Grassmann Algebra

ExploringGrassmannAlgebra.nb 38
�3; GrassmannFunctionForm�f��Xb, Yb�, �Xs, Ys���
f�Xb, Yb�� Ys f �0,1� �Xb, Yb�� 1 2 �0,2�
���� Ys f �Xb, Yb�� 2
Xs f�1,0��Xb, Yb�� Xs � Ys f�1,1��Xb, Yb�� 1 2 �1,2�
���� Xs � Ys f �Xb, Yb�� 2 1 2 �2,0�
���� Xs f �Xb, Yb�� 2
1 2 ���� Xs � Ys f
2 �2,1� �Xb, Yb�� 1 2 2 �2,2� ���� Xs � Ys f �Xb, Yb�
4
�5; GrassmannFunctionForm�f��Xb, Yb�, �Xs, Ys���
f�Xb, Yb� � Ys f �0,1� �Xb, Yb� � 1
����
2 Y s 2 f �0,2� �Xb, Yb� �
1
����
6 Y s 3 f �0,3� �Xb, Yb� � Xs f �1,0� �Xb, Yb� � Xs � Ys f �1,1� �Xb, Yb� �
1 2 �1,2�
���� Xs � Ys f �Xb, Yb�� 2 1 3 �1,3�
���� Xs � Ys f �Xb, Yb�� 6
1 2 �2,0�
���� Xs f �Xb, Yb�� 2 1 2
���� Xs � Ys f
2 �2,1��Xb, Yb�� 1 2 2 �2,2�
���� Xs � Ys f �Xb, Yb�� 4 1
�������
12 Xs 2 3 �2,3�
� Ys f �Xb, Yb�� 1 3 �3,0�
���� Xs f �Xb, Yb�� 6 1 3
���� Xs � Ys f
6 �3,1� �Xb, Yb�� 1
�������
12 Xs 3 2 �3,2� � Ys f �Xb, Yb�� 1
�������
36 Xs 3 3 �3,3� � Ys f �Xb, Yb�
� Calculating functions of Grassmann numbers
To calculate functions of Grassmann numbers, GrassmannAlgebra uses the
GrassmannFunction operation based on GrassmannFunctionForm.
? GrassmannFunction
GrassmannFunction��f�x�,x�,X� or GrassmannFunction�f�X��
computes the function f of a single Grassmann number X in
a space of the currently declared number of dimensions.
In the first form f�x� may be any formula; in the second,
f must be a pure function. GrassmannFunction��f�x,y,z,..�,
�x,y,z,..��,�X,Y,Z,..�� or GrassmannFunction�f�X,Y,Z,..��
computes the function f of Grassmann numbers X,Y,Z,...
In the first form f�x,y,z,..� may be any formula �with
xi corresponding to Xi�; in the second, f must be a pure
function. In both these forms the result of the function
may be dependent on the ordering of the arguments X,Y,Z,...
In the sections below we explore the GrassmannFunction operation and give some
examples. We will work in a 3-space with the general Grassmann number X.
2001 4 5
�3; X� CreateGrassmannNumber�Ξ�
Ξ0 � e1 Ξ1 � e2 Ξ2 � e3 Ξ3 �Ξ4 e1 � e2 �
Ξ5 e1 � e3 �Ξ6 e2 � e3 �Ξ7 e1 � e2 � e3