Grassmann Algebra

TheComplement.nb 25
�3;
X � ��e1 � e2���e2 � e3���e3 � e1�, �e1 � e2 � e3���e1 � e2 � e3��e1�;
ToComplementForm�X�
������������� ��������������������������������������������
�e1 � e2 �
�������������
e2 � e3 �
�������������
e3 �
��������������������
e1,e1�
e2 � e3 �
��������������������
e1 � e2 � e3 � e1
�����
����������������������������������������������������
�
Converting symbolic expressions
ToComplementForm may also be used with elements of symbolic grade. Note that the sign
of the result will depend on the dimension of the space, and for it to calculate correctly the
grades must be declared scalars.
For example, here is a symbolic expression in 3-space. The grades a, b, and c are by default
declared scalars. Since the sign ��1� m��3�m� is positive for all elements in a 3-space we have a
result independent of the grades of the elements.
�3; ToComplementForm�Α � Β
a b
��������������
Α
� �
� Βb �
�
Γc
a
� Γ� c
However, in a 4-space, the sign depends on the grades.
�4; ToComplementForm�Α a � Β b
��1� a�b�c Α a
�������������� � �
� Βb �
�
Γc
� Γ� c
Converting to complement form in an arbitrary dimension
It is possible to do calculations of this type in spaces of arbitrary dimension by declaring a
symbolic dimension with DeclareSymbolicDimension[n] (make sure n has been
declared a scalar). However, be aware that few functions (particularly those involving basis
elements) will give a correct result because the linear spaces in GrassmannAlgebra are by
default assumed finite. Remember to redeclare a finite basis when you have finished using the
arbitrary dimension!
2001 4 5
DeclareExtraScalars�Ω�; DeclareSymbolicDimension�Ω�
Ω
ToComplementForm�Α � Β
a b
��1� �a�b�c��1�� ��������������
Α
� �
� Βb �
�
Γc
a
� Γ� c