Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 35
Short�GrassmannMatrixFunction�f�A��, 3�
���1��, ��f�1� � f�3� � 1
���� x � f�1� �
2 1
���� x � f�3� ��22� �
2
x � z �2 f � �1� � 5
���� xf
2 � �1� � zf � �1� � f � �3� � 1 ���� xf
2 � �3� � 2zf � �3��,
f�3� ��9��x � z � 1
���� xf
2 � �1� � f � �3� � 1
���� zf
2 � �3����
We see from this that the derivatives in question are the zeroth and first, f[_], and f'[_].
We therefore declare these patterns to be scalars so that the derivatives evaluated at any
scalar arguments will be considered by GrassmannSimplify as scalars.
DeclareExtraScalars��f�_�, f � �_���
�a, b, c, d, e, f, g, h, �, f�_�, �_ � _�? InnerProductQ, _,f 0
� �_��
fA � GrassmannMatrixFunction�f�A��
��f�1� � xf � �1� � 1
���� x � z �f�1� � f�3� � 2f
2 � �1��, � 1
���� �f�1� � f�3�� x � z�,
2
��f�1� � 1
���� xf�1�� 2 1
���� zf�1�� f�3� �
2 1
���� xf�3�� 2 1
���� zf�3�� 2
xf � �1� � zf � �3� � 3
���� x � z �f�1� � f�3� � f
2 � �1� � f � �3��,
f�3� � zf��3� � 1
���� x � z �f�1� � f�3� � 2f
2 ��3���� The function f may be replaced by any specific function. We replace it here by Exp and
check that the result is the same as that given above.
expA � �fA �. f� Exp� ��Simplify
True
13.11 Supermatrices
To be completed.
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