Grassmann Algebra

ExploringCliffordAlgebra.nb 35
The decomposition of a Clifford product
Finally, therefore, we can write:
Α m �Β k
� �Α m �Β k
� e
��1� mk �Β �Α � �Α �Β
k m m k
which we can add and subtract to give:
�Α m �Β k
�Α m �Β k
� e
� o
� 1
����
2 ��Α m �Β k
� 1
����
2 ��Α m �Β k
Example: An m-element and a 1-element
Putting k equal to 1 gives:
� �Α m �Β k
� e
� o
� �Α m �Β k
� o
� ��1� mk �Β �Α �
k m
� ��1� mk �Β �Α �
k m
Α m � Β� 1
����
2 ��Α m �Β ���1� m �Β � Α m �
1
����
2 ��Α m �Β ���1� m �Β � Α m � ����1� m �Α m ���� Β
Example: An m-element and a 2-element
Putting k equal to 2 gives:
2001 4 26
1
����
2 ��Α m �Β 2
�Β�Α � �Α�Β 2 m m 2
�Α m ���� Β 2
12.57
12.58
12.59
12.60
12.61
12.62
12.63