Grassmann Algebra

TheRegressiveProduct.nb 39
We can calculate the dual of this formula by applying the GrassmannAlgebra function Dual:
Dual��Α � Β�� x � �Α � x ��Β � ��1�
m k n�1 m n�1 k
m �Α ��Β� x ��
m k n�1
�Α � Β��x �� ��1�
m k
�m�n Α � Β � x �Α� x � Β
m k m k
which we rearrange slightly to make it more readable as:
Note that here x is a 1-element.
�Α � Β��x �� �Α � x��Β � ��1�
m k
m k
n�m Α ��Β� x�
m k
The General Product Formula
If x is of a grade higher than 1, then similar relations hold, but with extra terms on the righthand
side. For example, if we replace x by x1 � x2 and note that:
�Α � Β���x1 � x2� � ��Α � Β��x1��x2 m k
m k
then the right-hand side may be expanded by applying the Product Formula [3.42] successively
to obtain:
�Α � Β���x1 � x2� � �Α ��x1 � x2�� � Β �Α��Β��x1 � x2��
m k
m k m k
���1� n�m ���Α � x2���Β� x1� � �Α � x1���Β� x2��
m k
m k
Continued reapplication of this process expresses for simple x the expression �Α � Β��x as
p m k p
the sum of 2p terms.
�Α m � Β k
x p � x1
r
p
��x p � �
r��0
���1� r��n�m� � �
� x1 � x2 � x2 � � � xΝ
p�r r p�r
r
We call formula 3.44 the General Product Formula.
Ν
���Α � xi
i��1 m p�r
� �
�
����
��Β
� k
� xi�
r
� xΝ , Ν��
p�r
p
r �
The dual of the General Product Formula may be obtained following the duality algorithm, or
by using the Dual function. Input to the Dual function requires the summations to be
expressed in such a way as to frustrate Mathematica interpreting them as the Sum function. To
make the formula more readable we have modified the actual dual by replacing r with nÐr.
2001 4 5
3.42
3.43
3.44