Grassmann Algebra

TheExteriorProduct.nb 13
� Factoring scalars
GrassmannAlgebra provides the function FactorScalars to factor out any scalars in a
product so that they are collected at the beginning of the product. For example:
FactorScalars�5��2 x���3�y���bz��
30 b x � y � z
If any of the factors in the product are scalars, they will also be collected at the beginning. Here
a is a scalar since it has been declared as one, either by default or by DeclareScalars.
FactorScalars�5��2�x���3�a���bz��
30abx� z
FactorScalars works with any Grassmann expression, or list of Grassmann expressions:
FactorScalars�5��2�x���3�y���bz� � 5��2�x���3�a���bz��
30abx� z � 30 b x � y � z
FactorScalars��5��2�x���3�y���bz�, 5��2�x���3�a���bz���
�30 b x � y � z, 30 a b x � z�
� Grassmann expressions
A Grassmann expression is any valid element of a Grassmann algebra. To check whether an
expression is a Grassmann expression, we can use the GrassmannAlgebra function
GrassmannExpressionQ:
GrassmannExpressionQ�x � y�
True
GrassmannExpressionQ also works on lists of expressions. Below we determine that
whereas the product of a scalar a and a 1-element x is a valid element of the Grassmann
algebra, the ordinary multiplication of two 1-elements x y is not.
GrassmannExpressionQ��x � y, a x, x y��
�True, True, False�
� Calculating the grade of a Grassmann expression
The grade of an m-element is m.
At this stage we have not yet begun to use expressions for which the grade is other than obvious
by inspection. However, as will be seen later, the grade will not always be obvious, especially
2001 4 5