Grassmann Algebra

ExploringGrassmannAlgebra.nb 34
? GrassmannFactor
GrassmannFactor�X,F� attempts to factorize the Grassmann number X
into the form given by F. F must be an expression involving the
symbols of X, together with sufficient scalar coefficients whose
values are to be determined. GrassmannFactor�X,F,S� attempts to
factorize the Grassmann number X by determining the scalars S.
If the scalars S are not already declared as scalars, they will
be automatically declared. If no factorization can be effected
in the form given, the original expression will be returned.
� Example: Factorizing a Grassmann number in 2-space
First, consider a general Grassmann number in 2-space.
�2; X� CreateGrassmannNumber�Ξ�
Ξ0 � e1 Ξ1 � e2 Ξ2 �Ξ3 e1 � e2
By using GrassmannFactor we find a general factorization into two Grassmann numbers of
lower maximum grade. We want to see if there is a factorization in the form
�a � be1 � ce2���f � ge1 � he2�, so we enter this as the second argument. The scalars
that we want to find are {a,b,c,f,g,h}, so we enter this list as the third argument.
Xf � GrassmannFactor�X,
�a � be1 � ce2���f � ge1 � he2�, �a, b, c, f, g, h��
� Ξ0
������� �
f e2 ��h Ξ0 � f Ξ2�
�������������������������������� ��������
f2 � e1 ��h Ξ0 Ξ1 � f Ξ1 Ξ2 � f Ξ0 Ξ3�
�������������������������������� �������������������������������� ���������
f2 ��
Ξ2
�f � he2 � e1 �h Ξ1 � f Ξ3�
������������������������������������� �
Ξ2
This factorization is valid for any values of the scalars {a,b,c,f,g,h} which appear in the
result, as we can verify by applying GrassmannSimplify to the result.
��Xf�
Ξ0 � e1 Ξ1 � e2 Ξ2 �Ξ3 e1 � e2
� Example: Factorizing a 2-element in 3-space
Next, consider a general 2-element in 3-space. We know that 2-elements in 3-space are simple,
hence we would expect to be able to obtain a factorization.
�3; Y� CreateElement�Ψ� 2
Ψ1 e1 � e2 �Ψ2 e1 � e3 �Ψ3 e2 � e3
By using GrassmannFactor we can determine a factorization. In this example we obtain
two classes of solution, each with different parameters.
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