Grassmann Algebra

ExploringGrassmannAlgebra.nb 9
��X � Y�
Ξ0 Ψ0 � e1 �Ξ1 Ψ0 �Ξ0 Ψ1� � e2 �Ξ2 Ψ0 �Ξ0 Ψ2� �
e3 �Ξ3 Ψ0 �Ξ0 Ψ3� � �Ξ4 Ψ0 �Ξ2 Ψ1 �Ξ1 Ψ2 �Ξ0 Ψ4� e1 � e2 �
�Ξ5 Ψ0 �Ξ3 Ψ1 �Ξ1 Ψ3 �Ξ0 Ψ5� e1 � e3 �
�Ξ6 Ψ0 �Ξ3 Ψ2 �Ξ2 Ψ3 �Ξ0 Ψ6� e2 � e3 �
�Ξ7 Ψ0 �Ξ6 Ψ1 �Ξ5 Ψ2 �Ξ4 Ψ3 �Ξ3 Ψ4 �Ξ2 Ψ5 �Ξ1 Ψ6 �Ξ0 Ψ7� e1 � e2 � e3
When the bodies of the numbers are zero we get only components of grades two and three.
��Soul�X��Soul�Y��
��Ξ2 Ψ1 �Ξ1 Ψ2� e1 � e2 �
��Ξ3 Ψ1 �Ξ1 Ψ3� e1 � e3 � ��Ξ3 Ψ2 �Ξ2 Ψ3� e2 � e3 �
�Ξ6 Ψ1 �Ξ5 Ψ2 �Ξ4 Ψ3 �Ξ3 Ψ4 �Ξ2 Ψ5 �Ξ1 Ψ6� e1 � e2 � e3
� The regressive product of Grassmann numbers
The regressive product of X and Y is again obtained by multiplying out the numbers termwise
and simplifying.
Z � ��X � Y�
e1 � e2 � e3 ��Ξ7 Ψ0 �Ξ6 Ψ1 �Ξ5 Ψ2 �Ξ4 Ψ3 �Ξ3 Ψ4 �Ξ2 Ψ5 �Ξ1 Ψ6 �Ξ0 Ψ7 �
e1 �Ξ7 Ψ1 �Ξ5 Ψ4 �Ξ4 Ψ5 �Ξ1 Ψ7� � e2 �Ξ7 Ψ2 �Ξ6 Ψ4 �Ξ4 Ψ6 �Ξ2 Ψ7� �
e3 �Ξ7 Ψ3 �Ξ6 Ψ5 �Ξ5 Ψ6 �Ξ3 Ψ7� � �Ξ7 Ψ4 �Ξ4 Ψ7� e1 � e2 �
�Ξ7 Ψ5 �Ξ5 Ψ7� e1 � e3 � �Ξ7 Ψ6 �Ξ6 Ψ7� e2 � e3 �Ξ7 Ψ7 e1 � e2 � e3�
We cannot obtain the result in the form of a pure exterior product without relating the unit nelement
to the basis n-element. In general these two elements may be related by a scalar
multiple which we have denoted �. In 3-space this relationship is given by
1 � � e1 � e2 � e3 . To obtain the result as an exterior product, but one that will also involve
3
the scalar �, we use the GrassmannAlgebra function ToCongruenceForm.
Z1 � ToCongruenceForm�Z�
1
�����
� �Ξ7 Ψ0 �Ξ6 Ψ1 �Ξ5 Ψ2 �Ξ4 Ψ3 �Ξ3 Ψ4 �Ξ2 Ψ5 �Ξ1 Ψ6 �Ξ0 Ψ7 �
e1 �Ξ7 Ψ1 �Ξ5 Ψ4 �Ξ4 Ψ5 �Ξ1 Ψ7� � e2 �Ξ7 Ψ2 �Ξ6 Ψ4 �Ξ4 Ψ6 �Ξ2 Ψ7� �
e3 �Ξ7 Ψ3 �Ξ6 Ψ5 �Ξ5 Ψ6 �Ξ3 Ψ7� � �Ξ7 Ψ4 �Ξ4 Ψ7� e1 � e2 �
�Ξ7 Ψ5 �Ξ5 Ψ7� e1 � e3 � �Ξ7 Ψ6 �Ξ6 Ψ7� e2 � e3 �Ξ7 Ψ7 e1 � e2 � e3�
In a space with a Euclidean metric � is unity. GrassmannSimplify looks at the currently
declared metric and substitutes the value of �. Since the currently declared metric is by default
Euclidean, applying GrassmannSimplify puts � to unity.
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