Grassmann Algebra

ExploringCliffordAlgebra.nb 28
� Α, Β, and Γ
m k p
����Γ
�p
† ���� Γp
are totally orthogonal
����Α
�
� ��� �
�m
p�
�
�
���à p
�
� ��� �
k�
������Α
� Β� Αi ���� Βj �Αi���� Γs �Βj���� Γs � 0
� m k
Calculating with Clifford products
12.45
The previous section summarizes some alternative formulations for a Clifford product when its
factors contain a common element, and when they have an orthogonality relationship between
them. In this section we discuss how these relations can make it easy to calculate with these type
of Clifford products.
� The Clifford product of totally orthogonal elements reduces to the
exterior product
If we know that all the factors of a Clifford product are totally orthogonal, then we can
interchange the Clifford product and the exterior product at will. Hence, for totally orthogonal
elements, the Clifford and exterior products are associative, and we do not need to include
parentheses.
Α m �Β k
�Γ p
�Α m �Β k
� Γ �Α� Β �Γ�Α�Β � Γ
p m k p m k p
Αi ���� Βj �Αi ���� Γs �Βj ���� Γs � 0
12.46
Note carefully however, that this associativity does not extend to the factors of the m-, k-, or pelements
unless the factors of the m-, k-, or p-element concerned are mutually orthogonal. In
which case we could for example then write:
Example
Α1 �Α2 � � �Αm �
Α1 � Α2 � � � Αm Αi ���� Αj � 0 i � j
12.47
For example if x�y and z are totally orthogonal, that is x����z � 0 and y����z � 0, then we can write
2001 4 26
�x � y� � z � x � y � z � x ��y � z� ��y ��x � z�