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I A new vector-product for geometric algebra?

  1. Mar 15, 2017 #1

    N88

    User Avatar

    I am investigating the mathematical properties of a vector-product. I am wondering if it might be old-hat in GA (which is new to me)?

    I am using the working-title "spin-product" for a vector-product that combines RANDOM rotation-only of a direction-vector [a unit 1-vector; say [itex]\boldsymbol{\sigma_1}[/itex]] ONTO another direction-vector [say [itex]\hat{a}[/itex]] FOLLOWED BY an inner product of the now-aligned vectors. The randomness follows from this fact: I am here treating the "spun" vector (the one rotated) as an unknown (ie, hidden) direction-vector associated with complex dynamics.* By way of example:

    Let: [itex]\boldsymbol{\sigma_1}+\boldsymbol{\sigma_2}=0.[/itex] (1)
    Given: [itex]\hat{a}\circ\boldsymbol{\sigma_1}=+1[/itex]; (2)
    Then: [itex]\hat{a}\circ\boldsymbol{\sigma_2}=-\hat{a}\circ\boldsymbol{\sigma_1}=-1[/itex]. (3)

    In general: [itex]\hat{a}\circ\boldsymbol{\sigma}=\pm\hat{a}.\hat{a}
    =\sigma_{\hat{a}}=\pm1[/itex]. (4)
    Expectations: [itex]\left\langle \hat{a}\!\circ\!\boldsymbol{\sigma}\right\rangle =0;
    \left\langle (\hat{a}\!\circ\!\boldsymbol{\sigma})^{2}\right\rangle =1
    [/itex]. (5)

    Question: Does such a vector-product exist in GA? More cheekily: Should it?
    --------------------------
    * The proposed "spin-product" allows all sorts of interesting combinations: eg, combine random or non-random rotation-and-reduction, rotation-only, rotation-and-dilation of a vector onto another followed by another vector-product of the now-aligned vectors; etc.

    Clearly, a user of this product needs to specify the variant being utilised; maybe with variant spin-symbols like:
    a[itex]\circ[/itex]b, a[itex]\circ[/itex]b, a[itex]\circ[/itex]b, a[itex]\bullet[/itex]b; etc. But I digress!
     
  2. jcsd
  3. Mar 15, 2017 #2

    jedishrfu

    Staff: Mentor

    GA meaning Genetic Algorithms?

    Oops got it Geometric Algebra.
     
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