Grassmann Algebra

ExploringGrassmannAlgebra.nb 32
Division by a scalar
Division by a scalar is just the Grassmann number divided by the scalar as expected.
�X �a, X� a�
� Ξ0
�������
�������������
�
a e1 Ξ1
a
� e2 Ξ2
�������������
a
� Ξ3 e1 � e2
����������������������� ,
a
Ξ0
�������
Division of a Grassmann number by itself
Division of a Grassmann number by itself give unity.
�X �X, X� X�
�1, 1�
Division by scalar multiples
�������������
�
a e1 Ξ1
a
Division involving scalar multiples also gives the results expected.
��a X��X, X� �a X��
�a, 1
���� �
a
Division by an even Grassmann number
� e2 Ξ2
�������������
a
� Ξ3 e1 � e2
����������������������� �
a
Division by an even Grassmann number will give the same result for both division operations,
since exterior multiplication by an even Grassmann number is commutative.
�X ��1 � e1 � e2�, X��1 � e1 � e2��
�Ξ0 � e1 Ξ1 � e2 Ξ2 � ��Ξ0 �Ξ3� e1 � e2,
Ξ0 � e1 Ξ1 � e2 Ξ2 � ��Ξ0 �Ξ3� e1 � e2�
� The non-uniqueness of exterior division
Because of the nilpotency of the exterior product, the division of Grassmann numbers does not
necessarily lead to a unique result. To take a simple case consider the division of a simple 2element
x�y by one of its 1-element factors x.
�x � y� �x
y � xC1 � C2 x � y
The quotient is returned as a linear combination of elements containing arbitrary scalar
constants C1 and C2 . To see that this is indeed a left quotient, we can multiply it by the
denominator from the left and see that we get the numerator.
2001 4 5
��x ��y � xC1 � C2 x � y��
x � y