Grassmann Algebra

ExpTheGeneralizedProduct.nb 14
between two equal expressions. On the other hand, expansion of the B form requires expansion
to the scalar product level, often involving significantly more computation.
We take the same generalized product Α 4 ����� 2 �Β 3
that we discussed in the previous section, and
show that we need to expand the product to the scalar product level before the identity of the
two expansions become evident.
Reduction to interior products
A � ToInteriorProductsB�Α����� �Β� 4 2 3
Α1 � Α2 � Α3 � Α4 � Β1 � Β2 � Β3 �
Α1 � Α2 � Α3 � Α4 � Β2 � Β1 � Β3 �Α1 � Α2 � Α3 � Α4 � Β3 � Β1 � Β2
To expand with respect to Α we reverse the order of the generalized product to give Β����� �Α and
4 3 2 4
multiply by ��1� �m�� �k�� (which we note for this case to be 1) to give
B � ToInteriorProductsB�Β����� �Α� 3 2 4
Α1 � Α2 � Β1 � Β2 � Β3 � Α3 � Α4 �Α1 � Α3 � Β1 � Β2 � Β3 � Α2 � Α4 �
Α1 � Α4 � Β1 � Β2 � Β3 � Α2 � Α3 �Α2 � Α3 � Β1 � Β2 � Β3 � Α1 � Α4 �
Α2 � Α4 � Β1 � Β2 � Β3 � Α1 � Α3 �Α3 � Α4 � Β1 � Β2 � Β3 � Α1 � Α2
Reduction to inner products
2001 4 26
A1 � ToInnerProducts�A�
�Α4 � Β1 � Β2 � Β3� Α1 � Α2 � Α3 � �Α4 � Β2 � Β1 � Β3� Α1 � Α2 � Α3 �
�Α4 � Β3 � Β1 � Β2� Α1 � Α2 � Α3 � �Α3 � Β1 � Β2 � Β3� Α1 � Α2 � Α4 �
�Α3 � Β2 � Β1 � Β3� Α1 � Α2 � Α4 � �Α3 � Β3 � Β1 � Β2� Α1 � Α2 � Α4 �
�Α3 � Α4 � Β2 � Β3� Α1 � Α2 � Β1 � �Α3 � Α4 � Β1 � Β3� Α1 � Α2 � Β2 �
�Α3 � Α4 � Β1 � Β2� Α1 � Α2 � Β3 � �Α2 � Β1 � Β2 � Β3� Α1 � Α3 � Α4 �
�Α2 � Β2 � Β1 � Β3� Α1 � Α3 � Α4 � �Α2 � Β3 � Β1 � Β2� Α1 � Α3 � Α4 �
�Α2 � Α4 � Β2 � Β3� Α1 � Α3 � Β1 � �Α2 � Α4 � Β1 � Β3� Α1 � Α3 � Β2 �
�Α2 � Α4 � Β1 � Β2� Α1 � Α3 � Β3 � �Α2 � Α3 � Β2 � Β3� Α1 � Α4 � Β1 �
�Α2 � Α3 � Β1 � Β3� Α1 � Α4 � Β2 � �Α2 � Α3 � Β1 � Β2� Α1 � Α4 � Β3 �
�Α1 � Β1 � Β2 � Β3� Α2 � Α3 � Α4 � �Α1 � Β2 � Β1 � Β3� Α2 � Α3 � Α4 �
�Α1 � Β3 � Β1 � Β2� Α2 � Α3 � Α4 � �Α1 � Α4 � Β2 � Β3� Α2 � Α3 � Β1 �
�Α1 � Α4 � Β1 � Β3� Α2 � Α3 � Β2 � �Α1 � Α4 � Β1 � Β2� Α2 � Α3 � Β3 �
�Α1 � Α3 � Β2 � Β3� Α2 � Α4 � Β1 � �Α1 � Α3 � Β1 � Β3� Α2 � Α4 � Β2 �
�Α1 � Α3 � Β1 � Β2� Α2 � Α4 � Β3 � �Α1 � Α2 � Β2 � Β3� Α3 � Α4 � Β1 �
�Α1 � Α2 � Β1 � Β3� Α3 � Α4 � Β2 � �Α1 � Α2 � Β1 � Β2� Α3 � Α4 � Β3