Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 4
X � a � be1 � ce2 � de1 � e2
a � be 1 � ce 2 � de 1 � e 2
We declare the 2-space, and then create a 2�2 matrix of the form X.
�2; M� CreateMatrixForm��2, 2���X�; MatrixForm�M�
� a1,1 � e1 b1,1 � e2 c1,1 � d1,1 e1 � e2 a1,2 � e1 b1,2 � e2 c1,2 � d1,2 e1 � e2
�
a2,1 � e1 b2,1 � e2 c2,1 � d2,1 e1 � e2 a2,2 � e1 b2,2 � e2 c2,2 � d2,2 e1 � e2 We need to declare the extra scalars that we have formed. This can be done with just one
pattern (provided we wish all symbols with this pattern to be scalar).
DeclareExtraScalars��Subscript�_, _, _���
�a, b, c, d, e, f, g, h, �, �_ � _� ?InnerProductQ, __,_ ,_� 0
� Creating matrices with specific coefficients
To enter a Grassmann matrix with specific coefficients, enter a placeholder as the kernel
symbol. An output will be returned which will be copied to an input form as soon as you
attempt to edit it. You can select a placeholder and enter the value you want. The tab key
can be used to select the next placeholder. The steps are:
• Generate the form you want.
Y � CreateGrassmannNumber���
���e 1 ��e2 ��e1 � e2
CreateMatrixForm��2, 2���Y�
�����e1 ��e2 ��e1 � e2 , ���e1 ��e2 ��e1 � e2�,
����e 1 ��e2 ��e1 � e2 , ���e1 ��e2 ��e1 � e2��
• Start editing the form above.
��2 ��e1 ��e2 ��e1 � e2 , ���e1 ��e2 ��e1 � e2�,
����e1 ��e2 ��e1 � e2 , ���e1 ��e2 ��e1 � e2��
• Finish off your editing and give the matrix a name.
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