Grassmann Algebra

ExpTheGeneralizedProduct.nb 36
In spaces of all dimensions, generalized products in which one of the factors is a scalar will
reduce to either the usual (field or linear space) product by the scalar quantity (when the order
of the product is zero) or else to zero (when the order of the product is greater than zero). Except
for a brief discussion of the implications of this for 0-space, further discussions will assume that
the factors of the generalized products are not scalar.
In spaces of dimension 0, 1, or 2, all generalized products reduce either to the ordinary field
product, the exterior product, the interior product, or zero. Hence they bring little new to a study
of these spaces.
These results will be clear upon reference to the simple properties of generalized products
discussed in previous sections. The only result that may not be immediately obvious is that of
the generalized product of order one of two 2-elements in a 2-space, for example Α����� �Β. In a 2-
2 1 2
space, Α and Β must be congruent (differ only by (possibly) a scalar factor), and hence by
2 2
formula 10.17 is zero.
If one of the factors is a scalar, then the only non-zero generalized product is that of order zero,
equivalent to the ordinary field product (and, incidentally, also to the exterior and interior
products).
� 0-space
In a space of zero dimensions (the underlying field of the Grassmann algebra), the generalized
product reduces to the exterior product and hence to the usual scalar (field) product.
ToScalarProducts�a����� 0 �b�
ab
Higher order products (for example a����� �b) are of course zero since the order of the product is
1
greater than the minimum of the grades of the factors (which in this case is zero).
In sum: The only non-zero generalized product in a space of zero dimensions is the product of
order zero, equivalent to the underlying field product.
� 1-space
Products of zero order
In a space of one dimension there is only one basis element, so the product of zero order (that is,
the exterior product) of any two elements is zero:
�a e1������ 0 ��b e1� � abe1 � e1 � 0
ToScalarProducts��ae1������ 0 ��b e1��
0
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