Grassmann Algebra

ExploringGrassmannAlgebra.nb 7
Grade�Z�
�0, 1, 2�
Because it may be necessary to expand out a Grassmann number into a sum of terms (each of
which has a definite grade) before the grades can be calculated, and because Mathematica will
reorganize the ordering of those terms in the sum according to its own internal algorithms, direct
correspondence with the original form is often lost. However, if a list of the grades of each term
in an expansion of a Grassmann number is required, we should:
¥ Expand and simplify the number into a sum of terms.
¥ Create a list of the terms.
¥ Map Grade over the list.
For example:
A � ��Z�
�4 x� x � y � 2 �x � z� � x � y � z � x � y � z � 1 � 2x� y
A � List �� A
��4 x,x� y, 2 �x � z�, x� y � z, ��x � y � z � 1�, �2 x� y�
Grade �� A
�1, 0, 0, 1, 0, 2�
To extract the components of a Grassmann number of a given grade or grades, we can use the
GrassmannAlgebra ExtractGrade[m][X] function which takes a Grassmann number and
extracts the components of grade m.
Extracting the components of grade 1 from the number Z defined above gives:
ExtractGrade�1��Z�
�4 x� x � y � z
ExtractGrade also works on lists or tensors of GrassmannNumbers.
DeclareExtraScalars��Ψ_ , Ζ_ ��
�a, b, c, d, e, f, g, h, �, �_ � _�? InnerProductQ, Ζ_, Ξ_, Ψ_, _� 0
��������
ExtractGrade�1��
�
2001 4 5
��������
�
e1 Ξ1 � e2 Ξ2
e1 Ψ1 � e2 Ψ2
e1 Ζ1 � e2 Ζ2
��������
�
Ξ0 � e1 Ξ1 � e2 Ξ2 �Ξ3 e1 � e2
Ψ0 � e1 Ψ1 � e2 Ψ2 �Ψ3 e1 � e2
Ζ0 � e1 Ζ1 � e2 Ζ2 �Ζ3 e1 � e2
��������
���MatrixForm
�