Grassmann Algebra

TheInteriorProduct.nb 17
�Α1 � Α2 � � � Αm � ���� �Β1 � Β2 � � � Βm� � Det�Αi ���� Βj�
� Calculating inner products
6.36
The expression for the inner product of two simple elements can be developed in
GrassmannAlgebra by using the operations DevelopScalarProductMatrix and
ToScalarProducts. DevelopScalarProductMatrix[Α,Β] or
DevelopScalarProductMatrix[Α�Β] develops the inner product of Α and Β into the
matrix of the scalar products of their factors. The determinant of the scalar product matrix is the
inner product.
We can use ScalarProductMatrix purely symbolically. If we apply it to elements of
(unspecified) symbolic grade we get:
DevelopScalarProductMatrix�Α ���� Β���MatrixForm m m
� Α1 � Β1 Α1 � Β2 Α1 � � Α1 � Βm
������������
�������������
Α2 � Β1 Α2 � Β2 Α2 � � Α2 � Βm
� � Β1 � � Β2 � � � ��Βm � Αm � Β1 Αm � Β2 Αm � � Αm � Βm �
Or we can use the GrassmannAlgebra operation DeterminantForm to present the result as a
determinant.
DevelopScalarProductMatrix�Α ���� Β���DeterminantForm m m
��������
Α1 � Β1 Α1 � Β2 Α1 � � Α1 � Βm
��������
Α2 � Β1 Α2 � Β2 Α2 � � Α2 � Βm
�
�������
�
� � Β1 � � Β2 � � � ��Βm������� � Αm � Β1 Αm � Β2 Αm � � Αm � Βm �
If the grade of the elements is specified as an integer, we get a more specific result.
DevelopScalarProductMatrix�Ξ 3
���� ���MatrixForm 3
� Ξ1 � Ψ1 Ξ1 � Ψ2 Ξ1 � Ψ3
�������
��������
Ξ2 � Ψ1 Ξ2 � Ψ2 Ξ2 � Ψ3
� Ξ3 � Ψ1 Ξ3 � Ψ2 Ξ3 � Ψ3 �
Displaying the inner product of two elements as the determinant of the scalar product of their
factors shows clearly how the symmetry of the inner product depends on the symmetry of the
scalar product (which in turn depends on the symmetry of the metric tensor).
To obtain the actual inner product we take the determinant of this matrix.
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