Grassmann Algebra

ExploringCliffordAlgebra.nb 22
12.8 The Clifford Product of Orthogonal
Elements
The Clifford product of totally orthogonal elements
In Chapter 10 on generalized products we showed that if Α m and Β k
are totally orthogonal (that is,
Αi ���� Βj � 0 for each Αi belonging to Α and Βj belonging to Β, then Α����� �Β �� 0, except when Λ
m k m Λ k
= 0.
Thus we see immediately from the definition of the Clifford product that the Clifford product of
two totally orthogonal elements is equal to their exterior product.
Α m �Β k
�Α m � Β k
Αi ���� Βj � 0
The Clifford product of partially orthogonal elements
Suppose now we introduce an arbitrary element Γ
p
of generalized products.
Α �
m ���Γ
� p
Min�m,k�p�
�
� �� � �
k�
Λ�0
���1�
But from formula 10.35 we have that:
Hence:
Α����� �
m Λ ���Γ
�p
�
� �� �
k�
�
�
Α �
m ����Γ
�p
��Α m ����� Λ �Γ p
���
� Β
� k
1
Λ��m� Λ������ Λ �Λ�1�
2 �Αm ����� �
Λ ���Γ
p
�
� ��� �
k�
����Α
�
�Γ��� � Β
�m
p�
k
12.25
into Α �Β, and expand the expression in terms
m k
�
�
� �� k�
Αi ���� Βj � 0 Λ�Min�m, p�
Αi ���� Βj � 0
12.26
Similarly, expressing ���Α
�
� �� � in terms of generalized products and substituting from
�m
p�
k
equation 10.37 gives:
2001 4 26
���Α
�
� Γ�� �Β
� m p�
k
�
Min�p,k�
� ���1�
Λ�0
1
��m�p�� � ����
2
Λ �Λ�1��m Λ
�Αm � �
�
��Γ ����� �Β
p Λ k
���
�