Grassmann Algebra

ExpTheGeneralizedProduct.nb 37
Products of the first order
The product of the first order reduces to the scalar product.
�a e1������ 1 ��b e1� � ab�e1 ���� e1�
ToScalarProducts��ae1������ 1 ��b e1��
ab�e1 � e1�
In sum: The only non-zero generalized product of elements (other than where one is a scalar) in
a space of one dimension is the product of order one, equivalent to the scalar product.
� 2-space
Products of zero order
The product of zero order of two elements reduces to their exterior product. Hence in a 2-space,
the only non-zero products (apart from when one is a scalar) is the exterior product of two 1elements.
�a e1 � be2������ 0 ��c e1 � de2� � �a e1 � be2���ce1 � de2�
Products of the first order
The product of the first order of two 1-elements is commutative and reduces to their scalar
product.
�a e1 � be2������ 1 ��c e1 � de2� � �a e1 � be2� ���� �c e1 � de2�
ToScalarProducts��ae1 � be2������ 1 ��c e1 � de2��
ac�e1 � e1� � bc�e1 � e2� � ad�e1 � e2� � bd�e2 � e2�
The product of the first order of a 1-element and a 2-element is also commutative and reduces to
their interior product.
�a e1 � be2������ 1 ��c e1 � e2� �
�c e1 � e2������ 1 ��a e1 � be2� � �c e1 � e2� ���� �a e1 � be2�
ToScalarProducts��ae1 � be2������ 1 ��c e1 � e2��
�a c�e1 � e2� e1 � bc�e2 � e2� e1 � ac�e1 � e1� e2 � bc�e1 � e2� e2
The product of the first order of two 2-elements is zero. This can be determined directly from
[10.17].
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Α m ����� Λ �Α m � 0, Λ�m