Grassmann Algebra

TheRegressiveProduct.nb 12
Example 4
Our fourth example is to generate the dual of the Common Factor Axiom. This axiom will be
introduced in the next section.
CommonFactorAxiom �
� ����Α
�
� ��� �
�m
p�
����Β
�
� ��� �
�k
p�
�
�
���Α m � Β k
Dual�CommonFactorAxiom�
���Α m � Γ p
� �
�
����
�
�
����Β
�
� ��� ��
�k
p�
�
�
���Α m � Β k
�
� ��� � à , m � k � p � n � 0�;
p�
p
�
� Γ��� � Γ, �k � m � 2n� p �� 0�
p�
p
We can verify that the dual of the dual of the axiom is the axiom itself.
CommonFactorAxiom � Dual�Dual�CommonFactorAxiom��
True
Example 5
Our fifth example is to obtain the dual of one of the product formulae derived in a later section.
F � �Α � Β�� x � �Α � x ��Β � ��1�
m k n�1 m n�1 k
m �Α ��Β� x �;
m k n�1
Dual�F�
�Α � Β��x �� ��1�
m k
�m�n Α � Β � x �Α� x � Β
m k m k
F � Dual�Dual�F��
True
Setting up your own input to the Dual function
The GrassmannAlgebra function Dual depends on the syntax of the statements given it as
arguments. The following are some suggestions for setting up your own input to Dual.
¥ Scalars under the usual field multiplication should be declared as scalars.
¥ Other symbols which you want to protect from change can be temporarily declared as scalars.
¥ A scalar denoted as a 0-element should be expressly multiplied using the exterior product.
¥ Separate statements should be combined into a list of statements.
¥ The statements should not include anything which will evaluate when entered.
¥ For example, summation signs should not use Mathematica's Sum function.
¥ If you use simple concatenated symbols, Mathematica will assume multiplication and arrange
the order as it pleases. Be prepared for this.
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