Grassmann Algebra

ExpTheGeneralizedProduct.nb 16
AB � CollectTerms�B1 � A1�
���Α4 � Β1 � Β2 � Β3� �Α4 � Β2 � Β1 � Β3 �Α4 � Β3 � Β1 � Β2� Α1 � Α2 � Α3 �
�Α3 � Β1 � Β2 � Β3 �Α3 � Β2 � Β1 � Β3 �Α3 � Β3 � Β1 � Β2� Α1 � Α2 � Α4 �
���Α2 � Β1 � Β2 � Β3� �Α2 � Β2 � Β1 � Β3 �Α2 � Β3 � Β1 � Β2� Α1 � Α3 � Α4 �
�Α2 � Α3 � Α4 � Β3 �Α2 � Α4 � Α3 � Β3 �Α2 � Β3 � Α3 � Α4� Α1 � Β1 � Β2 �
���Α2 � Α3 � Α4 � Β2� �Α2 � Α4 � Α3 � Β2 �Α2 � Β2 � Α3 � Α4� Α1 � Β1 � Β3 �
�Α2 � Α3 � Α4 � Β1 �Α2 � Α4 � Α3 � Β1 �Α2 � Β1 � Α3 � Α4� Α1 � Β2 � Β3 �
�Α1 � Β1 � Β2 � Β3 �Α1 � Β2 � Β1 � Β3 �Α1 � Β3 � Β1 � Β2� Α2 � Α3 � Α4 �
���Α1 � Α3 � Α4 � Β3� �Α1 � Α4 � Α3 � Β3 �Α1 � Β3 � Α3 � Α4� Α2 � Β1 � Β2 �
�Α1 � Α3 � Α4 � Β2 �Α1 � Α4 � Α3 � Β2 �Α1 � Β2 � Α3 � Α4� Α2 � Β1 � Β3 �
���Α1 � Α3 � Α4 � Β1� �Α1 � Α4 � Α3 � Β1 �Α1 � Β1 � Α3 � Α4� Α2 � Β2 � Β3 �
�Α1 � Α2 � Α4 � Β3 �Α1 � Α4 � Α2 � Β3 �Α1 � Β3 � Α2 � Α4� Α3 � Β1 � Β2 �
���Α1 � Α2 � Α4 � Β2� �Α1 � Α4 � Α2 � Β2 �Α1 � Β2 � Α2 � Α4� Α3 � Β1 � Β3 �
�Α1 � Α2 � Α4 � Β1 �Α1 � Α4 � Α2 � Β1 �Α1 � Β1 � Α2 � Α4� Α3 � Β2 � Β3 �
���Α1 � Α2 � Α3 � Β3� �Α1 � Α3 � Α2 � Β3 �Α1 � Β3 � Α2 � Α3� Α4 � Β1 � Β2 �
�Α1 � Α2 � Α3 � Β2 �Α1 � Α3 � Α2 � Β2 �Α1 � Β2 � Α2 � Α3� Α4 � Β1 � Β3 �
���Α1 � Α2 � Α3 � Β1� �Α1 � Α3 � Α2 � Β1 �Α1 � Β1 � Α2 � Α3� Α4 � Β2 � Β3 �
��2 �Α1 � Α3 � Α2 � Α4� � 2 �Α1 � Α2 � Α3 � Α4 �Α1 � Α4 � Α2 � Α3��
Β1 � Β2 � Β3
Reduction to scalar products
In order to show the equality of the two forms, we need to show that the above difference is
zero. As in the previous section we may do this by converting the inner products to scalar
products. Alternatively we may observe that the coefficients of the terms are zero by virtue of
the Zero Interior Sum Theorem discussed Section 10.8.
ToScalarProducts�AB�
0
This is an example of how the B form of the generalized product may be expanded in terms of
either factor.
10.6 The Zero Interior Sum Theorem
Generalized Grassmann products with the unit scalar
Suppose in the Generalized Product Theorem that Α m is unity. Then we can write:
1����� Λ �Β k
� k
Λ �
� � �1 ���� Βj j�1
Λ
�� j
k�Λ
� k
Λ �
� �
j�1
�1 � Βj � ���� Β
k�Λ
j
Λ
When Λ is equal to zero we have the trivial identity that:
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