Grassmann Algebra

ExploringGrassmannAlgebra.nb 28
? GrassmannScalarSolve
GrassmannScalarSolve�eqns� attempts to find the values of those �
declared� scalar variables which make the equations True.
GrassmannScalarSolve�eqns,scalars� attempts to find the
values of the scalars which make the equations True. If
not already in the DeclaredScalars list the scalars will
be added to it. GrassmannScalarSolve�eqns,scalars,elims�
attempts to find the values of the scalars which make the
equations True while eliminating the scalars elims. If
the equations are not fully solvable, GrassmannScalarSolve
will still find the values of the scalar variables which
reduce the number of terms in the equations as much as
possible, and will additionally return the reduced equations.
� Solving for an unknown Grassmann number
The function for solving for an unknown Grassmann number, or several unknown Grassmann
numbers is GrassmannSolve.
? GrassmannSolve
GrassmannSolve�eqns,vars� attempts to solve an equation or set
of equations for the Grassmann number variables vars. If
the equations are not fully solvable, GrassmannSolve will
still find the values of the Grassmann number variables
which reduce the number of terms in the equations as much as
possible, and will additionally return the reduced equations.
Suppose first that we have an equation involving an unknown Grassmann number, A, say. For
example if X is a general number in 3-space, and we want to find its inverse A as we did in the
Section 9.5 above, we can solve the following equation for A:
�3; X� CreateGrassmannNumber�Ξ�
Ξ0 � e1 Ξ1 � e2 Ξ2 � e3 Ξ3 �Ξ4 e1 � e2 �
Ξ5 e1 � e3 �Ξ6 e2 � e3 �Ξ7 e1 � e2 � e3
R � GrassmannSolve�X � A � 1, A�
��A � 1
�������
Ξ0
� e1 Ξ1 ������������� 2 Ξ0 Ξ6 e2 � e3
�����������������������
Ξ 0 2
As usual, we verify the result.
��X � A � 1 �. R�
�True�
� e2 Ξ2 ������������� 2 Ξ0 � e3 Ξ3 ������������� 2 Ξ0 � Ξ4 e1 � e2 ����������������������� 2 Ξ0 � Ξ5 e1 � e3 ����������������������� 2 Ξ0 � ��2 Ξ3 Ξ4 � 2 Ξ2 Ξ5 � 2 Ξ1 Ξ6 �Ξ0 Ξ7� e1 � e2 � e3
�������������������������������� �������������������������������� �������������������������������� ������������� ��
In general, GrassmannSolve can solve the same sorts of equations that Mathematica's
inbuilt Solve routines can deal with because it uses Solve as its main calculation engine. In
particular, it can handle powers of the unknowns. Further, it does not require the equations
2001 4 5
Ξ 0 3
�