Grassmann Algebra

ExploringGrassmannAlgebra.nb 20
� The inverse of a Grassmann number
Let X be a general Grassmann number in 3-space.
�3; X� CreateGrassmannNumber�Ξ�
Ξ0 � e1 Ξ1 � e2 Ξ2 � e3 Ξ3 �Ξ4 e1 � e2 �
Ξ5 e1 � e3 �Ξ6 e2 � e3 �Ξ7 e1 � e2 � e3
Finding an inverse of X is equivalent to finding the Grassmann number A such that X�A�1 or
A�X�1. In what follows we shall show that A commutes with X and therefore is a unique
inverse which we may call the inverse of X. To find A we need to solve the equation X�A-1�0.
First we create a general Grassmann number for A.
A � CreateGrassmannNumber�Α�
Α0 � e1 Α1 � e2 Α2 � e3 Α3 �Α4 e1 � e2 �
Α5 e1 � e3 �Α6 e2 � e3 �Α7 e1 � e2 � e3
The calculate and simplify the expression X�A-1.
��X � A � 1�
�1 �Α0 Ξ0 � e1 �Α1 Ξ0 �Α0 Ξ1� � e2 �Α2 Ξ0 �Α0 Ξ2� �
e3 �Α3 Ξ0 �Α0 Ξ3� � �Α4 Ξ0 �Α2 Ξ1 �Α1 Ξ2 �Α0 Ξ4� e1 � e2 �
�Α5 Ξ0 �Α3 Ξ1 �Α1 Ξ3 �Α0 Ξ5� e1 � e3 �
�Α6 Ξ0 �Α3 Ξ2 �Α2 Ξ3 �Α0 Ξ6� e2 � e3 �
�Α7 Ξ0 �Α6 Ξ1 �Α5 Ξ2 �Α4 Ξ3 �Α3 Ξ4 �Α2 Ξ5 �Α1 Ξ6 �Α0 Ξ7� e1 � e2 � e3
To make this expression zero we need to solve for the Αi in terms of the Ξi which make the
coefficients zero. This is most easily done with the GrassmannSolve function to be
introduced later. Here, to see more clearly the process, we do it directly with Mathematica's
Solve function.
S � Solve���1 �Α0 Ξ0 � 0, �Α1 Ξ0 �Α0 Ξ1� � 0, �Α2 Ξ0 �Α0 Ξ2� � 0,
�Α3 Ξ0 �Α0 Ξ3� � 0, �Α4 Ξ0 �Α2 Ξ1 �Α1 Ξ2 �Α0 Ξ4� � 0,
�Α5 Ξ0 �Α3 Ξ1 �Α1 Ξ3 �Α0 Ξ5� � 0,
�Α6 Ξ0 �Α3 Ξ2 �Α2 Ξ3 �Α0 Ξ6� � 0,
�Α7 Ξ0 �Α6 Ξ1 �Α5 Ξ2 �Α4 Ξ3 �Α3 Ξ4 �Α2 Ξ5 �Α1 Ξ6 �Α0 Ξ7� �
0�� �� Flatten
�Α7 � 2 Ξ3 Ξ4 � 2 Ξ2 Ξ5 � 2 Ξ1 Ξ6 �Ξ0 Ξ7
�������������������������������� �������������������������������� �����������
Α6 �� Ξ6
�������
Ξ 0 3
, Α4 �� Ξ4
�������
Ξ2 0 , Α1 �� Ξ1
�������
Ξ2 0 , Α2 �� Ξ2
�������
Ξ2 0 , Α3 �� Ξ3
������� 2 Ξ0 ,
Ξ 0 2 , Α5 �� Ξ5
�������
Ξ2 0 , Α0 � 1
������� �
We denote the inverse of X by Xr . We obtain an explicit expression for Xr by substituting the
values obtained above in the formula for A.
2001 4 5
Ξ0