Grassmann Algebra

TheRegressiveProduct.nb 40
�Α m � Β k
n�p
��x p � �
r��0
���1� mr � �
Ν
���Α � xi
i��1 m p�r
� �
�
����
��Β
� k
� xi �
n�r
n � p
x � x1 � x1 � x2 � x2 � � � xΝ � xΝ , Ν��
p n�r p�r n�r p�r
n�r p�r
r �
These formulae apply only for simple x, but may be extended to the non-simple case by
p
application to each component of a non-simple x. p
� Exploring the General Product Formula
The development of the General Product Formula [3.44] from the Product Formula [3.42] is a
good example of how Mathematica and GrassmannAlgebra may be used to explore and create
new results. The code below is typical of how you can combine functions from Mathematica
(for example Fold) with functions from GrassmannAlgebra.
ToProductFormula[(Α_�Β_)�x_,n_]:=
Module[{S,F},S=Scalars;DeclareExtraScalars[{Grade[Α],n}];
ExpandProductFormula[X_,y_]:=
SimplifySigns[Expand[FactorScalars[ExpandProducts[X�y]]
/.((u_�v_)�z_�(u�z)�v+(-1)^(n-Grade[Α]) u�(v�z))]];
F=Fold[ExpandProductFormula,Α�Β,List@@CreateVariableForm[x]];
DeclareScalars[S];F]
To get some immediate information from Mathematica on the syntax of functions to use, you
can type in a question mark ? followed by the name of the function.
? ToProductFormula
ToProductFormula��Α�Β��x,n� expands the product �Α�Β��x in terms
of the simple factors of x. The elements Α and Β may be
expressed as graded symbols with symbolic or numeric grade.
The element x must be either a simple element or a graded
symbol with a numeric grade. The expansion is independent
of the dimension of the currently declared space, but
requires the specification of a symbol n for this dimension.
Example: The General Product Formula for a 3-element
We take the example of the General Product Formula for a 3-element.
2001 4 5
3.45