Grassmann Algebra

ExploringCliffordAlgebra.nb 19
� 6. Add the terms and simplify the result by factoring out the scalars
U � FactorScalars�Plus �� T�
Β k
� Α4 � Α3 � Α2 � Α1 �Α1 ��Β� Α4 � Α3 � Α2 � �
k
Α2 ��Β� Α4 � Α3 � Α1 � �Α3 ��Β� Α4 � Α2 � Α1� �
k
k
Α4 ��Β k
Α1 � Α4 ��Β k
Α3 � Α4 ��Β k
� Α3 � Α2 � Α1 � �Α1 � Α2 ��Β� Α4 � Α3� �Α1 � Α3 ��Β� Α4 � Α2 � �
k
k
Α1 � Α3 � Α4 ��Β k
� Α3 � Α2 � �Α2 � Α3 ��Β� Α4 � Α1� �Α2 � Α4 ��Β� Α3 � Α1 � �
k
k
� Α2 � Α1 � �Α1 � Α2 � Α3 ��Β� Α4� �Α1 � Α2 � Α4 ��Β� Α3 � �
k
k
� 7. Compare to the original expression
X � U
True
� Α2 � �Α2 � Α3 � Α4 ��Β� Α1� �Α1 � Α2 � Α3 � Α4 � Β
k
k
12.7 The Clifford Product of Intersecting
Elements
General formulae for intersecting elements
Suppose two simple elements Γ � Α and Γ � Β which have a simple element Γ in common. Then
p m p k
p
by definition their Clifford product may be written as a sum of generalized products.
���Γ
� p
�
� Α�� �
m�
�
�
��à p
Min�m,k��p
�
� �� � �
k�
Λ�0
���1�
1
Λ��p�m�Λ� � ���� Λ �Λ�1� �
2 ��Γ
p
But it has been shown in Section 10.12 that for Λ ≥ p that:
���Γ
� p
�
� Α������� �
m�
Λ �
�
��à p
�
� �� � ��1�
k�
p���p� � ���
�
�
�
Substituting in the formula above gives:
���Γ
� p
�
� Α�� �
m�
�
�
��à p
Min�m,k��p
�
� �� � �
k�
Λ�0
��à p
��1�
�
� Α��� ����
m�
�p�Λ �Β k
���
���� Γ
� p
�
1
p�Λ��m�Λ� � ���� Λ �Λ�1� �
2 ��Γ
p
��
�
�
�
�
� Α������� �
m�
Λ �
�
��à p
�
� Α��� ����
m�
Λ�p �Β k
�
� �� k�
���
���� Γ
� p
Since the terms on the right hand side are zero for Λ < p, we can define Μ = Λ - p and rewrite the
right hand side as:
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