Grassmann Algebra

ExpTheGeneralizedProduct.nb 24
� Scalars may be factored out of generalized products
�a Α m ������ Λ �Β k
� The generalized product is quasi-commutative
Α m ����� Λ �Β k
This relationship was proven in Section 10.3.
� Α����� ��a Β� � a�� Α����� �Β� m Λ k
m Λ k
� ��1��m�Λ���k�Λ��Β����� �Α
k Λ m
10.10 The Triple Generalized Sum Conjecture
10.25
10.26
In this section we discuss an interesting conjecture associated with our definition of the Clifford
product in Chapter 12. As already mentioned, we have defined the generalized Grassmann
product to facilitate the definition of the Clifford product of general Grassmann expressions. As
is well known, the Clifford product is associative. But in general, a Clifford product will involve
both exterior and interior products. And the interior product is not associative. In this section we
look at a conjecture, which, if established, will prove the validity of our definition of the
Clifford product in terms of generalized Grassmann products.
� The generalized product is not associative
We take a simple example to show that the generalized product is not associative. First we set
up the assertion:
H � �x����� 0 �y������ 1 �z � x����� 0 ��y����� 1 �z�;
Then we convert the generalized products to scalar products.
ToScalarProducts�H�
y �x � z� � x �y � z� � x �y � z�
Clearly the two sides of the relation are not in general equal. Hence the generalized product is
not in general associative.
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