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\section{\usemenu{slacpub7205::context::slacpub72050010}{Commutation Relations}}\label{section::slacpub72050010}
If we consider three bitstrings which discriminate to the null string
\begin{equation}
{\bf a} \oplus {\bf b} \oplus {\bf h}_{ab} = {\bf \Phi}
\end{equation}
they can always be represented by three {\it orthogonal}
(and therefore {\it discriminately independent})
strings\cite{66,70}
\begin{eqnarray}
({\bf n}_a \oplus {\bf n}_b \oplus {\bf n}_{ab})\cdot
({\bf n}_a \oplus {\bf n}_b \oplus {\bf n}_{ab}) &=&
n_a+n_b+n_{ab}\nonumber\\
{\bf n}_a \cdot {\bf n}_b &=& 0\nonumber\\
{\bf n}_a \cdot {\bf n}_{ab} &=& 0\\
{\bf n}_b \cdot {\bf n}_{ab} &=& 0\nonumber
\end{eqnarray}
as follows
\begin{eqnarray}
{\bf a} &=& {\bf n}_a \oplus {\bf n}_{ab}\Rightarrow a=n_a+n_{ab}\nonumber\\
{\bf b} &=& {\bf n}_b \oplus {\bf n}_{ab}\Rightarrow b=n_b+n_{ab}\\
{\bf h}_{ab} &=& {\bf n}_a \oplus {\bf n}_b\Rightarrow h_{ab}=n_a+n_b\nonumber
\end{eqnarray}
It is then easy to see that the Hamming measures $a,b,h_{ab}$ satisfy the
triangle
inequalities, and hence that this configuration of bitstrings can be
interpreted as representing and integersided triangle. However,
if we are given only the three Hamming measures, and invert Eq.35
to obtain the three numbers $n_a,n_b,n_{ab}$, we find that
\begin{eqnarray}
n_{ab} &=& {1\over 2}[+a+b  h_{ab}]\nonumber\\
n_a &=& {1\over 2}[+ab + h_{ab}]\\
n_b &=& {1\over 2}[a+b + h_{ab}]\nonumber
\end{eqnarray}
Hence, if either one (or three) of the integers $a,b,h_{ab}$ is (are)
{\it odd}, then $n_a,n_b,n_{ab}$ are {\it halfintegers} rather than
integers, and we {\it cannot} represent them by bitstrings.
In order to interpret the angles in the triangle as {\it
rotations}, it is important to start with orthogonal
bitstrings rather than strings with arbitrary Hamming measures.
In the argument above, we relied on the theorem that
{\it if} ${\bf n_i}\cdot {\bf n}_j=n_i\delta _{ij}$ when $i,j \in
1,2,...,N$,
{\it then}
\begin{equation}
(\Sigma_{\oplus,i=1}^N {\bf n}_i)\cdot (\Sigma_{\oplus,i=1}^N {\bf n}_i)
= \Sigma_{i=1}^N n_i
\end{equation}
which is easily proved \cite{70}. Thus in the case of two
discriminately independent strings, under the evenodd constraint
derived above, we can always construct a representation of them simply
by concatenating three strings with Hamming measures $n_a,n_b,n_{ab}$.
This is clear from a second easily proved theorem:
\begin{equation}
({\bf a}\Vert {\bf b} \Vert {\bf c} \Vert ....)\cdot
({\bf a}\Vert {\bf b} \Vert {\bf c}....) = a+b+c+....
\end{equation}
Note that because we are relying on concatenation, in order to represent two
discriminately independent strings ${\bf a}$, ${\bf b}$ in this way
we must go to strings of length $W \geq a+b+h_{ab}$ rather than
simply $W \geq a+ b$, as one might have guessed simply from knowing the
Hamming measures and the Dirac inner product.
If we go to {\it three} discriminately independent strings,
the situation is considerably more complicated. We now need to know
the {\it seven} integers $a,b,c,h_{ab},h_{bc},h_{ca},h_{abc}$,
invert a $7\times 7$ matrix, and put further restrictions on
the initial choice in order to avoid quarterintegers as well as
halfintegers if we wish to construct an orthogonal representation
with strings of minimum length $W \geq
a+b+c+h_{ab}+h_{bc}+h_{ca}+h_{abc}$.
We have explored this situation to some extent in the references cited,
but a systematic treatment using the reference system provided by
tictoc clocks remains to be worked out in detail.
The problem with noncommutation now arises if we try to get away with
the scalars $a,a_R,\Delta,W$ arrived at in the last section when we
ask for a transformation either of the basis ${\bf R},{\bf L}
\rightarrow {\bf U},{\bf D}$ or the rotation of the string ${\bf a}$
under the constraint $a_R+a_L=a=a_U+a_D$ while keeping the two
sets of basis reference strings fixed. This changes $a_Ra_L$
to a different number $a_Ua_D$, or visa versa. If one examines this situation
in detail, this is exactly analagous to raising or lowering $j_z$
while keeping $j$ fixed in the ordinary quantum mechanical theory
of angular momentum. Consequently, if one wants to discuss a system in
which both $j$ and $j_z$ are conserved, one has to make a {\it second}
rotation restoring $j_z$ to its initial value. It turns out that,
representing rotations by $\oplus$ and bitstrings then gives
different results depending on whether $j_z$ is first raised and then
lowered or visa versa; finite and discrete commutation relations
of the standard form result. We will present the details of this
analysis on another occasion. In effect what it accomplishes is
a mapping of conventional quantum mechanics onto bitstrings in such
a way as to get rid of the need for continuum representations
(eg. Lie groups) while retaining {\it finite and discrete} commutation relations.
Then a new look at our recent results on the
Maxwell\cite{35} and Dirac\cite{36}
equations should become fruitful.
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