirac equation can be built using only bitstrings, and no
complex numbers. Nevertheless the patterns of composition of $i$
inform the inevitable structure of negative case counting
\cite{5,6} needed to build these solutions.
The paper is organized as follows. Section 2 reviews the Dirac
equation and expresses two versions (denoted RI, RII) in light cone
coordinates. The two versions depend upon two distinct
representations of the Dirac algebra. Section 3 reviews basic facts
about the discrete calculus and gives the promised solutions to the
Dirac equation. Section 4 interprets these solutions in terms of
lattice paths, Feynman checkerboard and bitstrings. Section 5
discusses the meaning of these results in the light of the
relationship between continuum and discrete physics.
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