Grassmann Algebra

ExploringGrassmannAlgebra.nb 37
? GrassmannFunctionForm
GrassmannFunctionForm��f�x�,x�,Xb,Xs� expresses the function f�x�
of a Grassmann number in a space of the currently declared
number of dimensions in terms of the symbols Xb and Xs, where
Xb represents the body, and Xs represents the soul of the
number. GrassmannFunctionForm��f�x,y,z,...�,�x,y,z,...��,�Xb,
Yb,Zb,...�,�Xs,Ys,Zs,...�� expresses the function f�x,y,z,...�
of Grassmann numbers x,y,z,... in terms of the symbols Xb,Yb,
Zb,... and Xs,Ys,Zs..., where Xb,Yb,Zb,... represent the
bodies, and Xs,Ys,Zs... represent the souls of the numbers.
GrassmannFunctionForm�f�Xb,Xs�� expresses the function f�X�
of a Grassmann number X�Xb�Xs in a space of the currently
declared number of dimensions in terms of body and soul
symbols Xb and Xs. GrassmannFunctionForm�f��Xb,Yb,Zb,...�,
�Xs,Ys,Zs,...��� is equivalent to the second form above.
GrassmannFunctionForm for a single variable
The form of a function will be different, depending on the dimension of the space it is in - the
higher the dimension, the potentially more terms there are in the series. We can generate a table
similar to the first of the pair above by using GrassmannFunctionForm.
Table��i, ��i; GrassmannFunctionForm�f�Xb, Xs����,
�i, 1, 8�� �� TableForm
1 f�Xb� � Xs f��Xb� 2 f�Xb� � Xs f��Xb� 3 f�Xb� � Xs f��Xb� � 1 ���� X2 2 s f���Xb� 4 f�Xb� � Xs f��Xb� � 1 ���� X2 2 s f���Xb� 5 f�Xb� � Xs f��Xb� � 1 ���� 2 X2 s f���Xb� � 1 ���� X3 6 s f�3��Xb� 6 f�Xb� � Xs f��Xb� � 1 ���� 2 X2 s f���Xb� � 1 ���� X3 6 s f�3��Xb� 7 f�Xb� � Xs f��Xb� � 1 ���� 2
2 �� Xs f �Xb� � 1 ���� 6
3 �3� Xs f �Xb� � 1 ������ 24 Xs 4 f�4��Xb 8 f�Xb� � Xs f��Xb� � 1 ���� 2
2 �� Xs f �Xb� � 1 ���� 6
3 �3� Xs f �Xb� � 1 ������ 24 Xs 4 f�4��Xb GrassmannFunctionForm for several variables
GrassmannFunctionForm will also give the form of the function for functions of several
Grassmann variables. Due to their length, we will have to display them individually rather than
in table form. We display only the odd dimensional results, since the following even
dimensional ones are the same.
2001 4 5
�1; GrassmannFunctionForm�f��Xb, Yb�, �Xs, Ys���
f�Xb, Yb�� Ys f �0,1� �Xb, Yb�� Xs f �1,0� �Xb, Yb�� Xs � Ys f �1,1� �Xb, Yb�