Grassmann Algebra

ExploringGrassmannAlgebra.nb 23
We can see this most easily by writing the expansion in the form below and noting that all the
intermediate terms cancel.
1 �� 2 � 3 � 4 � ... � q
�� 2 � 3 � 4 � ... ����  q � q�1
Alternatively, consider the product in the reverse order
�1 �� 2 � 3 � 4 � ... � q ���1 ��. Expanding this gives precisely the same
result. Hence a Grassmann number and its inverse commute.
Furthermore, it has been shown in the previous section that for a bodiless Grassmann number in
a space of n dimensions, the greatest non-zero power is pmax � 1 ���� �2 n� 1 � ��1� 4 n�. Thus
if q is equal to pmax , Β q�1 is equal to zero, and the identity becomes:
�1 ����1 �� 2 � 3 � 4 � ... � pmax � � 1
We have thus shown that 1 �Β�Β 2 �Β 3 �Β 4 ... �Β pmax is the inverse of 1+Β.
If now we have a general Grassmann number X, say, we can write X as X �Ξ0��1 �Β�, so that
if Xs is the soul of X, then
X �Ξ0��1 �Β� � Ξ0 � Xs
The inverse of X then becomes:
X�1 � 1
������� �
Ξ0
� Xs ���� 1 � ������� �
� Ξ0
���
�
Xs �
������� ��
Ξ0 �
2
Β� Xs
�������
Ξ0
� ���
�
Xs �
������� ��
�
Ξ0
3
� ... � ���
�
Xs
pmax
�
������� ��
�����
Ξ0 � �
We tabulate some examples for X �Ξ0 � Xs , expressing the inverse X �1 both in terms of Xs
and Ξ0 alone, and in terms of the even and odd components Σ and Σ 2 of Xs .
n pmax X �1 X �1
1 1
2 1
3 2
4 2
1
�����
Ξ0
1
�����
Ξ0
1
�����
Ξ0
1
�����
Ξ0
��1 � Xs ����� �
Ξ0
��1 � Xs ����� �
��1 � Xs �����
Ξ0
��1 � Xs �����
Ξ0
Ξ0
� � Xs ����� � 2 �
Ξ0
� � Xs ����� � 2 �
Ξ0
1
�����
Ξ0
1
�����
Ξ0
1
�����
Ξ0
1
�����
Ξ0
��1 � Xs �����
Ξ0
��1 � Xs �����
Ξ0
��1 � Xs ����� �
Ξ0
��1 � Xs ����� �
Ξ0
� 2
�������
Ξ0 2 Σ � Σ 2 �
� 1
�������
Ξ 0 2 �2 Σ�Σ 2 ��Σ 2 �
It is easy to verify the formulae of the table for dimensions 1 and 2 from the results above. The
formulae for spaces of dimension 3 and 4 are given below. But for a slight rearrangement of
terms they are identical to the results above.
2001 4 5
9.5
9.6