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This package allows a user to find the canonical form of expressions involving dummy variables. In that way, the simplification of polynomial expressions can be fully done. The indeterminates are general operator objects endowed with as few properties as possible. In that way the package may be used in a large spectrum of applications.
The possibility to handle dummy variables and to manipulate dummy summations are important features in many applications. In particular, in theoretical physics, the possibility to represent complicated expressions concisely and to realize simplifications efficiently depend on both capabilities. However, when dummy variables are used, there are many more ways to express a given mathematical objects since the names of dummy variables may be chosen almost arbitrarily. Therefore, from the point of view of computer algebra the simplification problem is much more difficult. Given a definite ordering, one is, at least, to find a representation which is independent of the names chosen for the dummy variables otherwise, simplifications are impossible. The package does handle any number of dummy variables and summations present in expressions which are arbitrary multivariate polynomials and which have operator objects eventually dependent on one (or several) dummy variable(s) as some of their indeterminates. These operators have the same generality as the one existing in REDUCE. They can be noncommutative, anticommutative or commutative. They can have any kind of symmetry property. Such polynomials will be called in the following dummy polynomials. Any monomial of this kind will be called dummy monomial. For any such object, the package allows to find a well defined normal form in one-to-one correspondance with it.
In section 2, the convention for writing dummy summations is explained and the available declarations to introduce or suppress dummy variables are given.
In section 3, the commands allowing to give various algebraic properties to the operators are described.
In section 4, the use of the function canonical is explained and illustrated.
For references, see [BL85, BC82, But82, Leo80, Leo84, Leo91, Lin91, McK78, RT89, Sim71b, Sim71a, BC94, Cap97].
The use of DUMMY requires that the package ASSIST version 2.2 be available. It is loaded automatically when DUMMY is loaded.
A dummy variable (let us name it dv) is an identifier which runs from the integer \(i_1\) to another integer \(i_2\). To the extent that no definite space is defined, \(i_1\) and \(i_2\) are assumed to be some integers which are the same for all dummy variables.
If f is any REDUCE operator, then the simplest dummy summation associated to dv is the sum \[ \sum _{dv=i_1}^{i_2} f(dv) \] and is simply written as \[ f(dv). \] No other rules govern the implicit summations. dv can appear as many times we want since the operator f may depend on an arbitrary number of variables. So, the package is potentially applicable to many contexts. For instance, it is possible to add rules of the kind one encounters in tensor calculus.
Obviously, there are as many ways we want to express the same quantity. If the name of another dummy variable is dum then the previous expression is written as \[ \sum _{dum=i_1}^{i_2} f(dum) \] and the computer algebra system should be able to find that the expression \[ f(dv)-f(dum); \] is equal to \(0\). A very special case which is allowed is when f is the identity operator. So, a generic dummy polynomial will be a sum of dummy monomials of the kind \[ \prod _i c_i*f_i(dv_1,\ldots ,dv_{k_i},fr_1,\ldots , fr_{l_i}) \] where \(dv_1,\ldots ,\) are dummy variables while \(fr_1, \ldots , \) are ordinary or free variables.
To declare dummy variables, two commands are available:
i.
where \(\langle \)idp\(\rangle \) is the name of any unassigned identifier.
ii.
The first one declares idp_1,…,idp_n as dummy variables i.e. all variables of the form idp_xxx where \(xxx\) is a number will be dummy variables, such as idp_1, idp_2,…, idp_23. The second one gives special names for dummy variables. All other identifiers which may appear are assumed to be free. However, there is a restriction: named and base dummy variables cannot be declared simultaneously. The above declarations are mutually exclusive. Here is an example showing that:
dummy_base dv; ==> dv % dummy indices are dv1, dv2, dv3, ... dummy_names i,j,k; ==> ***** The created dummy base dv must be cleared
When this is done, an expression like
op(dv1)*sin(dv2)*abs(i)*op(dv2)$
means a sum over \(dv_1,dv_2\). To clear the dummy base, and to create the dummy names \(i,j,k\) one is to do
clear_dummy_base; ==> t dummy_names i,j,k; ==> t % dummy indices are i,j,k.
When this is done, an expression like
op(dv1)*sin(dv2)*abs(x)*op(i)^3*op(dv2)$
means a sum over \(i\). Similarly, the command clear_dummy_names clears earlier defined named dummy variables.
One should keep in mind that every application of the above commands erases the previous ones. It is also possible to display the declared dummy names using show_dummy_names:
show_dummy_names(); ==> {i,j,k}
To suppress all dummy variables one can enter
clear_dummy_names; clear_dummy_base;
All dummy variables should appear at first level as arguments of operators. For instance, if \(i\) and \(j\) are dummy variables, the expression
rr:= op(i,j)-op(j,j)
is allowed but the expression
op(i,op(j)) - op(j,op(j))
is not allowed. This is because dummy variables are not detected if they appear at a level larger than 1. Apart from that there is no restrictions. Operators may be commutative, noncommutative or even anticommutative. Therefore they may be elements of an algebra, they may be tensors, spinors, grassman variables, etc. \(\ldots \) By default they are assumed to be commutative and without symmetry properties. The REDUCE command noncomdeclaration is taken into account and, in addition, the command
anticom at1, at2;
makes the operators \(at_1\) and \(at_2\) anticommutative.
One can also give symmetry properties to them. The usual declarations symmetric and antisymmetric are taken into account. Moreover and most important they can be endowed with a partial symmetry through the command symtree. Here are three illustrative examples for the \(r\) operator:
symtree (r,{!+, 1, 2, 3, 4});
symtree (r,{!*, 1, {!-, 2, 3, 4}});
symtree (r, {!+, {!-, 1, 2}, {!-, 3, 4}});
The first one makes the operator (fully) symmetric. The second one declares it antisymmetric with respect to the three last indices. The symbols !*, !+ and !- at the beginning of each list mean that the operator has no symmetry, is symmetric or is antisymmetric with respect to the indices inside the list. Notice that the indices are not denoted by their names but merely by their natural order of appearance. 1 means the first written argument of \(r\), 2 its second argument etc. The first command is equivalent to the declaration symmetric except that the number of indices of \(r\) is restricted to 4 i.e. to the number declared in symtree. In the second example \(r\) is stated to have no symmetry with respect to the first index and is declared to be antisymmetric with respect to the three last indices. In the third example, \(r\) is made symmetric with respect to the interchange of the pairs of indices 1,2 and 3,4 respectively and is made antisymmetric separately within the pairs \((1,2)\) and \((3,4)\). It is the symmetry of the Riemann tensor. The anticommutation property and the various symmetry properties may be suppressed by the commands remanticom and remsym. To eliminate partial symmetry properties one can also use symtree itself. For example, assuming that \(r\) has the Riemann symmetry, to eliminate it do
symtree (r,{!*, 1, 2, 3, 4});
However, notice that the number of indices remains fixed and equal to 4 while with remsym it becomes again arbitrary.
canonical is the most important functionality of the package. It can be applied on any polynomial whether it is a dummy polynommial or not. It returns a normal form uniquely determined from the current ordering of the system. If the polynomial does not contain any dummy index, it is rewriten taking into account the various operator properties or symmetries described above. For instance,
symtree (r, {!+, {!-, 1, 2}, {!-, 3, 4}});
aa:=r(x3,x4,x2,x1)$
canonical aa; ==> - r(x1,x2,x3,x4).
If it contains dummy indices, canonical takes also into account the various dummy summations, makes the relevant simplifications, eventually rename the dummy indices and returns the resulting normal form. Here is a simple example:
operator at1,at2;
anticom at1,at2;
dummy_names i,j,k; ==> t
show_dummy_names(); ==> {i,j,k}
rr:=at1(i)*at2(k) -at2(k)*at1(i)$
canonical rr; => 2*at1(i)*at2(j)
It is important to notice, in the above example, that in addition to the summations over indices \(i\) and \(k\), and of the anticommutativity property of the operators, canonical has replaced the index \(k\) by the index \(j\). This substitution is essential to get full simplification. Several other examples are given in the test file and, there, the output of canonical is explained.
As stated in the previous section, the dependence of operators on dummy indices is limited to first level. An erroneous result will be generated if it is not the case as the subsequent example illustrates:
operator op; dummy_names i,j; rr:=op(i,op(j))-op(j,op(j))$ canonical rr; ==> 0
Zero is obtained because, in the second term, canonical has replaced \(j\) by \(i\) but has left \(op(j)\) unchanged because it does not recognize the index \(j\) which is inside. This fact has also the consequence that it is unable to simplify correctly (or at all) expressions which contain some derivatives. For instance (\(i\) and \(j\) are dummy indices):
aa:=df(op(x,i),x) -df(op(x,j),x)$ canonical aa; ==> df(op(x,i),x) - df(op(x,j),x)
instead of zero. A second limitation is that canonical does not add anything to the problem of simplifications when side relations (like Bianchi identities) are present.
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