Grassmann Algebra

ExploringCliffordAlgebra.nb 21
����
�
����Α
�
� ��� �
�Β��� ���� Γ
�m
p�
k�
p
� ����Α
�
�m
�
�
���à p
�
� ��� k�
����
���� Γ
� p
12.23
The relations derived up to this point are completely general and apply to Clifford products in an
arbitrary space with an arbitrary metric. For example they do not require any of the elements to
be orthogonal.
Special cases of intersecting elements
We can derive special cases of the formulae derived above by putting Α and Β equal to unity.
m k
First, put Β equal to unity. Remembering that the Clifford product with a scalar reduces to the
k
ordinary product we obtain:
���Α
� Γ
†���
�Γ
� m p � p
� ��1� mp � �
�
��Α
�
��� ���� à � ��1�
m p�
p
mp � ���Α
�
� Γ�� ���� Γ
�m
p�
p
Next, put Α m equal to unity, and then, to enable comparison with the previous formulae, replace Β k
by Α m .
Γ p
† �
� ��Γ
� p
�
� Α�� �
m�
�
�
��à p
�
�Α�� ���� Γ
m�
p
� ���Γ
� p
�
� Α�� ���� Γ
m�
p
Finally we note that, since the far right hand sides of these two equations are equal, all the
expressions are equal.
����Α
� Γ
†����
�Γ
�m
p � p
�Γ p
† �
� ���Γ
�p
�
� Α��� m�
� ����Γ
�
�Α��� ���� Γ �
�p
m�
p
����Γ
�
� Α��� ���� Γ � ��1�
�p
m�
p
mp � ����Α
�
�Γ��� ���� Γ
�m
p�
p
12.24
By putting Α to unity in these equations we can recover the relation 12.11 between the Clifford
m
product of identical elements and their interior product.
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