Grassmann Algebra

ExpTheGeneralizedProduct.nb 12
� Example: Verification of the Generalized Product Theorem
As an example we take the same generalized product Α����� �Β that we explored in section [10.4].
4 2 3
First we convert it to interior products as we did in that section.
A � ToInteriorProducts�Α����� �Β� 4 2 3
�Α1 � Α2 � Α3 � Α4 � Β1 � Β2��Β3 �
�Α1 � Α2 � Α3 � Α4 � Β1 � Β3��Β2 � �Α1 � Α2 � Α3 � Α4 � Β2 � Β3��Β1
Next, we convert the same generalized product to its B form expression by using a modified
version of the GrassmannAlgebra function ToInteriorProducts. This modified version is
termed ToInteriorProductsB. (Note the 'B' at the end of the function name).
B � 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
Note that the A form and the B form at this first (interior product) level of their expansion both
have the same number of terms (in this case ����
�
3 ����
= 3).
2 �
We now convert these two expressions to their inner product forms.
2001 4 26
A1 � ToInnerProducts�A�
�Α3 � Α4 � Β2 � Β3� Α1 � Α2 � Β1 � �Α3 � Α4 � Β1 � Β3� Α1 � Α2 � Β2 �
�Α3 � Α4 � Β1 � Β2� Α1 � Α2 � Β3 � �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 � 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