Grassmann Algebra

ExpTheGeneralizedProduct.nb 13
B1 � ToInnerProducts�B�
�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
It can be seen that at this inner product level, these two expressions are not of the same form:
B1 contains additional terms to those of A1. The interior products of A1 are only of the form
�Αi � Αj ���� Βr � Βs�, whereas the extra terms of B1 are of the form �Αi � Βj ���� Βr � Βs�.
Calculating their difference (and simplifying by using the GrassmannAlgebra function
CollectTerms) gives:
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 �
���Α1 � Β1 � Β2 � Β3� �Α1 � Β2 � Β1 � Β3 �Α1 � Β3 � Β1 � Β2� Α2 � Α3 � Α4
To verify the equality of the two forms we need only show that this difference AB is zero. This
is equivalent to showing that the coefficient of each of the exterior products is zero. We can do
this most directly by converting the whole expression to its scalar product form.
ToScalarProducts�AB�
0
The fact that the expression AB is zero has shown us that certain sums of inner products are
zero. We generalize this result in Section 10.7.
� Verification that the B form may be expanded in terms of either
factor
In this section we verify that just as for the A form, the B form may be expanded in terms of
either of its factors.
The principal difference between an expansion using the A form, and one using the B form is
that the A form needs only to be expanded to the inner product level to show the identity
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