Grassmann Algebra

ExploringGrassmannAlgebra.nb 16
However, in the general case an even soul will not be nilpotent, but its powers will become zero
when the grades of the terms in the power expansion exceed the dimension of the space. We can
show this by calculating the powers of an even soul in spaces of successively higher
dimensions. Let
Zes � m � p � q � r � s � t � u � v � w � x � y � z;
Table���n; Dimension, ��Zes � Zes��, �n, 3, 8�� �� TableForm
3 0
4 0
5 0
6 2 m� p � q � r � s � t
7 2 m� p � q � r � s � t
8 2 m� p � q � r � s � t � 2m� p � u � v � w � x � y � z
� Powers of odd Grassmann numbers
Odd Grassmann numbers are nilpotent. Hence all positive integer powers except the first are
zero.
Xo � OddGrade�X�
e1 Ξ1 � e2 Ξ2 � e3 Ξ3 �Ξ7 e1 � e2 � e3
��Xo � Xo�
0
� Computing positive powers of Grassmann numbers
We can make use of the nilpotence of the odd component to compute powers of Grassmann
numbers more efficiently than the direct approach described at the beginning of this section by
noting that it will cause a binomial expansion of the sum of the even and odd components to
terminate after the first two terms. Let X � Xe � Xo , then:
X p � �Xe � Xo� p � Xe p � pXe p�1 � Xo � �
X p � Xe p�1 ��Xe � pXo�
where the powers of p and pÐ1 refer to the exterior powers of the Grassmann numbers. For
example, the cube of a general Grassmann number X in 3-space may then be calculated as
2001 4 5
�3; Xe � EvenGrade�X�; Xo � OddGrade�X�; ��Xe � Xe ��Xe � 3�Xo��
9.1
Ξ 0 3 � 3e1 Ξ 0 2 Ξ1 � 3e2 Ξ 0 2 Ξ2 � 3e3 Ξ 0 2 Ξ3 � 3 Ξ 0 2 Ξ4 e1 � e2 � 3 Ξ 0 2 Ξ5 e1 � e3 �
3 Ξ 0 2 Ξ6 e2 � e3 � �Ξ0 ��6 Ξ2 Ξ5 � 6 �Ξ3 Ξ4 �Ξ1 Ξ6�� � 3 Ξ 0 2 Ξ7� e1 � e2 � e3