Grassmann Algebra

ExploringGrassmannAlgebra.nb 3
�3; Basis���
�1, e1, e2, e3, e1 � e2, e1 � e3, e2 � e3, e1 � e2 � e3�
If we wish to generate a general symbolic Grassmann number in the currently declared basis, we
can use the function GrassmannAlgebra function CreateGrassmannNumber.
CreateGrassmannNumber[symbol] creates a Grassmann number with scalar
coefficients formed by subscripting the symbol given. For example, if we enter
GrassmannNumber[Ξ], we obtain:
CreateGrassmannNumber�Ξ�
Ξ0 � e1 Ξ1 � e2 Ξ2 � e3 Ξ3 �Ξ4 e1 � e2 �
Ξ5 e1 � e3 �Ξ6 e2 � e3 �Ξ7 e1 � e2 � e3
The positional ordering of the Ξi with respect to the basis elements in any given term is due to
Mathematica's internal ordering routines, and does not affect the meaning of the result.
The CreateGrassmannNumber operation automatically adds the pattern for the generated
coefficients Ξi to the list of declared scalars, as we see by entering Scalars.
Scalars
�a, b, c, d, e, f, g, h, �, �_ � _�? InnerProductQ, Ξ_, _� 0
If we wish to enter our own coefficients, we can use CreateGrassmannNumber with a
placeholder (�, �pl�) as argument, and then tab through the placeholders generated,
replacing them with the desired coefficients:
CreateGrassmannNumber���
���e1 ��e2 ��e3 ��e1 � e2 ��e1 � e3 ��e2 � e3 ��e1 � e2 � e3
Note that the coefficients entered here will not be automatically put on the list of declared
scalars.
If several Grassmann numbers are required, we can apply the operation to a list of symbols.
That is, CreateGrassmannNumber is Listable. Below we generate three numbers in 2space.
�2; CreateGrassmannNumber��Ξ, Ψ, Ζ�� �� MatrixForm
� Ξ1 � e1 Ξ3 � e2 Ξ4 �Ξ2 e1 � e2
�������
��������
Ψ1 � e1 Ψ3 � e2 Ψ4 �Ψ2 e1 � e2
� Ζ1 � e1 Ζ3 � e2 Ζ4 �Ζ2 e1 � e2 �
Grassmann numbers can of course be entered in general symbolic form, not just using basis
elements. For example a Grassmann number might take the form:
2 � x � 3�y � 5x� y � x � y � z
In some cases it might be faster to change basis temporarily in order to generate a template with
all but the required scalars formed.
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