Grassmann Algebra

ExploringCliffordAlgebra.nb 16
The Clifford product expressed by decomposition of both
factors
We can similarly decompose the Clifford product in terms of both Α and Β. The sign
m k
��1� 1
���� 2 Λ �Λ�1� i
can be absorbed by taking the reverse of either Α
Λ
Min�m,k�
�
Λ�0
� m
Λ �
� m
Λ �
� ���
i�1
���
i�1
� k
Λ �
j�1
Α m �Α 1
Λ
� k
Λ �
j�1
Α m �Β k
�
��1� Λ��m�Λ� ��Α i
Λ
Α m �Β k
Min�m,k�
� �
��1� Λ��m�Λ� ��Α i
Λ
Λ�0
† j
���� Β
Λ
�
or Βj .
Λ
� Α i
� Βj
m�Λ k�Λ
† i
���� Βj � Α � Βj
Λ m�Λ k�Λ
� Α1
m�Λ �Α2�
Α2
Λ m�Λ � � Β k
Β 1
Λ
� Β 1
k�Λ
� Β 2
Λ
� Β 2
� �
k�Λ
�
12.16
12.6 Writing Down a General Clifford Product
� The form of a Clifford product expansion
We take as example the Clifford product Α �Β and expand it in terms of the factors of its first
4 k
factor in both the D form and the C form. We see that all terms except the interior and exterior
products at the ends of the expression appear to be different. There are of course equal numbers
of terms in either form. Note that if the parentheses were not there, the expressions would be
identical except for (possibly) the signs of the terms. Although these central terms differ
between the two forms, we can show by reducing them to scalar products that their sums are the
same.
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