Grassmann Algebra

ExploringGrassmannAlgebra.nb 5
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Ξ0 � e1 Ξ1 � e2 Ξ2 �Ξ3 e1 � e2
Ψ0 � e1 Ψ1 � e2 Ψ2 �Ψ3 e1 � e2
Ζ0 � e1 Ζ1 � e2 Ζ2 �Ζ3 e1 � e2
e1 Ξ1 � e2 Ξ2 �Ξ3 e1 � e2
e1 Ψ1 � e2 Ψ2 �Ψ3 e1 � e2
e1 Ζ1 � e2 Ζ2 �Ζ3 e1 � e2
� Even and odd components
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Changing the order of the factors in an exterior product changes its sign according to the axiom:
Α � Β � ��1�
m k
mk �Β
k
� Α m
Thus factors of odd grade anti-commute; whereas, if one of the factors is of even grade, they
commute. This means that, if X is a Grassmann number whose components are all of odd grade,
then it is nilpotent.
The even components of a Grassmann number can be extracted with the function EvenGrade
and the odd components with OddGrade.
X
Ξ0 � e1 Ξ1 � e2 Ξ2 � e3 Ξ3 �Ξ4 e1 � e2 �
Ξ5 e1 � e3 �Ξ6 e2 � e3 �Ξ7 e1 � e2 � e3
EvenGrade�X�
Ξ0 �Ξ4 e1 � e2 �Ξ5 e1 � e3 �Ξ6 e2 � e3
OddGrade�X�
e1 Ξ1 � e2 Ξ2 � e3 Ξ3 �Ξ7 e1 � e2 � e3
It should be noted that the result returned by EvenGrade or OddGrade is dependent on the
dimension of the current space. For example, the calculations above for the even and odd
components of the number X were carried out in a 3-space. If now we change to a 2-space and
repeat the calculations, we will get a different result for OddGrade[X] because the term of
grade three is necessarily zero in this 2-space.
�2; OddGrade�X�
e1 Ξ1 � e2 Ξ2 � e3 Ξ3
For EvenGrade and OddGrade to apply, it is not necessary that the Grassmann number be in
terms of basis elements or simple exterior products. Applying EvenGrade and OddGrade to
the number Z below still separates the number into its even and odd components:
2001 4 5
Z � �x ���� y� � ��x ��y � 2�� ���� �z � 2�� � �z ��y � x��;
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