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\begin{document}
\thispagestyle{empty}
\title{{\tt CANTENS}\\
A Package for Manipulations \\
and Simplifications of Indexed Objects}
\date{}
\author{H. Caprasse\\
Institut de Physique \\
Sart-Tilman, B-4000 LIEGE\\
email: {\tt hubert.caprasse@ulg.ac.be}
}
\maketitle
\section{Introduction}
{\tt CANTENS} \ttindex{CANTENS} is a package that creates an environment
inside {\tt REDUCE}\ttindex{REDUCE}
which allows the user to
manipulate and simplify expressions containing various indexed objects
like tensors, spinors, fields and quantum fields.
Briefly said, it allows him
\begin{itemize}
\item[-] to define generic indexed quantities which can eventually depend
implicitly or explicitly on any number of variables;
\item[-] to define one or several affine or metric (sub-)spaces, and to work
within them without difficulty;
\item[-] to handle dummy indices and simplify adequatly expressions
which contain them.
\end{itemize}
Beside the above features, it offers the user:
\begin{enumerate}
\item Several invariant
elementary tensors which are always used in the applications involving
the use of indexed objects like {\tt delta, epsilon, eta} and the
generalized delta function.
\item The possibility to define any metric and to make it bloc-diagonal
if he wishes to.
\item The capability to symmetrize or antisymmetrize any expression.
\item The possibility to introduce any kind of symmetry (even partial symmetries)
for the indexed objects.
\item The choice to work with commutative, non-commutative or anticommutative
indexed objects.
\end{enumerate}
In this package, one cannot find algorithms or even specific objects
(i.e. like the covariant derivative or the SU(3) group structure constants)
which are of
used either in nuclear and particle physics.
The objective of the package is simply to allow
the user to easily formulate {\em his algorithms} in the {\em notations he likes
most}.
%\newpage
The package is also conceived so as to minimize the number of new commands.
However, the large number of new capabilities inherently implies that quite
a substantial number of new functions and commands must be used. On the other
hand, in order to avoid too many error or warning messages the package
assumes, in many cases, that the user is reponsible of the consistency of its
inputs.
The author is aware that the package is still perfectible and
he will be grateful to all people who shall spare some time to communicate
bugs or suggest improvements.
The documentation below is separated into four sections.
In the first one, the space(s) properties and definitions are described.
In the second one, the commands to geberate and handle
generic indexed quantities (called abusively tensors) are illustrated.
The manipulation and control of free and dummy indices is discussed.
In the third one, the special tensors are introduced and their properties
discussed especially with respect to their ability to work simultaneously within
several subspaces.
The last section, which is also the most important, is devoted entirely to
the simplification function \index{CANONICAL} CANONICAL. This function
originates from the package {\tt DUMMY} \ttindex{ DUMMY} and has been
substantially extended . It takes account of
all symmetries, make dummy summations and seeks a ``canonical''
form for any tensorial expression. Without it, the present
package would be much less useful.
Finally, an {\bf index} has been created. It contains numerous references
to the text. Different typings have been adopted to make a clear
distinction between them. The conventions are the following:
\begin{itemize}
\item Procedure keywords are typed in capital roman letters.
\item Package keywords are typed in typewriter capital letters.
\item Cantens package keywords are in small typewriter letters.
\item All other keywords are typed in small roman letters.
\end{itemize}
When {\tt CANTENS} \ttindex{CANTENS} is loaded, the packages
{\tt ASSIST} \ttindex{ASSIST} and {\tt DUMMY}\ttindex{DUMMY} are also
loaded.
\section{Handling of space(s)}
\index{space}
One can work either in a {\em single} space environment or in
a multiple space environment. After the package is loaded,
the single space environment is set and
a unique space is defined. It is euclidian, and has a symbolic
dimension equal to {\tt dim}.
The single space environment is determined by the switch
ONESPACE\index{ONESPACE} which is turned on.
One can verify the above assertions as follows \index{WHOLESPACE\_DIM}:
\begin{verbatim}
onespace ?; => yes
wholespace_dim ?; => dim
signature ?; => 0
\end{verbatim}
One can introduce a pseudoeuclidian metric for the above space by the
command SIGNATURE \index{signature} and verify that the signature is indeed~1:
\begin{verbatim}
signature 1;
signature ?; => 1
\end{verbatim}
In principle the signature may be set to any positive
integer. However, presently,
the package cannot handle signatures larger than 1.
One gets the Minkowski-like space metric\index{Minkowski}
$$
\left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{array}
\right)
$$
which corresponds to the convention of high energy physicists.
It is possible to change it into the astrophysicists convention using
the command GLOBAL\_SIGN \index{GLOBAL\_SIGN}:
\begin{verbatim}
global_sign ?; => 1
global_sign (-1);
global_sign ?; => -1
\end{verbatim}
This means that the actual metric is now $(-1,1,1,1)$.
The space dimension may, of course, be assigned at will using the function
\index{WHOLESPACE\_DIM} WHOLESPACE\_DIM. Below, it is assigned to 4:
\begin{verbatim}
wholespace_dim 4; ==> 4
\end{verbatim}
When the switch \index{ONESPACE}ONESPACE is turned off, the system {\em assumes} that this default
space is non-existent and, therefore, that the user is going to define the
space(s) in which he wants to work.
Unexpected error messages will occur if it is not done.
Once the switch is turned off many more functions become active. A few of them are
available in the algebraic mode to allow the user to properly conctruct
and control the properties of the various (sub-)spaces he is going to define
and, also, to assign symbolic indices to some of them.
\index{DEFINE\_SPACES}DEFINE\_SPACES is the space constructor and
\ttindex{wholespace}{\bf wholespace}
is a reserved identifier which is meant to be the name of the global space
if subspaces are introduced.
Suppose we want to define a unique space, we can choose for its any name but
choosing {\bf wholespace} will be more efficient. On the other hand, it leaves
open the possibility to introduce subspaces in a more transparent way.
So one writes, for instance,:
\ttindex{signature}\ttindex{indexrange}
\begin{verbatim}
define_spaces wholespace=
{6,signature=1,indexrange=0 .. 5}; ==>t
\end{verbatim}
The arguments inside the list, assign respectively the dimension, the signature
and the range of the numeric indices which is allowed.
Notice that the range starts from 0 and not from 1. This is made to conform with
the usual convention for spaces of signature equal to 1. However, this is not
compulsory.
Notice that the declaration of the indexrange may be omitted if this is the
only defined space.
There are two other options which may replace the signature option.
They are \ttindex{euclidian}{\bf euclidian} and \ttindex{affine}{\bf affine} they
have both an obvious significance.
In the subsequent example, an eleven dimension global space is defined
and two subspaces of this space are specified.
Notice that no indexrange has been declared for the entire space.
However, the indexrange declaration is compulsory for subspaces otherwise
the package will improperly work when dealing with numeric indices.
\begin{verbatim}
define_spaces wholespace={11,signature=1}; ==> t
define_spaces mink=
{4,signature=1,indexrange=0 .. 3}; ==> t
define_spaces eucl=
{6,euclidian,indexrange=4 .. 9}; ==> t
\end{verbatim}
To remind ones the space context in which one is working, the use of the
function \index{SHOW\_SPACES}SHOW\_SPACES is required. Its output is an
{\em algebraic value} from which the user can retrieve all the informations
displayed. After the declarations above, this function gives:
\begin{verbatim}
show_spaces(); ==>
{{wholespace,11,signature=1}
{mink,4,signature=1,indexrange=0..3},
{eucl,6,euclidian,indexrange=4..9}}
\end{verbatim}
If an input error is made or if one wants to change the space framework, one
cannot directly redefine the relevant space(s). For instance, the input
\begin{verbatim}
define_spaces eucl=
{7,euclidian,indexrange=4 .. 9}; ==>
*** Warning: eucl cannot be (or is already)
defined as space identifier
t
\end{verbatim}
whih aims to fill all dimensions present in {\tt wholespace}
tells that the space {\tt eucl} cannot be redefined. To redefine it effectively,
one is to {\em remove} the existing definition first using the function
\index{REM\_SPACES} REM\_SPACES\ which takes any number of space-names as
its argument. Here is the illustration:
%\end{document}
\begin{verbatim}
rem_spaces eucl; ==> t
show_spaces(); ==>
{{wholespace,11,signature=1},
{mink,4,signature=1,indexrange=0..3}}
define_spaces eucl=
{7,euclidian,indexrange=4 .. 10}; ==> t
show_spaces(); ==>
{{wholespace,11,signature=1},
{mink,4,signature=1,indexrange=0..3},
{eucl,7,euclidian,indexrange=4..10}}
\end{verbatim}
Here, the user is entirely responsible of the coherence of his construction.
The system does NOT verify it but will incorrectly run if there is a mistake
at this level.
When two spaces are direct product of each other (as the color and Minkowski
spaces in quantum chromodynamics), it is not necessary to introduce the
global space {\tt wholespace}\ttindex{wholespace}.
``Tensors'' and symbolic indices can be declared to belong to a specific space
or subspace. It is in fact an essential ingredient of the package and make it
able to handle expressions which involve quantities belonging to several
(sub-)spaces or to handle bloc-diagonal ``tensors''. This will be discussed in the next section.
Here, we just mention how to declare that some set of symbolic indices belong
to a specific (sub-)space\index{subspaces} or how to declare them to belong
to any space\index{spaces}.
The relevant command is \index{MK\_IDS\_BELONG\_SPACE} MK\_IDS\_BELONG\_SPACE
whose syntax is
\begin{verbatim}
mk_ids_belong_space(<list of indices>,
<space | subspace identifier>)
\end{verbatim}
For example, within the above declared spaces one could write:
\begin{verbatim}
mk_ids_belong_space({a0,a1,a2,a3},mink); ==> t
mk_ids_belong_space({x,y,z,u,v},eucl); ==> t
\end{verbatim}
The command \index{MK\_IDS\_BELONG\_ANYSPACE} MK\_IDS\_BELONG\_ANYSPACE
allows to remake them usable either in {\tt wholespace}\ttindex{wholespace}
if it is defined or in anyone among the defined spaces.
For instance, the declaration:
\begin{verbatim}
mk_ids_belong_anyspace a1,a2; ==> t
\end{verbatim}
tells that a1 and a2 belong either to {\tt mink} or to {\tt eucl} or
to {\tt wholespace}.
\section{Generic tensors and their manipulation}
\index{generic tensor}
\subsection{Definition}
The generic tensors handled by {\tt CANTENS}\ttindex{CANTENS} are objects
much more general than usual tensors. The reason is that they are not supposed to
obey well defined transformation properties under a change of coordinates.
They are only indexed quantities. The indices are either
contravariantly (upper indices) or covariantly (lower indices) placed.
They can be symbolic or numeric. When a given index is found both
in one upper and in one lower place, it is supposed to be summed over
all space-coordinates it belongs to viz. it is a {\em dummy}\index{dummy}
index
and {\em automatically recognized} as such.
So they are supposed to obey the summation rules of tensor calculus.
This why and only why they are called tensors. Moreover, aside from
indices they may also depend implicitly or explicitly on any number of
{\em variables}\index{variables}. Within this definition,
tensors may also be spinors, they can be non-commutative or anticommutative,
they may also be algebra generators and represent fields or quantum fields.
\subsection{Implications of \index{TENSOR}TENSOR\ declaration}
The procedure TENSOR which takes an arbitrary number of identifiers as argument
defines them as operator-like objects which admit an arbitrary number of indices.
Each component has a formal character and may or may not belong to a
specific (sub-)space. Numeric indices are also allowed. The way to distinguish
upper and lower indices is the same as the one in the package
\ttindex{EXCALC}{\tt EXCALC} e.g. $-a$ is a lower index and $a$ is an
upper index.
A special printing function has been created so as to mimic as much as possible
the way of writing such objects on a sheet of paper.
Let us illustrate the use of \index{TENSOR} TENSOR:
\begin{verbatim}
tensor te; ==> t
te(3,a,-4,b,-c,7); ==>
3 a b 7
te
4 c
te(3,a,{x,y},-4,b,-c,7); ==>
3 a b 7
te (x,y)
4 c
te(3,a,-4,b,{u,v},-c,7); ==>
3 a b 7
te (u,v)
4 c
te({x,y}); ==> te(x,y)
\end{verbatim}
Notice that the system distinguishes indices from variables on input
solely on the basis that the user puts variables {\em inside a list}.
The dependence can also be declared implicit through the \REDUCE\ command
\index{DEPEND}DEPEND which is generalized so as to allow to declare a tensor
to depend on another tensor irrespective of its components. It means that only {\em one}
declaration is enough to express the dependence with respect to
{\em all its components}.
A simple example:
\index{DF}
\begin{verbatim}
tensor te,x;
depend te,x;
df(te(a,-b),x(c)); ==>
a c
df(te ,x )
b
\end{verbatim}
Therefore, when {\em all} objects are tensors, the dependence declaration
is valid for all indices.
One can also avoid the trouble to place the explicit variables inside a list if
one declare them as variables through the command \index{MAKE\_VARIABLES}
MAKE\_VARIABLES.
This property can also be removed%
\footnote{One important feature of this package is its {\em reversibility}
viz. it gives the user the means to erase its previous operations
at any time. So, most functions described below do
possess ``removing'' action companions.} using
\index{REMOVE\_VARIABLES}REMOVE\_VARIABLES:
\begin{verbatim}
make_variables x,y; ==> t
te(x,y); ==> te(x,y)
te(x,y,a); ==>
a
te (x,y)
remove_variables x; ==> t
te(x,y,a); ==>
x a
te (y)
\end{verbatim}
If one does that one must be careful not to substitute a number to such
declared variables because this number would be considered as an index and
no longer as a variable. So it is only useful for {\em formal} variables%
\index{variables}.
A tensor can be easily eliminated using the function
\index{REM\_TENSOR}REM\_TENSOR. It has the syntax
\begin{verbatim}
rem_tensor t1,t2,t3 ....;
\end{verbatim}
\subsubsection{Dummy \index{dummy} indices recognition}
\index{dummy}
For all individual tensors met by the evaluator, the system will analyse
the written indices and will detect those which must be considered dummy
according to the usual rules of tensor calculus. Those indices will be given
the {\tt dummy} property and will no longer be allowed to play the role
of {\em free} indices unless the user removes this dummy property.
In that way, the system checks immediately the consistency of an input.
Three functions are at the disposal of the user to control dummy indices.
They are \index{DUMMY\_INDICES}DUMMY\_INDICES, \index{REM\_DUMMY\_INDICES}
REM\_DUMMY\_INDICES and \index{REM\_DUMMY\_IDS}REM\_DUMMY\_IDS.
The following illustrates their use as well as the behaviour of the
system:
\begin{verbatim}
dummy_indices(); ==> {} % In a fresh environment
te(a,b,-c,-a); ==>
a b
te
c a
dummy_indices(); ==> {a}
te(a,b,-c,a); ==>
***** ((c)(a b a)) are inconsistent lists of indices
% a cannot be found twice as an upper index
te(a,b,-b,-a); ==>
a b
te
b a
dummy_indices(); ==> {b,a}
te(d,-d,d); ==>
***** ((d)(d d)) are inconsistent lists of indices
dummy_indices(); ==> {d,b,a}
rem_dummy_ids d; ==> t
dummy_indices(); ==> {b,a}
te(d,d); ==>
d d
te % This is allowed again.
dummy_indices(); ==> {b,a}
rem_dummy_indices(); ==> t
dummy_indices(); ==> {}
\end{verbatim}
Other verifications of coherence are made when space specifications
are introduced both in the ON and OFF onespace environment. We shall
discuss them later\ttindex{onespace OFF}\ttindex{onespace ON}.
\subsubsection{Substitutions, assignements and rewriting rules}
\index{LET}\index{rewriting rules}
The user must be able to manipulate and give specific characteristics
to the generic tensors he has introduced. Since tensors are essentially
\REDUCE\ operators\ttindex{REDUCE}, the usual commands of the system are
available.
However, some limitations are implied by the fact that indices and,
especially numeric indices, must always be properly recognized before
any substitution or manipulation is done. We have gathered below a set of
examples which illustrate all the ``delicate'' points.
First, the substitutions\index{SUB}:
\begin{verbatim}
sub(a=-c,te(a,b)); ==>
b
te
c
sub(a=-1,te(a,b)); ==>
b
te
1
sub(a=-0,te(a,b)); ==>
0 b
te % sub has replaced -0 by 0. wrong!
sub(a=-!0,te(a,b)); ==>
b
te % right
0
\end{verbatim}
The substitution of an index by -0 is the {\em only one} case where
there is a problem. The function SUB replaces -0 by 0 because it does
not recognize 0 as an index of course. Such a recognition is
context dependent and implies a modification of SUB
for this {\em single} exceptional case. Therefore,we have opted, not do do so
and to use the index 0 which is simply !0 instead of 0.
Second, the assignements. Here, we advise the user to rely on the
operator\ttindex{ASSIST}\index{==}$==$%
\footnote{See the {\tt ASSIST} documentation for its description.}
instead of the operator \index{:=}$:=$. Again,
the reason is to avoid the problem raised above in the case of
substitutions. $:=$ does not evaluate its lefthandside so that -0 is
not recognized as an index and simplified to 0 while the $==$ evaluates both
its lefthandside and its righthandside and do recognize it.
The disadvantage of $==$ is that it demands that a second assignement
on a given component
be made only after having suppressed {\em explicitly} the first assignement.
This is done by the function \index{REM\_VALUE\_TENS}REM\_VALUE\_TENS
which can be applied on any component. We stress, however, that if one
is willing to use -!0 instead of -0 as the lower 0 index, the use of $:=$
is perfectly legitimate:
\begin{verbatim}
te({x,y},a,-0)==x*y*te(a,-0); ==>
a
te *x*y
0
te({x,y},a,-0); ==>
a
te *x*y
0
te({x,y},a,0); ==>
a 0
te (x,y)
te({x,y},a,-0)==x*y*te(a,-0); ==>
a
***** te *x*y invalid as setvalue kernel
0
rem_value_tens te({x,y},a,-0);
te({x,y},a,-0); ==>
a
te (x,y)
0
te({x,y},a,-0)==(x+y)*te(a,-0); ==>
a
te *(x + y)
0
\end{verbatim}
In the elementary application below, the use of a tensor avoids the
introduction of two different operators and makes the
calculation more readable.
\begin{verbatim}
te(1)==sin th * cos phi; ==> cos(phi)*sin(th)
te(-1)==sin th * cos phi; ==> cos(phi)*sin(th)
te(2)==sin th * sin phi; ==> sin(phi)*sin(th)
te(-2)==sin th * sin phi; ==> sin(phi)*sin(th)
te(3)==cos th ; ==> cos(th)
te(-3)==cos th ; ==> cos(th)
for i:=1:3 sum te(i)*te(-i); ==>
2 2 2 2 2
cos(phi) *sin(th) + cos(th) + sin(phi) *sin(th)
rem_value_tens te;
te(2); ==>
2
te
\end{verbatim}
There is no difference in the manipulation of numeric indices and numeric
{\em tensor} indices. The function \index{REM\_VALUE\_TENS}REM\_VALUE\_TENS
when applied to a tensor prefix suppresses the value of
{\em all its components}. Finally, there is no ``interference'' with
i as a dummy index and i as a numeric index in a loop.
Third, rewriting rules. They are either global or local and
can be used as in \REDUCE\ttindex{REDUCE}.
Again, here, the -0 index problem exists each time a substitution
by the index -0 must be made in a template.
\index{FOR ALL ... LET}
\begin{verbatim}
% LET:
let te({x,y},-0)=x*y;
te({x,y},-0); ==> x*y
te({x,y},+0); ==>
0
te (x,y)
te({x,u},-0); ==>
te (x,u)
0
% FOR ALL .. LET:
for all x,a let te({x},a,-b)=x*te(a,-b);
te({u},1,-b); ==>
1
te *u
b
te({u},c,-b); ==>
c
te *u
b
te({u},b,-b); ==>
b
te *u
b
te({u},a,-a); ==>
a
te (u)
a
for all x,a clear te({x},a,-b);
te({u},c,-b); ==>
c
te (u)
b
for all a,b let te({x},a,-b)=x*te(a,-b);
te({x},c,-b); ==>
c
te *x
b
te({x},a,-a); ==>
a
te *x
a
% The index -0 problem:
te({x},a,-0); ==> % -0 becomes +0 in the template
a
te (x) % the rule does not apply.
0
te({x},0,-!0); ==>
0
te *x % here it applies.
0
\end{verbatim}
\begin{verbatim}
% WHERE:
rul:={te(~a) => sin a}; ==>
a
rul := {te => sin(a)}
te(1) where rul; ==> sin(1)
te(1); ==>
1
te
\end{verbatim}
\index{variables}
\begin{verbatim}
% with variables:
rul1:={te(~a,{~x,~y}) => x*y*sin(a)}; ==>
~a
rul1 := {te (~x,~y) => x*y*sin(a)}
te(a,{x,y}) where rul1; ==> sin(a)*x*y
te({x,y},a) where rul1; ==> sin(a)*x*y
rul2:={te(-~a,{~x,~y}) => x*y*sin(-a)};
rul2 := {te (~x,~y) => x*y*sin(-a)}
~a
te(-a,{x,y}) where rul2; ==> -sin(a)*x*y
te({x,y},-a) where rul2; ==> -sin(a)*x*y
\end{verbatim}
Notice that the position of the list of variables inside the rule
may be chosen at will. It is an irrelevant feature of the template.
This may be confusing, so, we advise to write the rules not as
above but placing the list of variables {\em in front of all indices}
since it is in that canonical form which it is written by the simplification
function of individual tensors.
\subsection{Behaviour under space specifications}
\index{spaces}
The characteristics and the behaviour of generic tensors described up to now
are independent of all space specifications. They are complete as long as
we confine to the default space which is active when starting {\tt CANTENS}.
However, as soon as some space specification is introduced, it has some
consequences one the generic tensor properties. This is true both when
\index{ONESPACE}ONESPACE is switched ON or OFF. Here we shall describe
how to deal with these features.
When onespace is ON, if the space dimension is set to an integer, numeric
indices of any generic tensors are forced to be less or equal that integer
if the signature is 0 or less than that integer if the signature is equal to 1.
The following illustrates what happens.
\ttindex{onespace ON}
\begin{verbatim}
on onespace;
wholespace_dim 4; ==> 4
signature 0; ==> 0
te(3,a,-b,7); ==> ***** numeric indices out of range
te(3,a,-b,3); ==>
3 a 3
te
b
te(4,a,-b,4); ==>
4 a 4
te
b
sub(a=5,te(3,a,-b,3));
==> ***** numeric indices out of range
signature 1; ==> 1
% Now indices range from 0 to 3:
te(4,a,-b,4);
==> ***** numeric indices out of range
te(0,a,-b,3); ==>
0 a 3
te
b
\end{verbatim}
When onespace is OFF\ttindex{onespace OFF}, many more possibilities to control the input or to
give specific properties to tensors are open.
For instance, it is possible to declare that a tensor belongs
to one of them. It is also possible to declare that some indices
belongs to one of them. It is even possible to do that
for {\em numeric} indices thanks to the declaration
\ttindex{indexrange}indexrange
included optionally in the space definition generated by
\index{DEFINE\_SPACES}DEFINE\_SPACES.
First, when onespace is OFF, the run equivalent to the previous one is
like the following:
\index{DEFINE\_SPACES}\index{SHOW\_SPACES}\index{MAKE\_TENSOR\_BELONG\_SPACE}
\index{REM\_SPACES}
\begin{verbatim}
off onespace;
define_spaces wholespace={6,signature=1); ==> t
show_spaces(); ==> {{wholespace,6,signature=1}}
make_tensor_belong_space(te,wholespace);
==> wholespace
te(4,a,-b,6); ==>
***** numeric indices out of range
te(4,a,-b,5); ==>
4 a 5
te
b
rem_spaces wholespace;
define_spaces wholespace={4,euclidean}; ==> t
te(a,5,-b); ==> ***** numeric indices out of range
te(a,4,-b); ==>
a 4
te
b
define_spaces eucl={1,signature=0}; ==> t
show_spaces(); ==>
{{wholespace,5,signature=1},
{eucl,1,signature=0}}
make_tensor_belong_space(te,eucl); ==> eucl
te(1); ==>
1
te
te(2); ==> ***** numeric indices out of range
te(0); ==>
0
te
\end{verbatim}
In the run, the new function \index{MAKE\_TENSOR\_BELONG\_SPACE}
MAKE\_TENSOR\_BELONG\_SPACE
has been used.
One may be surprised that {\tt te(0)} is allowed in the end of
the previous run and, indeed,
it is incorrect that the system allows {\em two} different components to
{\tt te}.
This is due to an incomplete definition of the space. When one deals
with spaces of integer dimensions, if one wants to control numeric indices
correctly {\em when} onespace is switched off {\em one must also give the
indexrange}\ttindex{indexrange}.
So the previous run must be corrected to
\begin{verbatim}
define_spaces eucl=
{1,signature=0,indexrange=1 .. 1}; ==> t
make_tensor_belong_space(te,eucl); ==> eucl
te(0); ==>
***** numeric indices do not belong to (sub)-space
te(1); ==>
1
te
te(2); ==>
***** numeric indices do not belong to (sub)-space
\end{verbatim}
Notice that the error message has also changed accordingly.
So, now one can even constrain the 0 component to belong to
an euclidian space.
Let us go back to symbolic indices\index{symbolic indices}.
By default, any symbolic index belongs to the global space or to all
defined partial spaces. In many cases, this is, of course, not consistent.
So, the possibility exists to declare that one or several indices
belong to a specific (sub-)space. To this end, one is to use the
function \index{MK\_IDS\_BELONG\_SPACE}MK\_IDS\_BELONG\_SPACE.
Its syntax is
\begin{verbatim}
mk_ids_belong_space(<list of indices>,
<(sub-)space identifier>)
\end{verbatim}
The function \index{MK\_IDS\_BELONG\_ANYSPACE}MK\_IDS\_BELONG\_ANYSPACE
whose syntax is the same do the reverse operation.
Combined with the declaration \index{MAKE\_TENSOR\_BELONG\_SPACE}
MAKE\_TENSOR\_BELONG\_SPACE, it allows to express all problems
which involve tensors belonging to different spaces and do the dummy
summations correctly.
One can also define a tensor which has a \index{bloc-diagonal}
``bloc-diagonal'' structure.
All these features are illustrated in the next sections which describe specific
tensors and the properties of the extended function \index{CANONICAL}CANONICAL.
\section{Specific tensors}
The means provided in the two previous section to handle generic tensors
already allow to construct any specific tensor we may need. That the package contains
a certain number of them is already justified on the level of conviviality.
However, a more important justification is that some basic tensors are so
universaly and frequently used that a careful programming of these improves
considerably the robustness and the efficiency of most calculations.
The choice of the set of specific tensors is not clearcut. We have tried
to keep their number to a minimum but, experience, may lead us extend it
without dificulty. So, up to now, the list of specific tensors is:
\begin{list}{-}{\parsep 0in \itemsep 1pt}
\item {\tt delta} tensor\ttindex{delta},
\item {\tt eta} Minkowski tensor\index{Minkowski}\ttindex{eta},
\item {\tt epsilon} tensor,\ttindex{epsilon}
\item {\tt del} generalised delta tensor,\ttindex{del}
\item {\tt metric} generic tensor metric.\ttindex{metric}
\end{list}
It is important to realize that the typewriter font names in the list are
{\em keywords} for the corresponding tensors and do not necessarily correspond
to their {\em actual names}.
Indeed, the choice of the names of particular tensors is left to the user.
When startting \ttindex{CANTENS}{\tt CANTENS} specific tensors are NOT
available.
They must be activated by the user using the function
\index{MAKE\_PARTIC\_TENS}MAKE\_PARTIC\_TENS whose syntax is:
\begin{verbatim}
make_partic_tens(<tensor name> , <keyword>);
\end{verbatim}
The name chosen may be the same as the keyword.
As we shall see, it is never needed to define more than one {\tt delta}
tensor but it is often needed to define several {\tt epsilon} tensors.
Hereunder, we describe each of the above tensors especially their
behaviour in a multi-space environment.
\subsection{ DELTA tensor}
\ttindex{delta}
It is the simplest example of a bloc-diagonal\index{bloc-diagonal}
tensor we mentioned in the
previous section. It can also work in a space which is a direct product
of two spaces. Therefore, one never needs to introduce more than one
such tensor. If one is working in a graphic environment, it is advantageous
to choose the keyword as its name. Here we choose {\tt DELT}.
We illustrate how it works when the switch \index{ONESPACE}onespace is
successively switched ON and OFF.
\ttindex{onespace ON}
\begin{verbatim}
on onespace;
make_partic_tens(delt,delta); ==> t
delt(a,b); ==>
***** bad choice of indices for DELTA tensor
% order of upper and lower indices irrelevant:
delt(a,-b); ==>
a
delt
b
delt(-b,a); ==>
a
delt
b
delt(-a,b); ==>
b
delt
a
wholespace_dim ?; ==> dim
delt(1,-5); ==> 0
% dummy summation done:
delt(-a,a); ==> dim
wholespace_dim 4; ==> 4
delt(1,-5); ==> ***** numeric indices out of range
wholespace_dim 3; ==> 3
delt(-a,a); ==> 3
\end{verbatim}
There is a peculiarity of this tensor, viz. it can serve to represent
the Dirac {\em delta function}\ttindex{delta function}
when it has no indices and an explicit variable dependency as hereunder
\begin{verbatim}
delt({x-y}) ==> delt(x-y)
\end{verbatim}
Next we work in the context of several spaces:
\ttindex{onespace OFF}
\begin{verbatim}
off onespace;
define_spaces wholespace={5,signature=1}; ==> t
% we need to assign delta to wholespace when it exists:
make_tensor_belong_space(delt,wholespace);
delt(a,-a); ==> 5
delt(0,-0); ==>1
rem_spaces wholespace; ==> t
define_spaces wholespace={5,signature=0}; ==> t
delt(a,-a); ==> 5
delt(0,-a); ==>
***** bad value of indices for DELTA tensor
\end{verbatim}
The checking of consistency of chosen indices is made in the same way as for
generic tensor. In fact, all the previous functions which act on generic tensors
may also affect, in the same way, a specific tensor. For instance, it was
compulsory to explicitly tell that we want {\tt DELT} to belong to the
wholespace \index{MAKE\_TENSOR\_BELONG\_SPACE} overwise,
{\tt DELT} would remain defined on the {\em default space}.
In the next sample run, we display the bloc-diagonal property of
the \ttindex{delta} delta tensor\index{bloc-diagonal}.
\begin{verbatim}
onespace ?; ==> no
rem_spaces wholespace; ==> t
define_spaces wholespace={10,signature=1}$
define_spaces d1={5,euclidian}$
define_spaces d2={2,euclidian}$
mk_ids_belong_space({a},d1); ==> t
mk_ids_belong_space({b},d2); ==> t
% c belongs to wholespace so:
delt(c,-b); ==>
c
delt
b
delt(c,-c); ==> 10
delt(b,-b); ==> 2
delt(a,-a); ==> 5
% this is especially important:
delt(a,-b); ==> 0
\end{verbatim}
The bloc-diagonal property of \ttindex{delta}{\tt delt} is made
active under two conditions. The first is that the system knows
to which space it belongs, the second is that indices must be
declared to belong to a specific space\index{spaces}.
To enforce the same property on a generic tensor, we have to make
the \index{MAKE\_BLOC\_DIAGONAL}MAKE\_BLOC\_DIAGONAL declaration:
\begin{verbatim}
make_bloc_diagonal t1,t2, ...;
\end{verbatim}
and to make it active, one proceeds as in the above run.
Starting from a fresh environment, the following sample run
is illustrative\index{MAKE\_BLOC\_DIAGONAL}:
\begin{verbatim}
off onespace;
define_spaces wholespace={6,signature=1}$
define_spaces mink={4,signature=1,indexrange=0 .. 3}$
define_spaces eucl={3,euclidian,indexrange=4 .. 6}$
tensor te;
make_tensor_belong_space(te,eucl); ==> eucl
% the key declaration:
make_bloc_diagonal te; ==> t
% bloc-diagonal property activation:
mk_ids_belong_space({a,b,c},eucl); ==> t
mk_ids_belong_space({m1,m2},mink); ==> t
te(a,b,m1); ==> 0
te(a,b,m2); ==> 0
% bloc-diagonal property suppression:
mk_ids_belong_anyspace a,b,c,m1,m2; ==> t
te(a,b,m2); ==>
a b m2
te
\end{verbatim}
\subsection{ETA\ Minkowski tensor}
\ttindex{eta}\index{Minkowski}
The use of \index{MAKE\_PARTIC\_TENS}MAKE\_PARTIC\_TENS with the
keyword {\tt eta} allows to create a Minkowski diagonal metric tensor in a
one or multi-space context either with the convention of
high energy physicists or in the convention of astrophysicists.
Any {\tt eta}-like tensor is assumed to work within a space
of signature 1. Therefore, if the space whose metric, it is supposed
to describe has a signature 0, an error message follows if one is
working in an ON onespace \ttindex{onespace ON}\ttindex{onespace OFF}
context and a warning when in an OFF onespace context.
Illustration:
\index{SIGNATURE}
\begin{verbatim}
on onespace;
make_partic_tens(et,eta); ==> t
signature 0; ==> 0;
et(-b,-a); ==>
***** signature must be equal to 1 for ETA tensor
off onespace;
et(a,b); ==>
*** ETA tensor not properly assigned to a space
% it is then evaluated to zero:
0
on onespace;
signature 1; ==> 1
et(-b,-a); ==>
et
a b
\end{verbatim}
Since {\tt et(a,-a)} is evaluated to the corresponding {\tt delta} tensor,
one cannot define properly an {\tt eta}\ttindex{eta} tensor without a
simultaneous introduction of a {\tt delta} tensor. Otherwise one gets
the following message:
\begin{verbatim}
et(a,-a); ==> ***** no name found for (delta)
\end{verbatim}
So we need to issue, for instance,
\begin{verbatim}
make_partic_tens(delta,delta); ==> t
\end{verbatim}
The value of its diagonal elements depends on the chosen
{\index{GLOBAL\_SIGN}global sign. The next run illustrates this:
\begin{verbatim}
global_sign ?; ==> 1
et(0,0); ==> 1
et(3,3); ==> - 1
global_sign(-1); ==> -1
et(0,0); ==> - 1
et(3,3); ==> 1
\end{verbatim}
The tensor is of course symmetric \index{symmetric}.
Its indices are checked in the same way as for a generic tensor.
In a multi\_space context\index{spaces}, the {\tt eta} tensor must belong
to a well defined space of \index{signature}signature 1:
\ttindex{onespace OFF}
\begin{verbatim}
off onespace;
define_spaces wholespace={4,signature=1}$
make_tensor_belong_space(et,wholespace)$
et(a,-a); ==> 4
\end{verbatim}
If the space to which {\tt et} belongs to is a subspace\index{subspaces},
one must also
take care to give a space-identity to dummy indices which may appear inside
it. In the following run, the index {\tt a} belongs to {\tt wholespace}
\ttindex{wholespace}if it is not told to the system that it is a dummy
index of the space {\tt mink}:
\begin{verbatim}
make_tensor_belong_anyspace et; ==> t
rem_spaces wholespace; ==> t
define_spaces wholespace={8,signature=1}; ==> t
define_spaces mink={5,signature=1}; ==> t
make_tensor_belong_space(et,mink); ==> mink
% a sits in wholespace:
et(a,-a); ==> 8
mk_ids_belong_space({a},mink); ==> t
% a sits in mink:
et(a,-a); ==> 5
\end{verbatim}
\subsection{EPSILON tensors}
\ttindex{epsilon}
It is an antisymmetric \index{antisymmetric} tensor which
is the invariant tensor for the unitary group transformations in
n-dimensional complex space which are continuously connected to the
identity transformation. The number of their indices are always stricty
equal to the number of space dimensions.
So, to each specific space is associated a specific {\tt epsilon} tensor.
Its properties are also dependent on the signature of the space.
We describe how to define and manipulate it in the context of a
unique space and, next, in a multi-space context.
\subsubsection{ONESPACE is ON}
\ttindex{onespace ON}
The use of \index{MAKE\_PARTIC\_TENS} MAKE\_PARTIC\_TENS places it, by default, in an
euclidian space if the signature is 0 and in a Minkowski-type space if
the signature\index{signature} is 1.
For higher signatures it is not constructed.
For a space of symbolic dimension, the number of its indices is not
constrained. When it appears inside an expression, its indices are {\em all}
currently upper or lower indices. However, the system allows for
mixed positions of the indices. In that case, the output of the system
is changed compared to the input only to place all contravariant indices
to the left of the covariant ones.
\begin{verbatim}
make_partic_tens(eps,epsilon); ==> t
eps(a,d,b,-g,e,-f); ==>
a d b e
- eps
g f
eps(a,d,b,-f,e,-f); ==> 0
% indices have all the same variance:
eps(-b,-a); ==>
- eps
a b
signature ?; ==> 0
eps(1,2,3,4); ==> 1
eps(-1,-2,-3,-4); ==> 1
wholespace_dim 3; ==> 3
eps(-1,-2,-3); ==> 1
eps(-1,-2,-3,-4); ==>
***** numeric indices out of range
eps(-1,-2,-3,-3); ==>
***** bad number of indices for (eps) tensor
eps(a,b); ==>
***** bad number of indices for (eps) tensor
eps(a,b,c); ==>
a b c
eps
eps(a,b,b); ==> 0
\end{verbatim}
When the signature\index{signature} is equal to 1, it is known that
there exists
two {\em conventions} which are linked to the chosen value 1 or -1 of
the $(0,1,\ldots,n)$ component. So, the sytem does evaluate all components
in terms of the $(0,1,\ldots,n)$ upper index component. It is left to the user
to assign it to 1 or -1\index{GLOBAL\_SIGN}.
\begin{verbatim}
signature 1; ==> 1
eps(0,1,2); ==>
0 1 2
eps
eps(-0,-1,-2); ==>
0 1 2
eps
wholespace_dim 4; ==> 4
eps(0,1,2,3); ==>
0 1 2 3
eps
eps(-0,-1,-2,-3); ==>
0 1 2 3
- eps
% change of the global_sign convention:
global_sign(-1);
wholespace_dim 3; ==> 3
% compare with second input:
eps(-0,-1,-2); ==>
0 1 2
- eps
\end{verbatim}
\subsubsection{ONESPACE is OFF}
\ttindex{onespace OFF}
As already said, several epsilon tensors may be defined. They {\em must}
be assigned to a well defined (sub-)space otherwise the simplifying
function \index{CANONICAL}CANONICAL will not properly work.
The set of epsilon tensors defined associated to their space-name
may be retrieved using the function
\index{SHOW\_EPSILONS}SHOW\_EPSILONS.
An important word of caution here. The output of this function does NOT
show the epsilon tensor one may have defined in the ON onespace context.
This is so because the default space has {\em NO} name.
Starting from a fresh environment, the following run illustrates
this point:
\begin{verbatim}
show_epsilons(); ==> {}
onespace ?; ==> yes
make_partic_tens(eps,epsilon); ==> t
show_epsilons(); ==> {}
\end{verbatim}
To make the {\tt epsilon} tensor defined in the single space environment
visible in the multi-space environment, one needs to associate it to
a space.
For example:
\begin{verbatim}
off onespace;
define_spaces wholespace={7,signature=1}; ==> t
show_epsilons(); ==> {} % still invisible
make_tensor_belong_space(eps,wholespace); ==>
wholespace
show_epsilons(); ==> {{eps,wholespace}}
\end{verbatim}
Next, let us define an {\em additional} {\tt epsilon}-type tensor:
\begin{verbatim}
define_spaces eucl={3,euclidian}; ==> t
make_partic_tens(ep,epsilon); ==>
*** Warning: ep MUST belong to a space
t
make_tensor_belong_space(ep,eucl); ==> eucl
show_epsilons(); ==> {{ep,eucl},{eps,wholespace}}
% We show that it is indeed working inside eucl:
ep(-1,-2,-3); ==> 1
ep(1,2,3); ==> 1
ep(a,b,c,d); ==>
***** bad number of indices for (ep) tensor
ep(1,2,4); ==>
***** numeric indices out of range
\end{verbatim}
As previously, the discrimation between symbolic indices
\index{symbolic indices}may be introduced
by assigning them to one or another space\index{spaces}:
\begin{verbatim}
rem_spaces wholespace;
define_spaces wholespace={dim,signature=1}; ==> t
mk_ids_belong_space({e1,e2,e3},eucl); ==> t
mk_ids_belong_space({a,b,c},wholespace); ==> t
ep(e1,e2,e3); ==>
e1 e2 e3
ep % accepted
ep(e1,e2,z); ==>
e1 e2 z
ep % accepted because z
% not attached to a space.
ep(e1,e2,a);==>
***** some indices are not in the space of ep
eps(a,b,c); ==>
a b c
eps % accepted because *symbolic*
% space dimension.
\end{verbatim}
{\tt epsilon}-like tensors can also be defined on disjoint spaces.
The subsequent sample run starts from the environment of the previous one.
It suppresses the space {\tt wholespace}\ttindex{wholespace} as well as the
space-assignment of the indices {\tt a,b,c}. It defines the new space
{\tt mink}. Next, the previously defined {\tt eps} tensor is attached
to this space. {\tt ep} remains unchanged and {\tt e1,e2,e3} still
belong to the space {\tt eucl}.
\index{SHOW\_SPACES}\index{SHOW\_EPSILONS}
\begin{verbatim}
rem_spaces wholespace; ==> t
make_tensor_belong_anyspace eps; ==> t
show_epsilons(); ==> {{ep,eucl}}
show_spaces(); ==> {{eucl,3,signature=0}}
mk_ids_belong_anyspace a,b,c; ==> t
define_spaces mink={4,signature=1}; ==> t
show_spaces(); ==>
{{eucl,3,signature=0},
{mink,4,signature=1}}
make_tensor_belong_space(eps,mink); ==> mink
show_epsilons(); ==> {{eps,mink},{ep,eucl}}
eps(a,b,c,d); ==>
a b c d
eps
eps(e1,b,c,d); ==>
***** some indices are not in the space of eps
ep(e1,b,c,d); ==>
***** bad number of indices for (ep) tensor
ep(e1,b,c); ==>
b c e1
ep
ep(e1,e2,e3); ==>
e1 e2 e3
ep
\end{verbatim}
\subsection{{\tt DEL} generalized delta tensor}
\ttindex{del}
The generalized delta function comes from the contraction of two epsilons.
It is totally antisymmetric. Suppose its name has been chosen to be $gd$,
that the space to which it is
attached has dimension n while the name of the chosen delta tensor
is $\delta$, then
one can define it as follows:
$$
gd^{a_1,a_2,\ldots,\a_n}_{b_1,b_2,\ldots, b_n}=
\left|\begin{array}{cccc}
\delta^{a_1}_{b_1} & \delta^{a_1}_{b_2} & \ldots & \delta^{a_1}_{b_n} \\
\delta^{a_2}_{b_1} & \delta^{a_2}_{b_2} & \ldots & \delta^{a_2}_{b_n} \\
\vdots & \vdots & \ddots & \vdots \\
\delta^{a_n}_{b_1} & \delta^{a_n}_{b_1} & \ldots & \delta^{a_n}_{b_1}
\end{array}
\right|
$$
It is, in general uneconomical to explicitly write that determinant except
for particular {\em numeric} values of the indices
\index{numeric indices} or when almost all
upper and lower indices are recognized as dummy indices.
In the sample run below, {\tt gd} is defined as the generalized delta function
in the default space. The main automatic evaluations are illustrated.
The indices which are summed over are always simplified:
\begin{verbatim}
onespace ? ==> yes
make_partic_tens(delta,delta); ==> t
make_partic_tens(gd,del); ==> t
% immediate simplifications:
gd(1,2,-3,-4); ==> 0
gd(1,2,-1,-2); ==> 1
gd(1,2,-2,-1); ==> -1 % antisymmetric
gd(a,b,-a,-b);
==> dim*(dim - 1) % summed over dummy indices
gd(a,b,c,-a,-d,-e); ==>
b c
gd *(dim - 2)
d e
gd(a,b,c,-a,-d,-c); ==>
b 2
delta *(dim - 3*dim + 2)
d
% no simplification:
gd(a,b,c,-d,-e,-f); ==>
a b c
gd
d e f
\end{verbatim}
One can force evaluation in terms of the determinant in all cases.
To this end, the switch \index{EXDELT}EXDELT is provided. It is initially
OFF. Switching it On will most often give inconveniently large outputs:
\begin{verbatim}
on exdelt;
gd(a,b,c,-d,-e,-f); ==>
a b c a b c
delta *delta *delta - delta *delta *delta
d e f d f e
a b c a b c
- delta *delta *delta + delta *delta *delta
e d f e f d
a b c a b c
+ delta *delta *delta - delta *delta *delta
f d e f e d
\end{verbatim}
In a multi-space environment, it is never necessary to define several
such tensor. The reason is that \index{CANONICAL}CANONICAL uses it
always from the contraction of a pair of {\tt epsilon}-like tensors.
Therefore the control of indices is already done, the space-dimension
in which {\tt del} is working is also well defined.
\subsection{{\tt METRIC} tensors}
\ttindex{metric}
Very often, one has to define a specific metric. The {\tt metric}-type
of tensors include all generic properties. The first one is their symmetry,
the second one is their equality to the {\tt delta} \ttindex{delta} tensor
when they get mixed indices, the third one is their optional bloc-diagonality.
So, a metric (generic) tensor is generated by the declaration
\index{MAKE\_PARTIC\_TENS}
\begin{verbatim}
make_partic_tens(<tensor-name>,metric);
\end{verbatim}
By default, when one is working in a multi-space environment, it is
defined in {\tt wholespace}{\ttindex{wholespace}
One uses the usual means of \REDUCE\ to give it specific values.
In particular, the metric 'delta' tensor of the euclidian space
can be defined that way.
Implicit or explicit dependences on variables are allowed.
Here is an illustration in the single space environment:
\begin{verbatim}
make_partic_tens(g,metric); ==> t
make_partic_tens(delt,delta); ==> t
onespace ?; ==> yes
g(a,b); ==>
a b
g
g(b,a); ==>
a b
g
g(a,b,c); ==>
***** bad choice of indices for a METRIC tensor
g(a,b,{x,y}); ==>
a b
g (x,y)
g(a,-b,{x,z}); ==>
a
delt
b
let g({x,y},1,1)=1/2(x+y);
g({x,y},1,1); ==>
x + y
-------
2
rem_value_tens g({x,y},1,1);
g({x,y},1,1); ==>
1 1
g (x,y)
\end{verbatim}
\section{The simplification function CANONICAL}
\index{CANONICAL}
\subsection{Tensor expressions}
\index{tensor polynomial}
Up to now, we have described the behaviour of individual
tensors and how they simplify themselves whenever possible.
However, this is far from being sufficient. In general, one is
to deal with objects which involve several tensors together
with various dummy summations between them.
We define a tensor expression as an arbitrary multivariate
polynomial. The indeterminates of such a polynomial may be
either an indexed object, an operator, a variable or a rational number.
A tensor-type indeterminate cannot appear to a degree larger
than one except if it is a trace\index{trace}.
The following is a tensor expression:
\begin{verbatim}
aa:= delt({x - y})*delt(a, - g)*delt(d, - g)*delt(g, -r)
*eps( - d, - e, - f)*eps(a,b,c)*op(x,y) + 1; ==>
a d g
aa := delt(x - y)*delt *delt *delt *eps
g g r d e f
a b c
*eps *op(x,y) + 1
\end{verbatim}
In the above expression, {\tt delt} and {\tt eps} are, respectively, the
{\tt delta}\ttindex{delta} and the {\tt epsilon}\ttindex{epsilon} tensors,
{\tt op} is an operator\index{operator}.
and {\tt delt(x-y)} is the Dirac delta function.\ttindex{delta function}
Notice that the above expression is not coh\'erent since the first term
has a variance while the second term is a scalar. Moreover, the
dummy index {\tt g} appears {\em three} times in the first term.
In fact, on input, each factor is simplified and each
factor is checked for coherence not more.
Therefore, if a dummy summation appears inside one factor, it will
be done whenever possible. Hereunder {\tt delt(a,-a)} is
summed over:
\index{SUB}
\begin{verbatim}
sub(g=a,aa); ==>
a d a b c
delt(x - y)*delt *delt *eps *eps
r a d e f
*op(x,y)*dim + 1
\end{verbatim}
\subsection{The use of CANONICAL}
\index{CANONICAL}CANONICAL is an offspring of the function
with the same name of the package {\tt DUMMY}
\ttindex{DUMMY}. It applies to tensor expressions as defined above.
When it acts, this functions has several features which are
worth to realise:
\begin{enumerate}
\item It tracks the free indices in each term and checks their
identity. It identifies and verify the coherence
of the various dummy index summations\index{dummy indices}.
\item Dummy indices summations are done on tensor products whenever
possible since it recognises the particular tensors
defined above or defined by the user.
\item It seeks a canonical form for the
various simplified terms, makes the comparison between them.
In that way it maximises simplifications and generates a canonical form
for the output polynomial.
\end{enumerate}
Its capabilities have been extended in four directions:
\begin{itemize}
\item It is able to work within {\em several} spaces\index{spaces}.
\item It manages correctly expressions where
formal tensor {\em derivatives} are present%
\footnote{In {\tt DUMMY}\ttindex{DUMMY} it does not take them into account}.
\item It takes into account all symmetries even if partial.
\item As its parent function, it can deal with non-commutative
and anticommutative indexed objects\index{anticommutative}.
So, Indexed objects may be spinors\index{spinor} or quantum fields.
\end{itemize}
We describe most of these features in the rest of this
documentation.
\subsection{Check of tensor indices}
\index{indices}
Dummy indices for individual tensors are kept in the memory of
the system. If they are badly distributed over several tensors,
it is CANONICAL\ which gives an error message:
\begin{verbatim}
tensor te,tf; ==> t
bb:=te(a,b,b)*te(-b); ==>
a b b
bb := te *te
b
canonical bb; ==>
***** ((b)(a b b)) are inconsistent lists of indices
aa:=te(b,-c)*tf(b,-c); ==>
b b
aa := te *tf % b and c are free.
c c
canonical aa; ==>
b b
te *tf
c c
bb:=te(a,c,b)*te(-b)*tf(a)$
canonical bb; ==>
a c b a
te *te *tf
b
delt(a,-a); ==> dim % a is now a dummy index
canonical bb; ==>
***** wrong use of indices (a)
\end{verbatim}
The message of canonical is clear, the first sublist contains the
list of all lower indices, and the second one the list of all upper
indices. The index {\tt b} is repeated {\em three} times.
In the second example, {\tt b} and {\tt c} are considered as free
indices, so they may be repeated.
The last example shows the interference between the check on
individual tensors and the one of canonical. The use of {\tt a}
as dummy index inside {\tt delt} does no longer allow {\tt a}
to be used as a free index in expression {\tt bb}.
To be usable, one must explicitly remove it as dummy index
using REM\_DUMMY\_INDICES \index{REM\_DUMMY\_INDICES}.
Dans le quatri\`eme cas, il n'y a pas de probl\`eme puisque {\tt b} et
{\tt c} sont tous les deux des indices {\em libres}.
CANONICAL\index{CANONICAL} checks that in a tensor polynomial
all do possess the {\em same} variance:
\begin{verbatim}
aa:=te(a,c)+x^2; ==>
a c 2
aa := te + x
canonical aa; ==>
***** scalar added with tensor(s)
aa:=te(a,b)+tf(a,c); ==>
a b a c
aa := te + tf
canonical aa; ==>
***** mismatch in free indices : ((a c) (a b))
\end{verbatim}
In the message the first two lists of incompatible indices are
explicitly indicated. So, it is not an exhaustive message and a more
complete correction may be needed.
Of course, no message of that kind appears if the indices are inside
ordinary operators%
\footnote{This is the case inside the {\tt DUMMY}\ttindex{DUMMY}\ package.}
\begin{verbatim}
dummy_names b; ==> t
cc:=op(b)*op(a,b,b); ==> cc := op(a,b,b)*op(b)
canonical cc; ==> op(a,b,b)*op(b)
clear_dummy_names; ==> t
\end{verbatim}
\subsection{Position and renaming of dummy indices}
\index{dummy indices}
For a specific tensor, contravariant dummy indices are place
in front of covariant ones. This already leads to some useful
simplifications. For instance:
\begin{verbatim}
pp:=te(a,-a)+te(-a,a)+1; ==>
a a
pp := te + te + 1
a a
canonical pp; ==>
a
2*te + 1
a
pp:=te(a,-a)+te(-b,b); ==>
b a
pp := te + te
b a
canonical pp; ==>
a
2*te
a
pp:=te(r,a,c,d,-a,f)+te(r,-b,c,d,b,f); ==>
r c d b f r a c d f
pp := te + te
b a
canonical pp; ==>
r a c d f
2*te
a
\end{verbatim}
In the second and third example, there is also a renaming of the
dummy variable {\tt b} whih becomes {\tt a}.
There is a loophole at this point. For some expressions one will
never reach a stable expression.
This is the case for the following very simple monom:
\begin{verbatim}
tensor nt; ==> t
a1:=nt(-a,d)*nt(-c,a); ==>
d a
nt *nt
a c
canonical a1; ==>
a d
nt *nt
c a
a12:=a1-canonical a1; ==>
d a a d
a12 := nt *nt - nt *nt
a c c a
canonical a12; ==>
d a a d
- nt *nt + nt *nt % changes sign.
a c c a
\end{verbatim}
In the above example, no canonical form can be reached. When applied twice
on the tensor monom {\tt a1} it gives back {\tt a1}!
No change of dummy index position is allowed if a tensor belongs
to an {\tt AFFINE}\ttindex{affine} space.
With the tensor polynomial {\tt pp} introduced above one has:
\ttindex{onespace OFF}
\begin{verbatim}
off onespace;
define_spaces aff={dd,affine}; ==> t
make_tensor_belong_space(te,aff); ==> aff
mk_ids_belong_space({a,b},aff); ==> t
canonical pp; ==>
r c d a f r a c d f
te + te
a a
\end{verbatim}
The renaming of {\tt b} has been made however.
\subsection{Contractions and summations with particular tensors}
\index{tensor contractions}
This is a central part of the extension of CANONICAL.
The required contractions and summations can be done in a
multi-space environment as well in a single space environment.
\begin{center}
The case of {\tt DELTA}\ttindex{delta}
\end{center}
Dummy indices are recognized contracted and summed over whenever possible:
\begin{verbatim}
aa:=delt(a,-b)*delt(b,-c)*delt(c,-a) + 1; ==>
a b c
aa := delt *delt *delt + 1
b c a
canonical aa; ==> dim + 1
aa:=delt(a,-b)*delt(b,-c)*delt(c,-d)*te(d,e)$
canonical aa; ==>
a e
te
\end{verbatim}
CANONICAL will not attempt to make contraction with
dummy indices included inside ordinary operators:
\index{OPERATOR}
\begin{verbatim}
operator op;
aa:=delt(a,-b)*op(b,b)$
canonical aa; ==>
a
delt *op(b,b)
b
dummy_names b; ==> t
canonical aa; ==>
a
delta *op(b,b)
b
\end{verbatim}
\begin{center}
The case of {\tt ETA}\ttindex{eta}
\end{center}
First, we introduce {\tt ETA}:
\begin{verbatim}
make_partic_tens(eta,eta); ==> t
signature 1; ==> 1 % necessary
aa:=delta(a,-b)*eta(b,c); ==>
a b c
aa := delt *eta
b
canonical aa; ==>
a c
eta
canonical(eta(a,b)*eta(-b,c)); ==>
a c
eta
canonical(eta(a,b)*eta(-b,-c)); ==>
a
delt
c
canonical(eta(a,b)*eta(-b,-a)); ==> dim
canonical (eta(-a,-b)*te(d,-e,f,b)); ==>
d f
te
e a
aa:=eta(a,b)*eta(-b,-c)*te(-a,c)+1; ==>
a b c
aa := eta *eta *te + 1
b c a
canonical aa; ==>
a
te + 1
a
aa:=eta(a,b)*eta(-b,-c)*delta(-a,c)+
1+eta(a,b)*eta(-b,-c)*te(-a,c)$
canonical aa; ==>
a
te + dim + 1
a
\end{verbatim}
Let us add a generic metric \index{metric tensor} tensor:
\begin{verbatim}
aa:=g(a,b)*g(-b,-d); ==>
a b
aa := g *g
b d
canonical aa; ==>
a
delt
d
aa:=g(a,b)*g(c,d)*eta(-c,-e)*eta(e,f)*te(-f,g); ==>
e f a b c d g
aa := eta *eta *g *g *te
c e f
canonical aa; ==>
a b d g
g *te
\end{verbatim}
\begin{center}
The case of {\tt EPSILON}\ttindex{epsilon}\ttindex{del}
\end{center}
The epsilon tensor plays an important role in many contexts.
CANONICAL\ realises the contraction of two epsilons if and only if
they belong to the same space. The proper use of CANONICAL\ on expressions
which contains it requires a preliminary definition of the tensor {\tt DEL}.
When the signature\ttindex{signature} is 0; the contraction of
two epsilons gives a {\tt DEL}-like tensor. When the
signature is equal to 1, it is equal to {\em minus} a {\tt DEL}-like tensor.
Here we choose 1 for the signature\index{signature} and we work in a
single space\index{spaces}.
\ttindex{onespace ON}
We define the {\tt DEL} tensor:
\begin{verbatim}
on onespace;
wholespace_dim dim; ==> dim
make_partic_tens(gd,del); ==> t
signature 1; ==> 1
\end{verbatim}
We define the {\tt EPSILON} tensor and show how CANONICAl\ contracts
expression containing {\em two}%
\footnote{No contractions are done on expressions containing
three or more epsilons which sit in the {\em same} space.
We are not sure whether it is useful
to be more general than we are presently.}
of them:
\begin{verbatim}
aa:=eps(a,b)*eps(-c,-d); ==>
a b
aa := eps *eps
c d
canonical aa; ==>
a b
- gd
c d
aa:=eps(a,b)*eps(-a,-b); ==>
a b
aa := eps *eps
a b
canonical aa; ==> dim*( - dim + 1)
on exdelt;
gd(-a,-b,a,b); ==> dim*(dim - 1)
aa:=eps(a,b,c)*eps(-b,-d,-e)$
canonical aa; ==>
a c a c
delt *delt *dim - 2*delt *delt -
d e d e
a c a c
- delt *delt *dim + 2*delt * delt
e d e d
\end{verbatim}
Several expressions which contain the epsilon tensor together
with other special tensors are given below as examples to treat
with CANONICAL:
\begin{verbatim}
aa:=eps( - b, - c)*eta(a,b)*eta(a,c); ==>
a b a c
eps *eta *eta
b c
canonical aa; ==> 0
aa:=eps(a,b,c)*te(-a)*te(-b); ==> % te is generic.
a b c
aa := eps *te *te
a b
canonical aa; ==> 0
tensor tf,tg;
aa:=eps(a,b,c)*te(-a)*tf(-b)*tg(-c)
+ eps(d,e,f)*te(-d)*tf(-e)*tg(-f); ==>
canonical aa; ==>
a b c
2*eps *te *tf *tg
a b c
aa:=eps(a,b,c)*te(-a)*tf(-c)*tg(-b)
+ eps(d,e,f)*te(-d)*tf(-e)*tg(-f)$
canonical aa; ==> 0
\end{verbatim}
Since \index{CANONICAL}CANONICAL is able to work inside several
spaces, we can introduce also several epsilons and make the relevant
simplifications on each (sub)-spaces. This is the goal of the
next illustration.
\ttindex{onespace OFF}\index{SHOW\_EPSILONS}
\begin{verbatim}
off onespace;
define_spaces wholespace=
{dim,signature=1}; ==> t
define_spaces subspace=
{3,signature=0}; ==> t
show_spaces(); ==>
{{wholespace,dim,signature=1},
{subspace,3,signature=0}}
make_partic_tens(eps,epsilon); ==> t
make_partic_tens(kap,epsilon); ==> t
make_tensor_belong_space(eps,wholespace);
==> wholespace
make_tensor_belong_space(kap,subspace);
==> subspace
show_epsilons(); ==>
{{eps,wholespace},{kap,subspace}}
off exdelt;
aa:=kap(a,b,c)*kap(-d,-e,-f)*eps(i,j)*eps(-k,-l)$
canonical aa; ==>
a b c i j
- gd *gd
d e f k l
\end{verbatim}
If there are no index summation, as in the expression above, one can
develop both terms into the delta tensor with EXDELT
\index{EXDELT} switched ON.
In fact, the previous calculation is correct {\em only if there are no
dummy index} inside the two {\tt gd}'s.
If some of the indices are
dummy, then
we must
take care of the respective spaces in which the two {\tt gd} tensors
are considered. Since, the tensor themselves do not belong
to a given space, the space identification can only be made through
the indices. This is enough since the {\tt DELTA}-like tensor
is bloc-diagonal. With {\tt aa} the result of the above illustration,
one gets, for example,:
\index{MK\_IDS\_BELONG\_SPACE}\index{indices}
\begin{verbatim}
mk_ids_belong_space({a,b,c,d,e,f},wholespace)$
mk_ids_belong_space({i,j,k,l},subspace)$
sub(d=a,e=b,k=i,aa); ==>
c j 2
2*delt *delt *( - dim + 3*dim - 2)
f l
sub(k=i,l=j,aa); ==>
a b c
- 6*gd
d e f
\end{verbatim}
\subsection{CANONICAL\ and symmetries}
\index{symmetries}
Most of the time, indexed objects have some symmetry property.
When this property is either full symmetry or antisymmetry, there
is no difficulty to implement it using the declarations
SYMMETRIC\ \index{SYMMETRIC} or \index{ANTISYMMETRIC}ANTISYMMETRIC
of \REDUCE. However, most often, indexed objects are neither
fully symmetric nor fully antisymmetric: they have {\em partial}
or {\em mixed} symmetries\index{partial symmetry}\index{mixed symmetry}.
In the {\tt DUMMY}\ttindex{DUMMY}\ package, the declaration
\index{SYMTREE}SYMTREE\ allows to impose such type of symmetries
on operators. This command has been improved and extended to
apply to tensors.
In order to illustrate it, we shall take the example of the
wellknown Riemann \index{Rieman tensor}tensor in general relativity.
Let us remind the reader
that this tensor has four indices. It is separately {\em antisymmetric} with
respect to the interchange of the first two indices and with respect to the
interchange of the last two indices. It is {\em symmetric} with respect to
the interchange of the first two and the last two indices.
In the illustration below, we show how to express this and how
CANONICAL\ is able to recognize mixed symmetries:
\begin{verbatim}
tensor r; ==> t
symtree(r,{!+,{!-,1,2},{!-,3,4}});
rem_dummy_indices a,b,c,d; % free indices
ra:=r(b,a,c,d); ==>
b a c d
ra := r
canonical ra; ==>
a b c d
- r
ra:=r(c,d,a,b); ==>
c d a b
ra := r
canonical ra; ==>
a b c d
r
canonical r(-c,-d,a,b); ==>
a b
r
c d
r(-c,-c,a,b); ==> 0
ra:=r(-c,-d,c,b); ==>
c b
ra := r
c d
canonical ra; ==>
b c
- r
c d
\end{verbatim}
In the last illustration, contravariant indices are placed in front
of covariant indices and the contravariant indices are transposed.
The superposition of the two partial symmetries gives a minus sign.
There exists an important (though natural) restriction on the use of
SYMTREE\ which is linked to the algorithm itself: Integer used to localize
indices must start from 1, be {\em contiguous} and monotoneously increasing.
For instance, one is not allow to introduce
\begin{verbatim}
symtree(r,{!*,{!+,1,3},{!*,2,4}});
symtree(r,{!*,{!+,1,2},{!*,4,5}};
symtree(r,{!*,{!-,1,3},{!*,2}});
\end{verbatim}
but the subsequent declarations are allowed:
\begin{verbatim}
symtree(r,{!*,{!+,1,2},{!*,3,4}});
symtree(r,{!*,{!+,1,2},{!*,3,4,5}});
symtree(r,{!*,{!-,1,2},{!*,3}});
\end{verbatim}
The first declaration endows {\tt r} with a {\em partial} symmetry
with respect to the first two indices.
A side effect of SYMTREE\ is to restrict the number of indices of
a generic tensor. For instance, the second declaration in the above
illustrations makes {\tt r} depend on 5 indices as illustrated below:
\begin{verbatim}
symtree(r,{!*,{!+,1,2},{!*,3,4,5}});
canonical r(-b,-a,d,c); ==>
***** Index `5' out of range for
((minus b) (minus a) d c) in nth
canonical r(-b,-a,d,c,e); ==>
d c e
r % correct
a b
canonical r(-b,-a,d,c,e,g); ==>
d c e
r % The sixth index is forgotten!
a b
\end{verbatim}
Finally, the function REMSYM\index{REMSYM}\ applied on any tensor
identifier removes all symmetry properties.
Another related question is the frequent need to symmetrize
a tensor polynomial.
To fulfill it, the function \index{SYMMETRIZE}SYMMETRIZE
of the package {\tt ASSIST} \ttindex{ASSIST} has been improved and
generalised. For any kernel\index{kernel} ( which may be either
an operator or a tensor) that function generates
\begin{itemize}
\item[-] the sum over the cyclic permutations of indices,
\item[-] the symetric or antisymetric sums over all permutations
of the indices.
\end{itemize}
Moreover, if it is given a list of indices, it generates a new list
which contains sublists wich contain the relevant permutations of
these indices
\begin{verbatim}
symmetrize(te(x,y,z,{v}),te,cyclicpermlist); ==>
x y z y z x z x y
te (v) + te (v) + te (v)
symmetrize(te(x,y),te,permutations); ==>
x y y x
te + te
symmetrize(te(x,y),te,permutations,perm_sign); ==>
x y y x
te - te
symmetrize(te(y,x),te,permutations,perm_sign); ==>
x y y x
- te + te
\end{verbatim}
If one wants to symmetrise an expression which is not a kernel, one
can also use SYMMETRIZE\ to obtain the desired result as the next
example shows:
\begin{verbatim}
ex:=te(a,-b,c)*te1(-a,-d,-e); ==>
a c
ex := te *te1
b a d e
ll:=list(b,c,d,e)$ % the chosen relevant indices
lls:=symmetrize(ll,list,cyclicpermlist); ==>
lls := {{b,c,d,e},{c,d,e,b},{d,e,b,c},{e,b,c,d}}
% The sum over the cyclic permutations is:
excyc:=for each i in lls sum
sub(b=i.1,c=i.2,d=i.3,e=i.4,ex); ==>
a c a d
excyc := te *te1 + te *te1
b a d e c a e b
a e a b
+ te *te1 + te *te1
d a b c e a c d
\end{verbatim}
\subsection{CANONICAL and tensor derivatives}
\index{tensor derivatives}
Only ordinary (partial) derivatives are fully correctly handled
by CANONICAL. This is enough, to explicitly construct covariant
derivatives. We recognize here that extensions should still be made.
The subsequent illustrations show how CANONICAL\ does indeed
manage to find the canonical form and simplify expressions
which contain derivatives. Notice, the use of
the (modified) \index{DEPEND}DEPEND\ declaration.
\ttindex{onespace ON}
\begin{verbatim}
on onespace;
tensor te,x; ==> t
depend te,x;
aa:=df(te(a,-b),x(-b))-df(te(a,-c),x(-c))$
canonical aa; ==> 0
make_partic_tens(eta,eta); ==> t
signature 1;
aa:=df(te(a,-b),x(-b))$
aa:=aa*eta(-a,-d);
a
aa := df(te ,x )*eta
b b a d
canonical aa; ==>
a a
df(te ,x )
d
\end{verbatim}
In the last example, after contraction, the covariant dummy index {\tt b}
has been changed into the contravariant dummy index {\tt a}. This is allowed
since the space is metric.
\newpage
\printindex
\end{document}