Grassmann Algebra

ExploringCliffordAlgebra.nb 5
� The grade of a Clifford product
It can be seen that, from its definition 12.1 in terms of generalized products, the Clifford product
of Α and Β is a sum of elements with grades ranging from m + k to | m - k | in steps of 2. All the
m k
elements of a given Clifford product are either of even grade (if m + k is even), or of odd grade
(if m + k is odd). To calculate the full range of grades in a space of unspecified dimension we
can use the GrassmannAlgebra function RawGrade. For example:
RawGrade�Α �Β� 3 2
�1, 3, 5�
Given a space of a certain dimension, some of these elements may necessarily be zero (and thus
result in a grade of Grade0), because their grade is larger than the dimension of the space. For
example, in a 3-space:
�3; Grade�Α �Β� 3 2
�1, 3, Grade0�
In the general discussion of the Clifford product that follows, we will assume that the dimension
of the space is high enough to avoid any terms of the product becoming zero because their grade
exceeds the dimension of the space. In later more specific examples, however, the dimension of
the space becomes an important factor in determining the structure of the particular Clifford
algebra under consideration.
� Clifford products in terms of generalized products
As can be seen from the definition, the Clifford product of a simple m-element and a simple kelement
can be expressed as the sum of signed generalized products.
For example, Α �Β may be expressed as the sum of three signed generalized products of grades
3 2
1, 3 and 5. In GrassmannAlgebra we can effect this conversion by applying the
ToGeneralizedProducts function.
ToGeneralizedProducts�Α �Β� 3 2
Α � Β �Α� Β �Α� Β
3 0 2 3 1 2 3 2 2
Multiple Clifford products can be expanded in the same way. For example:
2001 4 26
ToGeneralizedProducts�Α �Β�Γ� 3 2
�Α � Β� � Γ��Α � Β� � Γ���Α � Β� � Γ��Α � Β� � Γ��Α � Β� � Γ���Α � Β� � Γ
3 0 2 0 3 1 2 0 3 2 2 0 3 0 2 1 3 1 2 1 3 2 2 1