Grassmann Algebra

ExploringCliffordAlgebra.nb 12
12.5 Alternate Forms for the Clifford Product
Alternate expansions of the Clifford product
The Clifford product has been defined in Section 12.2 as:
Α m �Β k
�
Min�m,k�
� ���1�
Λ�0
1
��m��� ����
2
Λ �Λ�1�
�Αm ����� �Β
Λ k
Alternate forms for the generalized product have been discussed in Chapter 10. The generalized
product, and hence the Clifford product, may be expanded by decomposition of Α m , by
decomposition of Β, or by decomposition of both Α and Β. k
m k
� The Clifford product expressed by decomposition of the
first factor
The generalized product, expressed by decomposition of Α m is:
Α m ����� Λ �Β k
� m
Λ �
� �
i�1
Αi m�Λ ��Β���� Α
k
i
Λ � Α m �Α1 � Α1
Λ m�Λ �Α2 � Α2 � �
Λ m�Λ
Substituting this into the expression for the Clifford product gives:
Α m �Β k
Α m �Α 1
Λ
Min�m,k�
� �
Λ�0
� m
Λ �
�� ��1�
i�1
� Α1
m�Λ �Α2 � Α2 � �
Λ m�Λ
�
1
Λ��m�Λ�� ���� Λ �Λ�1�
2 i
Α
m�Λ ��Β k
���� Α i
Λ �
Our objective is to rearrange the formula for the Clifford product so that the signs are absorbed
into the formula, thus making the form of the formula independent of the values of m and Λ. We
can do this by writing ��1� 1
����
2
Λ �Λ�1� i �Α
Λ �Αi
† i † i
(where Α is the reverse of Α ) and
Λ
Λ
Λ
interchanging the order of the decomposition of Α into a Λ-element and a (m-Λ)-element to
m
absorb the ��1� Λ��m�Λ� factor: ��1� Λ��m�Λ� �Α �Α
m 1 � Α1
m�Λ Λ �Α2 � Α2 � � .
m�Λ Λ
2001 4 26
Α m �Β k
Min�m,k�
� �
Λ�0
� m
Λ �
��
i�1
Α i
m�Λ ��Β k
Α �Α
m 1
� Α1
m�Λ Λ �Α2 � Α2 � �
m�Λ Λ
���� Α i †
�
Λ