Grassmann Algebra

ExploringGrassmannAlgebra.nb 29
necessarily to be expressed in terms of basis elements. For example, we can take a quadratic
equation in Α:
Q �Α� Α�2��1 � x � x � y��Α��1 � 2�x � 2�x � y� � 0;
S � GrassmannSolve�Q, Α�
��Α ��1 � xC1 � C2 x � y��
Here there are two arbitrary parameters C1 and C2 introduced by the GrassmannSolve
function. (C is a symbol reserved by Mathematica for the generation of unknown coefficients in
the solution of equations.)
Now test whether Α is indeed a solution to the equation:
��Q �. S�
�True�
Note that here we have a double infinity of solutions, parametrized by the pair {C1 ,C2 }. This
means that for any values of C1 and C2 , S is a solution. For example for various C1 and C2 :
���Q �. Α��1 � x�, ��Q �. Α��1�, ��Q �. Α��1 � 9�x � y��
�True, True, True�
GrassmannSolve can also take several equations in several unknowns. As an example we
take the following pair of equations:
Q � ��1 � x��Β�x � y � Α�3 � 2x� y,
�1 � y��Α � Α��2 � y � x � 5�x � y� � 7 � 5�x�;
First, we solve for the two unknown Grassmann numbers Α and Β.
S � GrassmannSolve�Q, �Α, Β��
��Α �� �����
5 � 3x
��������� ����� �
5 2y x � y
��������� ����� � �������������
5 2 �����
�����
, Β�3 � 3x� �2 � 5 � x � y�,
5
�Α � �����
5 � 3x
��������� ����� �
5 2y x � y
��������� ����� � �������������
5 2 �����
�����
, �3 � 3x� �2 � 5 � x � y��
5
Here we get two solutions, which we verify with GrassmannSimplify.
��Q �. S�
��True, True�, �True, True��
Note that GrassmannAlgebra assumes that all other symbols (here, x and y) are 1-elements. We
can add the symbol x to the list of Grassmann numbers to be solved for.
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