Grassmann Algebra

ExploringGrassmannAlgebra.nb 43
GrassmannFunction��Exp�x� Exp�y�, �x, y��, �X, Y��
� Ξ0 �Ψ0 �� Ξ0 �Ψ0 e1 �Ξ1 �Ψ1� �� Ξ0 �Ψ0 e2 �Ξ2 �Ψ2� �
� Ξ0 �Ψ0 �Ξ3 �Ξ2 Ψ1 �Ξ1 Ψ2 �Ψ3� e1 � e2
GrassmannFunction��Exp�x� Exp�y�, �x, y��, �Y, X��
� Ξ0 �Ψ0 �� Ξ0 �Ψ0 e1 �Ξ1 �Ψ1� �� Ξ0 �Ψ0 e2 �Ξ2 �Ψ2� �
� Ξ0 �Ψ0 �Ξ3 �Ξ2 Ψ1 �Ξ1 Ψ2 �Ψ3� e1 � e2
On the other hand, if we wish to compute the exponential of the sum X + Y we need to use
GrassmannFunction, because there can be two possibilities for its definition. That is,
although Exp[x+y] appears to be independent of the order of its arguments, an interchange in
the 'ordering' argument from {X,Y} to {Y,X} will change the signs in the result.
GrassmannFunction��Exp�x � y�, �x, y��, �X, Y��
� Ξ0 �Ψ0 �� Ξ0 �Ψ0 e1 �Ξ1 �Ψ1� �� Ξ0 �Ψ0 e2 �Ξ2 �Ψ2� �
� Ξ0 �Ψ0 �Ξ3 �Ξ2 Ψ1 �Ξ1 Ψ2 �Ψ3� e1 � e2
GrassmannFunction��Exp�x � y�, �x, y��, �Y, X��
� Ξ0 �Ψ0 �� Ξ0 �Ψ0 e1 �Ξ1 �Ψ1� �� Ξ0 �Ψ0 e2 �Ξ2 �Ψ2� �
� Ξ0 �Ψ0 �Ξ3 �Ξ2 Ψ1 �Ξ1 Ψ2 �Ψ3� e1 � e2
In sum: A function of several Grassmann variables may have different results depending on the
ordering of the variables in the function, even if their usual (scalar) form is commutative.
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