Grassmann Algebra

ExpTheGeneralizedProduct.nb 8
��� � � �� � � �� � � �
Tabbing through the placeholders and entering factors and product orders, we might obtain
something like:
��X����� 4 �Y������ 3 �Z������ 2 �W
Here, X, Y, Z, and W represent any Grassmann expressions. Using Mathematica's FullForm
operation shows how this is internally represented.
FullForm���X����� 4 �Y������ 1 �Z������ 0 �W�
GeneralizedProduct�0��
GeneralizedProduct�1��GeneralizedProduct�4��X, Y�, Z�, W�
You can edit the expression at any stage to group the products in a different order.
FullForm�X����� 4 ���Y����� 3 �Z������ 2 �W��
GeneralizedProduct�4��X,
GeneralizedProduct�2��GeneralizedProduct�3��Y, Z�, W��
� Reduction to interior products
If the generalized product Α����� �Β
m Λ k
it will result in a sum of ����
�
k �
Λ �
is expanded in terms of the first factor Α it will result in a sum of
m ����
�
m �
Λ �
of simple elements is expanded in terms of the second factor Β
k
��� terms (0 ≤ Λ ≤ k) involving just exterior and interior products. If it
��� terms (0 ≤ Λ ≤ m).
These sums, although appearing different because expressed in terms of interior products, are of
course equal due to the result in the previous section.
The GrassmannAlgebra function ToInteriorProducts will take any generalized product
and convert it to a sum involving exterior and interior products by expanding the second factor
as in the definition. If the elements are given in symbolic grade-underscripted form,
ToInteriorProducts will first create a corresponding simple exterior product before
expanding.
For example, suppose the generalized product is given in symbolic form as Α����� �Β. Applying
4 2 3
ToInteriorProducts expands with respect to Β 3
2001 4 26
and gives:
A � ToInteriorProducts�Α����� �Β� 4 2 3
�Α1 � Α2 � Α3 � Α4 � Β1 � Β2��Β3 �
�Α1 � Α2 � Α3 � Α4 � Β1 � Β3��Β2 � �Α1 � Α2 � Α3 � Α4 � Β2 � Β3��Β1