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Grassmann numbers

  1. Aug 21, 2007 #1
    i was not sure where to put this topic since I don't know which subject of math grassmann math constitutes. Is there an actual grassman number or is it symbolically represented by generators?
  2. jcsd
  3. Aug 21, 2007 #2

    Chris Hillman

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    Grassmann algebras

    Well, the answer depends upon what you mean by an "actual" number!

    If for you an "actual number" is one which can be added and multiplied, then Grassmann numbers are actual numbers.

    If an "actual number" is one which has a geometric interpretation (e.g. complex numbers can be interpreted as plane vectors), then Grassmann numbers are actual numbers, although their interpretation is rather different from what you might have in mind if you are only familiar with real and complex numbers.

    If for you an "actual number" is one which has in some suitable linear algebra-inspired sense only one "component", then a Grassmann number is not an actual number.

    The natural place for a question about Grassmann numbers would be the "Linear and Algebra" subforum.

    Grassmann numbers are elements of a Grassmann algebra, or exterior algebra, and they can be used to compute with geometric quantities---roughly, "area" elements. If you want to compute an integral over some "surface", then exterior calculus, which is based upon the notion of a Grassmann algebra, is just what you want. A nice exposition which stresses the geometric interpretation is

    Desmond Fearnsley-Sander, "A Royal Road to Geometry", Mathematics Magazine 53 (1980): 259--268.

    Those interested in "flat two-dimensional Galilean spacetime" (one of the nine planar homogeneous geometries found by Klein) should note that this can be identified with the Grassmann algebra on one generator (a trivial example!) considered as a two-dimensional real algebra.

    Someone--- I guess me--- should say that a Grassmann algebra is simply a real linear associative algebra generated by n elements (which we can think of as "vectors" in R^n)
    [tex]\vec{e}_1, \; \vec{e}_2, \dots \vec{e}_n[/tex]
    where we have a wedge product (bilinear, associative) such that
    [tex]\vec{e}_j \wedge \vec{e}_k = -\vec{e}_k \wedge \vec{e}_j[/tex]
    which implies
    [tex]\vec{e}_j \wedge \vec{e}_j = 0[/tex]
    Then wedge products of k distinct generators are "k-multivectors" in R^n; we can think of a k-multivector as a k-dimensional rhombus in R^n, modulo an affine transformation (so that only the "orientation in space" and "k-volume" of the rhombus have geometric significance, not the directions of its edges). A Grassmann number is then a linear combination of k-multivectors. The Grassmann algebra generated by n "vectors" as above has dimension 2^n, with a vector basis consisting of unity, the n basis vectors, the n choose 2 bivectors, ... and the volume element
    [tex]\omega = \vec{e}_1 \wedge \vec{e}_2 \wedge \dots \vec{e}_n[/tex]
    Here, summing the binomial coefficients gives
    [tex]1+n + \left( \begin{matrix} n \\ 2 \end{matrix} \right)
    + \left( \begin{matrix} n \\ 3 \end{matrix} \right) + \dots
    + \left( \begin{matrix} n \\ n-1 \end{matrix} \right) + 1 = 2^n[/tex]
    Last edited: Aug 21, 2007
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