Grassmann Algebra

ExploringCliffordAlgebra.nb 10
By rewriting equation 12.7 as:
��1� m �Α m � x � x � Α m �Α m ���� x
Α m � x �Α m � x � ��1� m �Α m ���� x
12.7
we can add and subtract equations 12.6 and 12.7 to express the exterior product and interior
products of a 1-element and an m-element in terms of Clifford products
x � Α m � 1
����
2 ��x �Α m � ��1� m �Α m � x�
Α m ���� x � 1
����
2 ��x �Α m � ��1� m �Α m � x�
The Clifford product of an m-element and a 2-element
12.8
12.9
The Clifford product of an arbitrary m-element and a 2-element is given by just three terms, an
exterior product, a generalized product of order 1, and an interior product.
Α m �Β 2
�Α�Β � ��1�
m 2
m �Α����� �Β
m 1 2
� The Clifford product of two 2-elements
�Α m ���� Β 2
12.10
By way of example we explore the various forms into which we can cast a Clifford product of
two 2-elements. The highest level is into a sum of generalized products. From this we can
expand the terms into interior, inner or scalar products where appropriate.
We expect the grade of the Clifford product to be a composite one.
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RawGrade��x � y� � �u � v��
�0, 2, 4�
ToGeneralizedProducts��x � y� � �u � v��
x � y � 0 u � v � x � y � 1 u � v � x � y � 2 u � v