Grassmann Algebra

ExpTheGeneralizedProduct.nb 26
Similarly the triple generalized sums of form B for grades of 0, 1 and 2 are
Table��i, TripleGeneralizedSumB�i��x,y,z��, �i, 0, 2�� ��
m k p
TableForm
0 x �
m 0 ���y
� z
���
�k
0 p�
1�k
1 ��1�
���x
�
�m
0 � ��y � z
���
�k
1 p�
���
1�m
� ��1�
���x
�
�
�m
1 ���y
� z
���
�k
0 p�
���
�
2 � ���x
�
�m
0 ���y
� z
���
�k
2 p�
���
2�k�m
� ��1�
���x
�
�
�m
1 ���y
� z
���
�k
1 p�
���
� x �
� m 2 ���y
� z
���
�k
0 p�
The triple generalized sum conjecture
As we have shown above, a single triple generalized product is not in general associative. That
is, the A form and the B form are not in general equal. However we conjecture that the triple
generalized sum of form A is equal to the triple generalized sum of form B.
n
�
Λ�0
���1� sA � �
�
��x m ����� Λ �y k
n
�z � �
n�Λ p
Λ�0
����
����
�
���1�sB �xm ����� �
Λ ���y�
����
�k
n�Λ �z p
���
�
10.27
where sA � m Λ� 1 ���� Λ �Λ�1� � �m � k� �n �Λ�� 2 1 ���� �n �Λ� �n �Λ�1�
2
and sB � m Λ� 1 ���� Λ �Λ�1� � k �n �Λ�� 2 1 ���� �n �Λ� �n �Λ�1�.
2
If this conjecture can be shown to be true, then the associativity of the Clifford product of a
general Grassmann expression can be straightforwardly proven using the definition of the
Clifford product in terms of exterior and interior products (or, what is equivalent, in terms of
generalized products). That is, the rules of Clifford algebra may be entirely determined by the
axioms and theorems of the Grassmann algebra, making it unnecessary, and indeed potentially
inconsistent, to introduce any special Clifford algebra axioms.
� Exploring the triple generalized sum conjecture
We can explore the triple generalized sum conjecture by converting the generalized products
into scalar products.
For example first we generate the triple generalized sums.
2001 4 26
A � TripleGeneralizedSumA�2��x,y,z� 3 2
���x � y� � z� � �x � y� � z � �x � y� � z
3 2 2 0 3 1 2 1 3 0 2 2