Grassmann Algebra

ExploringGrassmannAlgebra.nb 2
� Example: Factorizing a Grassmann number in 2-space
� Example: Factorizing a 2-element in 3-space
� Example: Factorizing a 3-element in 4-space
9.9 Functions of Grassmann Numbers
The Taylor series formula
� The form of a function of a Grassmann number
� Calculating functions of Grassmann numbers
� Powers of Grassmann numbers
� Exponential and logarithmic functions of Grassmann numbers
� Trigonometric functions of Grassmann numbers
� Functions of several Grassmann numbers
9.1 Introduction
The Grassmann algebra is an algebra of "numbers" composed of linear sums of elements from
any of the exterior linear spaces generated from a given linear space. These are numbers in the
same sense that, for example, a complex number or a matrix is a number. The essential property
that an algebra has, in addition to being a linear space is, broadly speaking, one of closure under
a product operation. That is, the algebra has a product operation such that the product of any
elements of the algebra is also an element of the algebra.
Thus the exterior product spaces � discussed up to this point, are not algebras, since products
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(exterior, regressive or interior) of their elements are not generally elements of the same space.
However, there are two important exceptions. � is not only a linear space, but an algebra (and a
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field) as well under the exterior and interior products. � is an algebra and a field under the
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regressive product.
Many of our examples will be using GrassmannAlgebra functions, and because we will often be
changing the dimension of the space depending on the example under consideration, we will
indicate a change without comment simply by entering the appropriate DeclareBasis
symbol from the palette. For example the following entry:
�3;
effects the change to a 3-dimensional linear or vector space.
9.2 Grassmann Numbers
� Creating Grassmann numbers
A Grassmann number is a sum of elements from any of the exterior product spaces �. They
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thus form a linear space in their own right, which we call �.
A basis for � is obtained from the current declared basis of � by collecting together all the
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basis elements of the various �. This may be generated by entering Basis�[].
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