Grassmann Algebra

TheInteriorProduct.nb 18
Det�DevelopScalarProductMatrix����� �� 3 3
��Ξ1 � Ψ3� �Ξ2 � Ψ2� �Ξ3 � Ψ1� � �Ξ1 � Ψ2� �Ξ2 � Ψ3� �Ξ3 � Ψ1� �
�Ξ1 � Ψ3� �Ξ2 � Ψ1� �Ξ3 � Ψ2� � �Ξ1 � Ψ1� �Ξ2 � Ψ3� �Ξ3 � Ψ2� �
�Ξ1 � Ψ2� �Ξ2 � Ψ1� �Ξ3 � Ψ3� � �Ξ1 � Ψ1� �Ξ2 � Ψ2� �Ξ3 � Ψ3�
If the elements of the determinant are given in terms of basis elements, we can obtain a final
result. For example, if we are in 3-space and we take the scalar product of the element
e 3 � e1 � e2 � e3 with itself we get:
�3; DevelopScalarProductMatrix�e 3 ���� e 3 ���MatrixForm
��������
�
e1 � e1 e1 � e2 e1 � e3
e2 � e1 e2 � e2 e2 � e3
e3 � e1 e3 � e2 e3 � e3
A � Det�DevelopScalarProductMatrix�e 3 ���� e 3 ��
��������
�
��e1 � e3� �e2 � e2� �e3 � e1� � �e1 � e2� �e2 � e3� �e3 � e1� �
�e1 � e3� �e2 � e1� �e3 � e2� � �e1 � e1� �e2 � e3� �e3 � e2� �
�e1 � e2� �e2 � e1� �e3 � e3� � �e1 � e1� �e2 � e2� �e3 � e3�
We can now substitute the metric elements gi,j for the scalar products ei ���� ej by using the
GrassmannAlgebra function ToMetricForm. Declaring a default general metric we get:
DeclareMetric���
���1,1, �1,2, �1,3�, ��1,2, �2,2, �2,3�, ��1,3, �2,3, �3,3��
ToMetricForm�A�
2
��1,3 2
�2,2 � 2 �1,2 �1,3 �2,3 � �1,1 �2,3 2
� �1,2 �3,3 � �1,1 �2,2 �3,3
This is Det�gi,j� as we would expect. Had we declared the default Euclidean metric we
would have obtained unity as expected.
To calculate an inner product directly we can use the GrassmannAlgebra function
ToScalarProducts.
ToScalarProducts��x � y � z� ���� �u � v � w��
��u � z� �v � y� �w � x� � �u � y� �v � z� �w � x� � �u � z��v � x� �w � y� �
�u � x� �v � z� �w � y� � �u � y� �v � x� �w � z� � �u � x� �v � y� �w � z�
In fact, ToScalarProducts works to convert any interior products in a Grassmann
expression to scalar products.
2001 4 5
X � 1 � x ���� a � b �x � y � z� ���� w � c��e1 � e2� ���� �e2 � e3�
1 � x � a � ce1 � e2 � e2 � e3 � bx� y � z � w