Grassmann Algebra

ExploringCliffordAlgebra.nb 30
���
�
���Α
�
� Ε � Θ � Γ�� �
� m s z p�
���Γ
�
� Θ � Β � ∆�� � p z k q�
���
� �∆ � Ε � Ω � Θ� � q s r z
� ���
�
���Θ
† † �
� Γp �� ����
� z �
���Θ
�
� �� � z p�
�����Α
� Ε � Β � ∆� � �∆ � Ε � Ω � Θ� � m s k q q s r z
� ���
�
���Θ
† † �
� Γp �� ����
� z �
���Θ
�
� �� � z p�
������1�
�
sk † †
���Ε � ∆q � ���� �Εs � ∆����Α � Β���Ω� Θ� s
q m k r z
� ��1� sk � � ��
�
���Θ
†
�
†�
Γp �� ����
� z �
���Θ
�
� �� �z
p�
������Ε
† †
� ∆q � ���� �Εs � ∆����Α � Β � Ω � Θ� � s
q m k r z
On the other hand we can associate the second pair to obtain the same result.
���Α
�
� Ε � Θ � Γ�� �
� m s z p�
���
�
���Γ
�
� Θ � Β � ∆�� � �∆ � Ε � Ω � Θ� � p z k q�
q s r z ���
�
� ���Α
�
� Ε � Θ � Γ�� � ��1�
�m
s z p�
zk�z��s�r� † †
���Θ � ∆q � ���� �Θz � ∆��� z
q � ��Γ
�
� �� ��� � � p k�
s r
� ��1� zk�z��s�r� † †
���Θ � ∆q � ���� �Θz � ∆��� z
q ���Α
�
� Ε � Θ � Γ�� �
�m
s z p�
���Γ
�
�  � Š� �� � p k s r�
� ��1� zk�z��s�r��sz�ks † †
���Θ � ∆q � ���� �Θz � ∆��� z
q ���Α
�
� Θ � Ε � Γ�� �
�m
z s p�
���Γ
�
� Š�  � �� � p s k r�
� ��1� zk�zr�ks † †
���Θ � ∆q � ���� �Θz � ∆��� z
q � ��
�
���Ε
†
�
†�
Γp �� ����
�s
�
���Ε
�
� �� � s p�
�����Α
� ���� � � m z k r
� ��1� sk � � ��
�
���Θ
†
�
†�
Γp �� ����
� z �
���Θ
�
� �� �z
p�
������Ε
† †
� ∆q � ���� �Εs � ∆����Α � Β � Ω � Θ� � s
q m k r z
In sum: We have shown that the Clifford product of possibly intersecting but otherwise totally
orthogonal elements is associative. Any factors which make up a given individual element are
not specifically involved, and hence it is of no consequence to the associativity of the element
with another whether or not these factors are mutually orthogonal.
A mnemonic formula for products of orthogonal elements
Because Θ z and Γ p
are orthogonal to each other, and Ε s and ∆ q are orthogonal to each other we can
rewrite the inner products in the alternative forms:
���Θ
† � †�
Γp �� ����
� z �
���Θ
�
� Γ�� † � �Θ ���� Θz ��
�z
p�
z
� ��Γ
� p
† ���� Γp
���
�
† † † †
�Ε � ∆q � ���� �Εs � ∆� � �Ε ���� Εs ���∆ ���� ∆q �
s
q s
q
The results of the previous section may then summarized as:
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