Grassmann Algebra

ExploringCliffordAlgebra.nb 11
The first of these generalized products is equivalent to an exterior product of grade 4. The
second is a generalized product of grade 2. The third reduces to an interior product of grade 0.
We can see this more explicitly by converting to interior products.
ToInteriorProducts��x � y� � �u � v��
��u � v � x � y� � �x � y � u��v � �x � y � v��u � u � v � x � y
We can convert the middle terms to inner (in this case scalar) products.
ToInnerProducts��x � y� � �u � v��
��u � v � x � y� � �v � y� u � x �
�v � x� u � y � �u � y� v � x � �u � x� v � y � u � v � x � y
Finally we can express the Clifford number in terms only of exterior and scalar products.
ToScalarProducts��x � y� � �u � v��
�u � y� �v � x� � �u � x� �v � y� � �v � y� u � x �
�v � x� u � y � �u � y� v � x � �u � x� v � y � u � v � x � y
The Clifford product of two identical elements
The Clifford product of two identical elements Γ is, by definition
p
Γ �Γ
p p
p
� �
Λ�0
���1�
1
Λ��p�Λ�� ���� Λ �Λ�1� �
2 ��Γ ����� �Γ
p Λ p
Since the only non-zero generalized product of the form Γ����� �Γ
p Λ p
Γ p
���� Γ, we have immediately that:
p
Γ �Γ
p p
Or, alternatively:
2001 4 26
� ��1� 1
����
2 p �p�1� �Γp
���� à p
Γ ���� Γ �Γ
p p p
�
† �Γp
���
�
��à p p
†
is that for which Λ = p, that is
12.11