Grassmann Algebra

ExploringGrassmannAlgebra.nb 41
tanX � GrassmannFunction�Tan�X��
Sec�Ξ0� 2 e1 Ξ1 � Sec�Ξ0� 2 e2 Ξ2 � Sec�Ξ0� 2 e3 Ξ3 � Tan�Ξ0� �
Sec�Ξ0� 2 Ξ4 e1 � e2 � Sec�Ξ0� 2 Ξ5 e1 � e3 � Sec�Ξ0� 2 Ξ6 e2 � e3 �
Sec�Ξ0� 2 �Ξ7 � ��2 Ξ2 Ξ5 � 2 �Ξ3 Ξ4 �Ξ1 Ξ6�� Tan�Ξ0�� e1 � e2 � e3
sinX � GrassmannFunction�Sin�X��
Sin�Ξ0� � Cos�Ξ0� e1 Ξ1 � Cos�Ξ0� e2 Ξ2 � Cos�Ξ0� e3 Ξ3 �
Cos�Ξ0� Ξ4 e1 � e2 � Cos�Ξ0� Ξ5 e1 � e3 � Cos�Ξ0� Ξ6 e2 � e3 �
�Sin�Ξ0� ��Ξ3 Ξ4 �Ξ2 Ξ5 �Ξ1 Ξ6� � Cos�Ξ0� Ξ7� e1 � e2 � e3
secX � GrassmannFunction�Sec�X��
Sec�Ξ0� � Sec�Ξ0� e1 Ξ1 Tan�Ξ0� � Sec�Ξ0� e2 Ξ2 Tan�Ξ0� �
Sec�Ξ0� e3 Ξ3 Tan�Ξ0� � Sec�Ξ0� Ξ4 Tan�Ξ0� e1 � e2 �
Sec�Ξ0� Ξ5 Tan�Ξ0� e1 � e3 � Sec�Ξ0� Ξ6 Tan�Ξ0� e2 � e3 �
�Sec�Ξ0� 3 �Ξ3 Ξ4 �Ξ2 Ξ5 �Ξ1 Ξ6� � Sec�Ξ0�
�Ξ7 Tan�Ξ0� � �Ξ3 Ξ4 �Ξ2 Ξ5 �Ξ1 Ξ6� Tan�Ξ0� 2 �� e1 � e2 � e3
tanX � ��sinX � secX� ��Simplify
True
� Functions of several Grassmann numbers
GrassmannFunction can calculate functions of several Grassmann variables. It is important
to realize however, that there may be several different results depending on the order in which
the exterior products occur in the series expansion for the function, a factor that does not arise
with functions of scalar variables. This means then, that the usual function notation for several
variables is not adequate for specifically denoting precisely which form is required. For
simplicity in what follows we work in 2-space.
�2; �X � CreateGrassmannNumber�Ξ�,
Y � CreateGrassmannNumber���
�Ξ0 � e1 Ξ1 � e2 Ξ2 �Ξ3 e1 � e2, Ψ0 � e1 Ψ1 � e2 Ψ2 �Ψ3 e1 � e2�
Products of two Grassmann numbers
As a first example, we look at the product. Calculating the exterior product of two Grassmann
numbers in 2-space gives:
��X � Y�
Ξ0 Ψ0 � e1 �Ξ1 Ψ0 �Ξ0 Ψ1� � e2 �Ξ2 Ψ0 �Ξ0 Ψ2� �
�Ξ3 Ψ0 �Ξ2 Ψ1 �Ξ1 Ψ2 �Ξ0 Ψ3� e1 � e2
Using GrassmannFunction in the form below, we get the same result.
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