Grassmann Algebra

TheComplement.nb 18
The default metric assumed by GrassmannAlgebra is the Euclidean metric. In the case that the
GrassmannAlgebra package has just been loaded, entering Metric will show the components
of the default Euclidean metric tensor.
Metric
��1, 0, 0�, �0, 1, 0�, �0, 0, 1��
These components are arranged as the elements of a 3�3 matrix because the default basis is 3dimensional.
Basis
�e1, e2, e3�
However, if the dimension of the basis is changed, the default metric changes accordingly.
DeclareBasis���, i,j,k��
��, i,j,k�
Metric
��1, 0, 0, 0�, �0, 1, 0, 0�, �0, 0, 1, 0�, �0, 0, 0, 1��
We will take the liberty of referring to this matrix as the metric.
� Declaring a metric
GrassmannAlgebra permits us to declare a metric as a matrix with any numeric or symbolic
components. There are just three conditions to which a valid matrix must conform in order to be
a metric:
1) It must be symmetric (and hence square).
2) Its order must be the same as the dimension of the declared linear space.
3) Its components must be scalars.
It is up to the user to ensure the first two conditions. The third is handled by GrassmannAlgebra
automatically adding any symbols in the matrix not already declared as scalars to the list of
declared scalars.
Example: Declaring a metric
We return to 3-space (by entering �3 ), create a 3�3 matrix M, and then declare it as the metric.
2001 4 5
�3; M� ��1, 0, Ν�, �0, �1, 0�, �Ν, 0,�1��; MatrixForm�M�
� 1 0 Ν
�������
��������
0 �1 0
� Ν 0 �1 �
DeclareMetric�M�
��1, 0, Ν�, �0, �1, 0�, �Ν, 0,�1��