Grassmann Algebra

TheRegressiveProduct.nb 45
�Α1 � � � Αm ���Β1 � � � Βm �
n�1 n�1
� �����Α1
�
� � � Αm��Βj
��� ����1�
�
�
j�1 �Β1
� �
�
���� i ���1� i�1 � �
n�1
n�1
� � � ���� j � � � Βm �
n�1
���Αi
�
� Βj
�����Α1 � � � ���� i � � � Αm�
� �
����
�
�
n�1
���1�j�1 �Β1 � � � ���� j � � � Βm �
n�1
n�1
�����1� i i�j � ����Αi
�
� Βj
� n�1�
By repeating this process one obtains finally that:
������Α1 � � � ���� i � � � Αm���Β1
n�1
�Α1 � � � Αm ���Β1 � � � Βm � � Det�Αi � Βj �
n�1 n�1
� � � ���� j � � � Βm ��
n�1
n�1
3.54
where Det�Αi � Βj � is the determinant of the matrix whose elements are the scalars Αi � Βj .
n�1
n�1
For the case m = 2 we have:
�Α1 � Α2���Β1
n�1
� Β2
n�1
� � �Α1 � Β1 ���Α2 � Β2 � � �Α1 � Β2 ���Α2 � Β1 �
n�1
Although the right-hand side is composed of products of scalars, if we want to obtain the correct
dual we need to use the exterior product instead of the usual field multiplication for these scalars.
Dual�
�Α1 � Α2���Β1
n�1
� Β2
n�1
n�1
n�1
n�1
� � �Α1 � Β1 ���Α2 � Β2 � � �Α1 � Β2 ���Α2 � Β1 ��
n�1
� Α1 � Α2 ��Β1 � Β2 �� Α1 � Β1 � Α2 � Β2 � Α1 � Β2 � Α2 � Β1
�1�n �1�n
�1�n �1�n �1�n �1�n
Formula 3.54 is of central importance in the computation of inner products to be discussed in
Chapter 6.
2001 4 5
n�1
n�1
n�1