Grassmann Algebra

ExpTheGeneralizedProduct.nb 33
The GrassmannAlgebra function OrthogonalSimplificationRules generates a list of
rules which put to zero all the scalar products of a 1-element from Α with a 1-element from Β. m k
� The generalized product of partially orthogonal elements
Consider the generalized product Α����� �
m Λ ����Γ
�
� Β��� of an element Α and another element Γ � Β in
�p
k�
m p k
which Α and Β are totally orthogonal, but Γ is arbitrary.
m k
p
But since Αi ���� Βj � 0, the interior products of Α and any factors of Γ � Β in the expansion of
m p k
the generalized product will be zero whenever they contain any factors of Β. That is, the only
k
non-zero terms in the expansion of the generalized product are of the form �Α ���� Γi�� Γ i � Βk .
m Λ p�Λ
Hence:
Α����� �
m Λ ����Γ
�p
�
� ��� �
k�
�
�
Α����� �
m Λ � ���Γ
�p
���Α m ����� Λ �Γ p
����
� Β
� k
Αi ���� Βj � 0 Λ�Min�m, p�
�
� Β��� � 0 Αi ���� Βj � 0 Λ�Min�m, p�
k�
Flatten�Table�ToScalarProducts�Α����� �
m Λ ����Γ
�
� ��� �
�p
k�
����Α
�
����� �Γ��� � Β
�m
Λ p�
k
OrthogonalSimplificationRules���Α, Β���, m k
�m, 0, 3�, �k, 0, m�, �p, 0, m�, �Λ, 0,Min�m, p����
��.
10.35
10.36
�0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0�
By using the quasi-commutativity of the generalized and exterior products, we can transform the
above results to:
2001 4 26
����Α
�
� Γ
�m
p�
�������� Λ �Β k
����Γ
�p
�
� Α
m�
� ��1�m Λ�Α �
m ����Γ
�p
�������� Λ �Β k
����� Λ �Β k
����
Αi ���� Βj � 0 Λ�Min�k, p�
�
� 0 Αi ���� Βj � 0 Λ�Min�k, p�
10.37
10.38