Grassmann Algebra

ExploringGrassmannAlgebra.nb 4
DeclareBasis��x, y, z��;
T � Table�CreateGrassmannNumber���, �3��; �3; MatrixForm�T�
� ��x ��y ��z ���x � y ��x � z ��y � z ��x � y � z
�������
��������
��x ��y ��z ���x � y ��x � z ��y � z ��x � y � z
� ��x ��y ��z ���x � y ��x � z ��y � z ��x � y � z �
� Body and soul
For certain operations (particularly the inverse) it is critically important whether or not a
Grassmann number has a (non-zero) scalar component.
The scalar component of a Grassmann number is called its body. The body may be obtained
from a Grassmann number by using the Body function. Suppose we have a general Grassmann
number X in a 3-space.
�3; X� CreateGrassmannNumber�Ξ�
Ξ0 � e1 Ξ1 � e2 Ξ2 � e3 Ξ3 �Ξ4 e1 � e2 �
Ξ5 e1 � e3 �Ξ6 e2 � e3 �Ξ7 e1 � e2 � e3
The body of X is then:
Body�X�
Ξ0
The rest of a Grassmann number is called its soul. The soul may be obtained from a Grassmann
number by using the Soul function. The soul of the number X is:
Soul�X�
e1 Ξ1 � e2 Ξ2 � e3 Ξ3 �Ξ4 e1 � e2 �Ξ5 e1 � e3 �Ξ6 e2 � e3 �Ξ7 e1 � e2 � e3
Body and Soul apply to Grassmann numbers whose components are given in any form.
Z � �x ���� y� � ��x ��y � 2�� ���� �z � 2�� � �z ��y � x��;
�Body�Z�, Soul�Z��
�x � y � 2 �x � z� � z � y � x, �4 x� x � y � z � 2x� y�
The first two terms of the body are scalar because they are interior products of 1-elements
(scalar products). The third term of the body is a scalar in a space of 3 dimensions.
Body and Soul are both Listable. Taking the Soul of the list of Grassmann numbers
generated in 2-space above gives
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