Grassmann Algebra

ExploringGrassmannAlgebra.nb 42
GrassmannFunction��x y,�x, y��, �X, Y��
Ξ0 Ψ0 � e1 �Ξ1 Ψ0 �Ξ0 Ψ1� � e2 �Ξ2 Ψ0 �Ξ0 Ψ2� �
�Ξ3 Ψ0 �Ξ2 Ψ1 �Ξ1 Ψ2 �Ξ0 Ψ3� e1 � e2
However, to allow for the two different exterior products that X and Y can have together, the
variables in the second argument {X,Y} may be interchanged. The parameters x and y of the
first argument {x y,{x,y}} are simply dummy variables and can stay the same. Thus Y�X
may be obtained by evaluating:
��Y � X�
Ξ0 Ψ0 � e1 �Ξ1 Ψ0 �Ξ0 Ψ1� � e2 �Ξ2 Ψ0 �Ξ0 Ψ2� �
�Ξ3 Ψ0 �Ξ2 Ψ1 �Ξ1 Ψ2 �Ξ0 Ψ3� e1 � e2
or using GrassmannFunction with {Y,X} as its second argument.
GrassmannFunction��x y,�x, y��, �Y, X��
Ξ0 Ψ0 � e1 �Ξ1 Ψ0 �Ξ0 Ψ1� � e2 �Ξ2 Ψ0 �Ξ0 Ψ2� �
�Ξ3 Ψ0 �Ξ2 Ψ1 �Ξ1 Ψ2 �Ξ0 Ψ3� e1 � e2
The exponential of a sum
As a final example we show how GrassmannFunction manages the two different exterior
product equivalents of the scalar identity Exp[x+y]� Exp[x] Exp[y].
First calculate Exp[X] and Exp[y].
expX � GrassmannFunction�Exp�X��
� Ξ 0 �� Ξ 0 e1 Ξ1 �� Ξ 0 e2 Ξ2 �� Ξ 0 Ξ3 e1 � e2
expY � GrassmannFunction�Exp�Y��
� Ψ0 �� Ψ0 e1 Ψ1 �� Ψ0 e2 Ψ2 �� Ψ0 Ψ3 e1 � e2
If we compute their product using the exterior product operation we observe that the order must
be important, and that a different result is obtained when the order is reversed.
��expX � expY�
� Ξ0 �Ψ0 �� Ξ0 �Ψ0 e1 �Ξ1 �Ψ1� �� Ξ0 �Ψ0 e2 �Ξ2 �Ψ2� �
� Ξ0 �Ψ0 �Ξ3 �Ξ2 Ψ1 �Ξ1 Ψ2 �Ψ3� e1 � e2
��expY � expX�
� Ξ0 �Ψ0 �� Ξ0 �Ψ0 e1 �Ξ1 �Ψ1� �� Ξ0 �Ψ0 e2 �Ξ2 �Ψ2� �
� Ξ 0 �Ψ 0 �Ξ3 �Ξ2 Ψ1 �Ξ1 Ψ2 �Ψ3� e1 � e2
We can compute these two results also by using GrassmannFunction. Note that the second
argument in the second computation has its components in reverse order.
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