Grassmann Algebra

ExpTheGeneralizedProduct.nb 9
To expand with respect to Α we reverse the order of the generalized product to give Β����� �Α and
m 3 2 4
multiply by ��1� �m�� �k�� (which we note for this case to be 1) to give:
B � ToInteriorProducts�Β����� �Α� 3 2 4
�Β1 � Β2 � Β3 � Α1 � Α2��Α3 � Α4 � �Β1 � Β2 � Β3 � Α1 � Α3��Α2 � Α4 �
�Β1 � Β2 � Β3 � Α1 � Α4��Α2 � Α3 � �Β1 � Β2 � Β3 � Α2 � Α3��Α1 � Α4 �
�Β1 � Β2 � Β3 � Α2 � Α4��Α1 � Α3 � �Β1 � Β2 � Β3 � Α3 � Α4��Α1 � Α2
Note that in the expansion A there are ����
�
3 ����
(= 3) terms while in expansion B there are
2 �
����
�
4 ����
(= 6)
2 �
terms.
� Reduction to inner products
At this stage the results do not look similar. Next we develop the inner product forms:
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
B1 � ToInnerProducts�B�
�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
By inspection we see that these two expressions are the same. Calculating the difference in
Mathematica verifies this.
A1 � B1
0
The identity of the forms A1 and B1 is an example of the expansion of a generalized product in
terms of either factor yielding the same result.
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