Grassmann Algebra

ExploringCliffordAlgebra.nb 7
� Clifford products in terms of inner products
Clifford products may also be expressed in terms of inner products, some of which may be
scalar products. From this form it is easy to read the grade of the terms.
ToInnerProducts�Α �Β� 3 2
��Α2 � Α3 � Β1 � Β2� Α1 � �Α1 � Α3 � Β1 � Β2 � Α2 �
�Α1 � Α2 � Β1 � Β2 � Α3 � �Α3 � Β2 � Α1 � Α2 � Β1 �
�Α3 � Β1� Α1 � Α2 � Β2 � �Α2 � Β2 � Α1 � Α3 � Β1 � �Α2 � Β1 � Α1 � Α3 � Β2 �
�Α1 � Β2� Α2 � Α3 � Β1 � �Α1 � Β1 � Α2 � Α3 � Β2 �Α1 � Α2 � Α3 � Β1 � Β2
� Clifford products in terms of scalar products
Finally, Clifford products may be expressed in terms of scalar and exterior products only.
ToScalarProducts�Α �Β� 3 2
�Α2 � Β2 ��Α3 � Β1 � Α1 � �Α2 � Β1 ��Α3 � Β2 � Α1 �
�Α1 � Β2��Α3 � Β1� Α2 � �Α1 � Β1 ��Α3 � Β2 � Α2 �
�Α1 � Β2��Α2 � Β1� Α3 � �Α1 � Β1 ��Α2 � Β2 � Α3 � �Α3 � Β2 � Α1 � Α2 � Β1 �
�Α3 � Β1� Α1 � Α2 � Β2 � �Α2 � Β2 � Α1 � Α3 � Β1 � �Α2 � Β1 � Α1 � Α3 � Β2 �
�Α1 � Β2� Α2 � Α3 � Β1 � �Α1 � Β1 � Α2 � Α3 � Β2 �Α1 � Α2 � Α3 � Β1 � Β2
12.3 The Reverse of an Exterior Product
Defining the reverse
We will find in our discussion of Clifford products that many operations and formulae are
simplified by expressing some of the exterior products in a form which reverses the order of the
1-element factors in the products.
We denote the reverse of a simple m-element Α m by Α m
† , and define it to be:
�Α1 � Α2 � � � Αm� † �Αm � Αm�1 � � � Α1
We can easily work out the number of permutations to achieve this rearrangement as
1 ��� 2 m �m � 1�. Mathematica automatically simplifies ��1� 1
���� 2 m �m�1� m �m�1� to � .
2001 4 26
12.2