Grassmann Algebra

ExploringGrassmannAlgebra.nb 14
� Checking for zero terms
ZeroSimplify looks at each term and decides if it is zero, either from nilpotency or from
expotency. A term is nilpotent if it contains repeated 1-element factors. A term is expotent if the
number of 1-element factors in it is greater than the dimension of the space. (We have coined
this term 'expotent' for convenience.) An interior product is zero if the grade of the first factor is
less than the grade of the second factor. ZeroSimplify then puts those terms it finds to be
zero, equal to zero.
A3 � ZeroSimplify�A2�
6 � 6x� 3y� 2 �x � z� � y � z � x � y � z � y � x � z � 3x� y � 3y� x
� Reordering factors
In order to simplify further it is necessary to put the factors of each term into a canonical order
so that terms of opposite sign in an exterior product will cancel and any inner product will be
transformed into just one of its two symmetric forms. This reordering may be achieved by using
the GrassmannAlgebra ToStandardOrdering function. (The rules for GrassmannAlgebra
standard ordering are given in the package documentation.)
A4 � ToStandardOrdering�A3�
6 � 6x� 3y� 2 �x � z� � y � z
� Simplifying expressions
We can perform all the simplifying operations above with GrassmannSimplify. Applying
GrassmannSimplify to our original expression A we get the expression A4 .
��A4�
6 � 6x� 3y� 2 �x � z� � y � z
9.5 Powers of Grassmann Numbers
� Direct computation of powers
The zeroth power of a Grassmann number is defined to be the unit scalar 1. The first power is
defined to be the Grassmann number itself. Higher powers can be obtained by simply taking the
exterior product of the number with itself the requisite number of times, and then applying
GrassmannSimplify to simplify the result.
To refer to an exterior power n of a Grassmann number X (but not to compute it), we will use
the usual notation X n .
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