Grassmann Algebra

TheComplement.nb 24
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2 2
��1,3 �2,2 � 2 �1,2 �1,3 �2,3 � �1,1 �2,3 � �1,2 �3,3 � �1,1 �2,2 �3,3
5.8 Calculating Complements
� Entering a complement
To enter the expression for the complement of a Grassmann expression X in
GrassmannAlgebra, enter GrassmannComplement[X]. For example to enter the expression
for the complement of x�y, enter:
GrassmannComplement�x � y�
���������
x � y
Or, you can simply select the expression x�y and click the � � button on the
GrassmannAlgebra palette.
Note that an expression with a bar over it symbolizes another expression (the complement of the
original expression), not a GrassmannAlgebra operation on the expression. Hence no
conversions will occur by entering an overbarred expression.
GrassmannComplement will also work for lists, matrices, or tensors of elements. For
example, here is a matrix:
M � ��a e1 � e2, be2�, ��b e2, ce3 � e1��; MatrixForm�M�
� ae1 � e2
�b e2
be2
�
ce3 � e1
And here is its complement:
�����
M �� MatrixForm
���������������� ����������������
� ae1 ����
� e2 be2 �����
����������������
� �b
����������������
e2 ce3 � e1 �
� Converting to complement form
In order to see what a Grassmann expression, or list of Grassmann expressions, looks like when
expressed only in terms of the exterior product and complement operations, we can use the
GrassmannAlgebra operation ToComplementForm. Essentially, this operation takes an
expression and repeatedly uses rules to replace any regressive products by exterior products and
complements. This operation also works on other products, like the interior product, which will
be defined in later chapters. Note that the signs will be calculated using the dimension of the
currently declared basis.
For example here is a pair of expressions in 3-space:
2001 4 5