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\label{sec:indepth}
\subsection{Pattern matching}
to be written
\subsection{The problem of dummy indices}
\FORM\ has a indices that can be automatically renumbered. With this we mean
that when we have an expression like
\begin{verbatim}
f(i)*g(i)*h(j)*k(j)-f(j)*g(j)*h(i)*k(i)
\end{verbatim}
we can say
\begin{verbatim}
Sum i,j;
\end{verbatim}
and \FORM\ will change the expression into
\begin{verbatim}
f(N1_?)*g(N1_?)*h(N2_?)*k(N2_?)-f(N2_?)*g(N2_?)*h(N1_?)*k(N1_?)
\end{verbatim}
in which \C{Ni\_?} are internal indices.
These internal indices follow a number of rules:
\begin{enumerate}
\item
their numbers (\C{AC.CurDum}) start at \C{AM.IndDum}, which again starts at
\C{AM.DumInd+WILDOFFSET} and \C{AM.DumInd} starts at \C{AM.OffsetIndex + 2*WILDOFFSET}.
Hence \C{AC.CurDum} starts at \C{AM.OffsetIndex +\\ 3*WILDOFFSET}.
Because we need this extra space \C{WILDOFFSET} cannot be too large and this
limits the number of indices that is allowed.
\item The dimension of the dummy indices is equal to the default dimension.
\item The internal (dummy) indices can be renamed at any time in order to
create uniquely minimal terms. In the above expression that would mean
that the second term would be 'rearranged' into
\begin{verbatim}
f(N2_?)*g(N2_?)*h(N1_?)*k(N1_?) -->
f(N1_?)*g(N1_?)*h(N2_?)*k(N2_?)
\end{verbatim}
and the expression becomes zero.
\end{enumerate}
There are problems with this concept.
\begin{enumerate}
\item Multiplying expressions with dummy indices could give a repetition of
the same indices as in \C{(f(N1\_?)*g(N1\_?))\^{}3}. This has been solved
partially as can be seen with the following program:
\begin{verbatim}
CF f,g;
L F = (f(N1_?)*g(N1_?))^3;
L G = f(N1_?)*g(N1_?);
.sort
L G3 = G^3;
Print;
.end
\end{verbatim}
The routine that takes care of the proper shifts in dummy numbers is
\C{MoveDummies()}. As one can see from the example, the \C{SUBEXPRESSION} to a
power isn't treated this way. It would have a serious impact on the
speed. With the \C{G\^{}3} it is different because that is slower to begin
with.
\item Keep Brackets is extremely dangerous. The problem here is
\begin{verbatim}
f(N1_?)*(g(N1_?)*h(N2_?)*k(N2_?)+g(N2_?)*h(N1_?)*k(N2_?))
\end{verbatim}
What is inside the brackets is invisible during the module. Hence a
renumbering that involves \C{f(N1\_?)} only can change \C{N1\_?} into \C{N2?\_}
(\FORM\ doesn't know there is already a \C{N2\_?}) and anyway, the
corresponding \C{N1\_?} remains as it is.
It means that there are complicatetions with \C{Sum}, \C{Trace4} and things like
\C{id p = f(?);} which can generate dummy indices.
\end{enumerate}
The second problem requires some action.
\begin{enumerate}
\item[A] When Keep Brackets is active, renumbering should not be allowed, until
the contents are multiplied with the outside of the brackets.
\item[B] The multiplying with the contents of the bracket should follow the same
procedure as the multiplication with a complete expression \\
(\C{MoveDummies()}).
\item[C] Introduction of new dummy indices should be above \C{AM.IndDum + WILDOFFSET/2}.
These should vanish when the term is renumbered after multiplying the
outside of the bracket with the inside.
\end{enumerate}
\C{Trace4} involves the creation of dummy indices, but these vanish again
without renumbering. Hence they don't cause problems.
In order to implement \C{A-C} we have to have a good look at all routines that
use \C{AR.CurDum} and call \C{ReNumber()} or \C{DetCurDum()}.
\subsection{Values of indices (and vectors)}
The indices and vectors share common use. That means that vectors can occur
in the places that are reserved for indices. In addition we have various
types of indices. Hence it is important to know what range of values in an
index location refers to what.
\begin{enumerate}
\item Special values:
\begin{tabular}{p{6em}rp{20em}}
\C{GAMMA1} & 0 & Dirac unit matrix \\
\C{GAMMA5} & -1 & Dirac gamma 5 (only defined in 4 dimensions) \\
\C{GAMMA6} & -2 & Dirac (1+gamma5) (only defined in 4 dimensions) \\
\C{GAMMA7} & -3 & Dirac (1-gamma5) (only defined in 4 dimensions)
\end{tabular}
The above 4 indices are to be used only inside the function \C{g\_}.
\begin{tabular}{p{6em}rp{20em}}
\C{FUNNYVEC} & -4 & Used in \C{replace\_} to indicate a vector with an
unspecified index. Hence \C{VECTOR,4,numvec,FUNNYVEC}
instead of \C{INDEX,3,numvec}. \\
\C{FUNNYWILD} & -5 & Used to indicate an argument field wildcard like
\C{?a} inside a tensor. \\
\C{SUMMEDIND} & -6 & Used in \C{DELTA} to indicate \C{d\_(mu,mu)-4} as generated
in traces. \\
\C{NOINDEX} & -7 & Used by \C{ExecArg()} in splitting a multi-delta or
multi-index. Taking out one to make a new argument
we leave the old one with two or one empty spots. \\
\C{FUNNYDOLLAR} & -8 & Used to indicate a dollar variable inside a tensor. \\
\C{EMPTYINDEX} & -9 & Used in the bracket statement to indicate a \C{d\_}.
Because \C{d\_} isn't a regular function we cannot use
the function notation and it needs two arguments. \\
\C{MINSPEC} & -10 &
\end{tabular}
\C{MINSPEC} must be smaller than all the other special values.
\item Fixed indices. They are in the range of 1 to \C{AM.OffsetIndex-1}.
\item Vectors are in the range from \\
\C{AM.OffsetVector = -2*WILDOFFSET+MINSPEC;} \\
to \\
\C{AM.OffsetVector + WILDOFFSET}
\item Wildcard vectors are in the range \\
\C{AM.OffsetVector + WILDOFFSET} \\
to \\
\C{AM.OffsetVector + 2*WILDOFFSET}
\item Regular indices are in the range from \\
\C{AM.OffsetIndex} to \C{AM.OffsetIndex + WILDOFFSET}
\item Wildcard indices are in the range \\
\C{AM.OffsetIndex + WILDOFFSET (=AM.WilInd)} \\
to \\
\C{AM.OffsetIndex + 2*WILDOFFSET (=AM.DumInd)}
\item Unused in the range of \\
\C{AM.OffsetIndex + 2*WILDOFFSET (=AM.DumInd)} \\
to \\
\C{AM.OffsetIndex + 3*WILDOFFSET (=AM.IndDum)}
\item Summed indices (\C{Ni\_?}) are in the range of \\
\C{AM.OffsetIndex + 3*WILDOFFSET (=AM.IndDum)}
to \\
\C{AM.OffsetIndex + 4*WILDOFFSET}
\item Unused in the range of \\
\C{AM.OffsetIndex + 4*WILDOFFSET} \\
to \\
\C{AM.OffsetIndex + 5*WILDOFFSET (=AM.mTraceDum)}
\item Summed indices as generated by the trace routines are above \\
\C{AM.OffsetIndex + 5*WILDOFFSET (=AM.mTraceDum)}
\end{enumerate}
{\it Note (JV)}: I am not sure why there are unused regions. I must have had a
reason for them, but I have forgotten about it (it was more than 20 years
ago). And then, maybe it is used somewhere in a totally untransparent way.
{\it Note 2 (JV)}: It was good to make this list. It turned out that in several
places the code that checks for wildcard indices was only limited from
below, not from above. It would of course be very rare to run into trouble
with this, but it is better to have the code formally correct. One never
knows. This was particularly the case in \C{FindRest()} (in \C{findpat.c}). There may
be more. It is best to repair this, whenever encountered.
From the above it should be clear that on a 32-bits computer \\
\C{5*WILDOFFSET+AM.OffsetIndex+nTraceDummies < 2\^{}{15}} \\
in which \C{nTraceDummies} is the number of dummies that can be introduced when
taking a 4-dimensional trace.
If we assume that we will not take traces of more than 200 gamma matrices
(each with a different index, because otherwise there are contractions)
\C{nTraceDummies} will be at most 100. \C{AM.OffsetIndex} is by default 128.
The value that we selected for \C{WILDOFFSET} is 6100 which allows a maximum
value of 2167 for \C{AM.OffsetIndex}.
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