Grassmann Algebra

ExpTheGeneralizedProduct.nb 6
Α m ���� Β j
Λ
� m
Λ �
� �
i�1
�Α i
Λ
���� Βj��Α
Λ
i
m�Λ
Substituting this in the above expression for the generalized product gives:
Α m ����� Λ �Β k
�������������
Λ
� � �
i�1
j�1
�
�
� k
Λ �
� m
�Α i
Λ
���� Βj��Α
Λ
i
�������������
� Β
m�Λ
�
j
k�Λ
The expansion of the generalized product in terms of both factors Α m and Β k
Α m ����� Λ �Β k
Α m � Α 1
Λ
� m
Λ �
� �
� Α1
m�Λ
i�1
� k
Λ �
�
j�1
� Α2
Λ
�Α i
Λ
���� Βj�
Α
Λ
i
� Βj
m�Λ k�Λ
� Α2
m�Λ � �, Β k
� Β 1
Λ
0 �Λ�Min�m, k�
� Β 1
k�Λ
� Β 2
Λ
The quasi-commutativity of the generalized product
is thus given by:
� Β2 � �
k�Λ
From the symmetric form of the generalized product [10.8] we can show directly that the
ordering of the factors in a product is immaterial except perhaps for a sign.
To show this we begin with the symmetric form [10.8], and rewrite it for the product of Β k
Α, that is, Β����� �Α . We then reverse the order of the exterior product to obtain the terms of Α����� �Β
m k Λ m m Λ k
except for possibly a sign. The inner product is of course symmetric, and so can be written in
either ordering.
�
����� �Α � �
Λ m k
Λ � �
��
m
Λ �
Β k
� k
Λ �
� m
Λ �
� ���
j�1 i�1
j�1
�Α i
Λ
i�1
�Β j
Λ
Comparison with [10.8] then gives:
2001 4 26
���� Αi� Βj � Α
Λ k�Λ
i
m�Λ
���� Βj���1�
Λ
�m�Λ���k�Λ� �Α i
� Βj
m�Λ k�Λ
Α m ����� Λ �Β k
� ��1� �m�Λ���k�Λ� �Β����� �Α
k Λ m
10.8
with
10.9