Grassmann Algebra

ExploringGrassmannAlgebra.nb 13
9.4 Simplifying Grassmann Numbers
� Elementary simplifying operations
All of the elementary simplifying operations that work with elements of a single exterior linear
space also work with Grassmann numbers. In this section we take a Grassmann number A in 2space
involving both exterior and interior products, and explore the application of those
component operations of GrassmannSimplify which apply to exterior and interior products.
A � ��1 � x���2 � y � y � x�� ���� �3 � z�
�1 � x���2 � y � y � x� � �3 � z�
� Expanding products
ExpandProducts performs the first level of the simplification operation by simply
expanding out any products. It treats scalars as elements of the Grassmann algebra, just like
those of higher grade.Taking the expression A and expanding out both the exterior and interior
products gives:
A1 � ExpandProducts�A�
1 � 2 � 3 � 1 � 2 � z � 1 � y � 3 � 1 � y � z � x � 2 � 3 � x � 2 � z � x � y � 3 �
x � y � z � 1 � y � x � 3 � 1 � y � x � z � x � y � x � 3 � x � y � x � z
� Factoring scalars
FactorScalars collects all the scalars in each term, which figure either by way of ordinary
multiplication or by way of exterior multiplication, and multiplies them as a scalar factor at the
beginning of each term. Remember that both the exterior and interior product of scalars is a
scalar. Mathematica will also automatically rearrange the terms in this sum according to its
internal ordering algorithm.
2001 4 5
A2 � FactorScalars�A1�
6 � 6x� 3y� 2 �1 � z� � 2 �x � z� � y � z � x � y � z �
y � x � z � x � y � x � z � 3x� y � 3y� x � 3x� y � x