Grassmann Algebra

ExploringGrassmannAlgebra.nb 8
� Working with complex scalars
The imaginary unit � is treated by Mathematica and the GrassmannAlgebra package just as if it
were a numeric quantity. This means that just like other numeric quantities, it will not appear in
the list of declared scalars, even if explicitly entered.
DeclareScalars��a, b, c, d, 2, �, ���
�a, b, c, d�
However, � or any other numeric quantity is treated as a scalar.
ScalarQ��a, b, c, 2, �, �, 2� 3����
�True, True, True, True, True, True, True�
This feature allows the GrassmannAlgebra package to deal with complex numbers just as it
would any other scalars.
����a ��b��x���� y��
�� a � b� x � y
9.3 Operations with Grassmann Numbers
� The exterior product of Grassmann numbers
We have already shown how to create a general Grassmann number in 3-space:
X � CreateGrassmannNumber�Ξ�
Ξ0 � e1 Ξ1 � e2 Ξ2 � e3 Ξ3 �Ξ4 e1 � e2 �
Ξ5 e1 � e3 �Ξ6 e2 � e3 �Ξ7 e1 � e2 � e3
We now create a second Grassmann number Y so that we can look at various operations applied
to two Grassmann numbers in 3-space.
Y � CreateGrassmannNumber�Ψ�
Ψ0 � e1 Ψ1 � e2 Ψ2 � e3 Ψ3 �Ψ4 e1 � e2 �
Ψ5 e1 � e3 �Ψ6 e2 � e3 �Ψ7 e1 � e2 � e3
The exterior product of two general Grassmann numbers in 3-space is obtained by multiplying
out the numbers termwise and simplifying the result.
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