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%% section 9 Finite and Discrete Lorentz Transformations [slacpub7205009 in slacpub7205009: slacpub72050010]
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\section{\usemenu{slacpub7205::context::slacpub7205009}{Finite and Discrete Lorentz Transformations}}\label{section::slacpub7205009}
We now consider a situation in which our laboratory is receiving
two independent input signals one of which, for segments of length
$2W$, repeatedly gives Hamming measure $a$ and the other $2W  a$. Because of
our experience with the Doppler shift, we leap to the conclusion
that our laboratory is situated between two standard clocks similar
to our own which are sending output signals to us. We assume that
they are at relative rest but that our own lab is moving toward
the one for which the recorded Hamming measure is larger than $W$ and
away from the second one for which the recorded Hamming measure
is smaller than $W$. We calculate our velocity relative to these
two stationary, signalling tictoc clocks as $v_{lab} = (aW)/W$
measured relative to the velocity of light. If our lab is in fact a
rocket ship and we have any fuel left, we can immediately test this
hypothesis by turning on the motors and seeing if, after they have been
on long enough to give us a known velocity increment $\Delta v$,
our velocity measured relative to these external clocks changes to
\begin{equation}
v' = {v +\Delta v\over 1 + v\Delta v}
\end{equation}
If so, we have established our motion relative to a given, external
framework. Rather than go on to develop the bitstring version
of finite and discrete Lorentz boosts, which is obviously already
implicitly available, I defer that development until we have discussed
the more general bitstring transformations developed in the section
below on commutation relations. For an earlier approach,
see\cite{64}.
This situation is not so far fetched as might seem at first glance.
Basically, this is how the motion of the earth, and of the solar
system as a whole, have been determined relative to the $2.73 ^oK$ cosmic
background radiation in calibrating the COBE satellite measurements that give us
such interesting information about the early universe.
To extend this ``calibration'' of our laboratory relative to the
universe to three dimensions, we need only find much simpler pairs
of signals than those corresponding to the background radiation,
namely pairs, which for the moment we will call $U$ and $D$ which have
the properties
\begin{eqnarray}
{\bf U}\cdot {\bf U} &=& W = {\bf D}\cdot {\bf D}\nonumber\\
{\bf U}\cdot {\bf I} &=& W = {\bf D}\cdot {\bf I}\nonumber\\
{\bf U}\cdot {\bf D} &=& 0\\
{\bf U}\oplus {\bf D} &=& {\bf I}\nonumber
\end{eqnarray}
These look just like our standard clock, but compared to it we find
that
\begin{eqnarray}
{\bf U}\cdot {\bf R} &=& W +\Delta = {\bf D}\cdot {\bf L}\nonumber\\
{\bf U}\cdot {\bf L} &=& W \Delta = {\bf D}\cdot {\bf R}
\end{eqnarray}
where (for ${\bf U},{\bf D}$ distinct from ${\bf R},{\bf L}$)
we have that $\Delta \in 1,2,...,W1$.
These form the starting point for defining directions and finite and
discrete rotations. As has been proved by McGoveran\cite{45},
using a statistical result obtained by Feller\cite{29},
at most three independent sequences which repeatedly have the same number
of tic's in synchrony can be expected to produce such recurrences
often enough to serve as a ``homogeneous and isotropic'' basis for
describing independent dimensions, showing that our tictoc
lab {\it necessarily} resides in a threedimensional space.
It is well known that finite and discrete rotations
of any macroscopic object of sufficient complexity, such as our
laboratory, {\it do not commute}. It is therefore useful
to develop the noncommutative bitstring transformations
before we construct the formalism for finite and discrete Lorentz
boosts {\it and} rotations as a unified theory.
Once we have done so, we expect to understand better
why finite and discrete commutation
relations imply the finite and discrete version
of the free space Maxwell\cite{35} and Dirac
\cite{36}equations, which we developed
in order to answer some of the conceptual questions raised by
Dyson's report\cite{20} and analysis\cite{21} of Feynman's
1948 derivation\cite{30} of the Maxwell equations
from Newton's second law and the nonrelativistic quantum mechanical
commutation relations; we intend to extend our analysis to gravitation
because we feel that Tanimura's extension of the Feynman derivation
in this direction\cite{89} raises more questions than it
answers.
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