Grassmann Algebra

ExploringCliffordAlgebra.nb 15
In order to get this into a form for direct comparison of our previously derived result in formula
12.12, we write ��1� Λ��m�Λ� �Α �Α
m 1 � Α1
m�Λ Λ �Α2 � Α2 � � , then interchange Λ and (m-Λ) as before
m�Λ Λ
to get finally:
Α m �Β k
Min�m,k�
� �
Λ�0
Α m �Α 1
Λ
� m
Λ �
��
i�1
��1� Λ��m�Λ� ��Α i
Λ � Β� ���� Α
k
i †
m�Λ
� Α1
m�Λ �Α2�
Α2 � �
Λ m�Λ
As before, the right hand side of this expression is a sum of interior products. In
GrassmannAlgebra we can develop the Clifford product Α �Β in this form by using
m k
ToInteriorProductsC. This is done in the Section 12.6 below.
12.13
The Clifford product expressed by decomposition of the second
factor
If we wish to expand a Clifford product in terms of the second factor Β, we can use formulas A
k
and B of the generalized product theorem (Section 10.5) and substitute directly to get either of:
2001 4 26
Α m �Β k
Α m �Β k
Min�m,k�
� �
Β k
Λ�0
� Β 1
Λ
Min�m,k�
� �
Β k
Λ�0
� Β 1
Λ
� k
Λ �
�� ��1� Λ��m�Λ� ��Α ���� Β
m j
j�1
� Β 1
k�Λ
� k
Λ �
� Β 2
Λ
� Β 2
� �
k�Λ
† j
��Β
Λ k�Λ
�� ��1� Λ��m�Λ� ��Α � Β
m j
� ����  j
j�1
� Β 1
k�Λ
� Β 2
Λ
k�Λ
� Β 2
� �
k�Λ
Λ
†
12.14
12.15