- #1
N88
- 225
- 12
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 \boldsymbol{\sigma_1}] ONTO another direction-vector [say \hat{a}] 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: \boldsymbol{\sigma_1}+\boldsymbol{\sigma_2}=0. (1)
Given: \hat{a}\circ\boldsymbol{\sigma_1}=+1; (2)
Then: \hat{a}\circ\boldsymbol{\sigma_2}=-\hat{a}\circ\boldsymbol{\sigma_1}=-1. (3)
In general: \hat{a}\circ\boldsymbol{\sigma}=\pm\hat{a}.\hat{a}<br /> =\sigma_{\hat{a}}=\pm1. (4)
Expectations: \left\langle \hat{a}\!\circ\!\boldsymbol{\sigma}\right\rangle =0;<br /> \left\langle (\hat{a}\!\circ\!\boldsymbol{\sigma})^{2}\right\rangle =1<br />. (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\circb, a\circb, a\circb, a\bulletb; etc. But I digress!
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 \boldsymbol{\sigma_1}] ONTO another direction-vector [say \hat{a}] 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: \boldsymbol{\sigma_1}+\boldsymbol{\sigma_2}=0. (1)
Given: \hat{a}\circ\boldsymbol{\sigma_1}=+1; (2)
Then: \hat{a}\circ\boldsymbol{\sigma_2}=-\hat{a}\circ\boldsymbol{\sigma_1}=-1. (3)
In general: \hat{a}\circ\boldsymbol{\sigma}=\pm\hat{a}.\hat{a}<br /> =\sigma_{\hat{a}}=\pm1. (4)
Expectations: \left\langle \hat{a}\!\circ\!\boldsymbol{\sigma}\right\rangle =0;<br /> \left\langle (\hat{a}\!\circ\!\boldsymbol{\sigma})^{2}\right\rangle =1<br />. (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\circb, a\circb, a\circb, a\bulletb; etc. But I digress!