Grassmann Algebra

ExploringGrassmannAlgebra.nb 36
9.9 Functions of Grassmann Numbers
The Taylor series formula
Functions of Grassmann numbers can be defined by power series as with ordinary numbers. Let
X be a Grassmann number composed of a body Ξ0 and a soul Xs such that X �Ξ0 � Xs . Then
the Taylor series expanded about the body Ξ0 (and truncated at the third term) is:
Normal�Series�f�X�, �X, Ξ0, 3���
f�Ξ0� � �X �Ξ0� f��Ξ0� � 1
���� �X �Ξ0�
2 2 f��Ξ0� � 1
���� �X �Ξ0�
6 3 f�3��Ξ0� It has been shown in Section 9.5 that the degree pmax of the highest non-zero power of a
Grassmann number in a space of n dimensions is given by pmax � 1 ���� 4 �2 n� 1 � ��1� n �.
If we expand the series up to the term in this power, it then becomes a formula for a function f
of a Grassmann number in that space. The tables below give explicit expressions for a function
f[X] of a Grassmann variable in spaces of dimensions 1 to 4.
n pmax f�X�
1 1 f�Ξ0��f��Ξ0�Xs 2 1 f�Ξ0��f��Ξ0�Xs 3 2 f�Ξ0� � f � �Ξ0� Xs � 1
����
2 f � �Ξ0��Xs 2
4 2 f�Ξ0� � f � �Ξ0� Xs � 1
����
2 f � �Ξ0��Xs 2
If we replace the square of the soul of X by its simpler formulation in terms of the components
of X of the first and second grade we get a computationally more convenient expression in the
case on dimensions 3 and 4.
n pmax f�X�
1 1 f�Ξ0��f��Ξ0�Xs 2 1 f�Ξ0��f��Ξ0�Xs 3 2 f�Ξ0��f��Ξ0�Xs� f��Ξ0� Σ � Σ
2
4 2 f�Ξ0� � f � �Ξ0� Xs � 1
����
2 f � �Ξ0���2 Σ�Σ 2 ��Σ 2
� The form of a function of a Grassmann number
To automatically generate the form of formula for a function of a Grassmann number, we can
use the GrassmannAlgebra function GrassmannFunctionForm.
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