Grassmann Algebra

TheComplement.nb 19
We can verify that this is indeed the metric by entering Metric.
Metric
��1, 0, Ν�, �0, �1, 0�, �Ν, 0,�1��
And verify that Ν has indeed been added to the list of declared scalars by entering Scalars.
Scalars
�a, b, c, d, e, f, g, h, �, Ν, �_ � _� ?InnerProductQ, _� 0
� Declaring a general metric
For theoretical calculations it is sometimes useful to be able to quickly declare a metric of
general symbolic elements. We can do this as described in the previous section, or we can use
the GrassmannAlgebra function DeclareMetric[g] where g is a symbol. We will often use
the 'double-struck' symbol � for the kernel symbol of the metric components.
�3; G� DeclareMetric���; MatrixForm�G�
� �1,1 �1,2 �1,3
��������
���������
�1,2 �2,2 �2,3
� �1,3 �2,3 �3,3 �
Note that GrassmannAlgebra has automatically declared the pattern for these components to be
scalars (rather than the components themselves). This means that any symbol of the form �i,j
is considered a scalar (for example �A,B ).
ScalarQ��A,B�
True
You can test to see if a symbol is a component of the currently declared metric by using
MetricQ.
MetricQ���2,3, g23, g3,4��
�True, False, False�
� Calculating induced metrics
The GrassmannAlgebra function for calculating the metric induced on � m by the metric in � 1 is
Metric�[m].
Example: Induced general metrics
Suppose we are working in 3-space with the general metric defined above. Then the metrics on
� 0 , � 1 , � 2 , � 3 can be calculated by entering:
2001 4 5