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I am interested in generating a list of n 4-vectors (t,x,y,z) such that -t^2+x^2+y^2+z^2=0 for each vector and the sum of the n 4-vectors equals zero. All of the t,x,y,z are real. I, particular, I am interested in solutions where none of the entries are zero, thus excluding the trivial solution and the case in which thing are collinear, e.g. the four vectors (plus/minus 1,0,0,plus/minus 1). I'm interested in this question because it is relevant to the study of scattering amplitudes in the center of mass frame. Is there an efficient algorithmic implementation of this, or, barring that, an implementation that works for n~10 to 20?

Thanks, Ning

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  • $\begingroup$ @AlexM. The last vector in your algorithm, the $-s$ that cancels the sum of the previous vectors, won't generally satisfy $-t^2+x^2+y^2+z^2=0$. $\endgroup$
    – Andreas Blass
    Feb 14, 2018 at 21:56
  • $\begingroup$ Unless I'm making a silly mistake, every time-like or space-like vector can be expressed as a sum of two null vectors (in infinitely many ways --- there seems to be a free paramater ranging over a 2-dimensional sphere). So use @AlexM's process for $n-2$ steps and then write the final $-s$ as the sum of two null vectors. $\endgroup$
    – Andreas Blass
    Feb 14, 2018 at 22:07
  • $\begingroup$ @AlexM Please un-delete your comment, because I refer to it in a corrected version. $\endgroup$
    – Andreas Blass
    Feb 14, 2018 at 22:08
  • $\begingroup$ What is the process that you referred to by AlexM? Can you recreate it? $\endgroup$
    – Ning Bao
    Feb 14, 2018 at 22:10
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    $\begingroup$ For the sake of convenience to @AndreasBlass and others, AlexM's comment read: "Let $s=(0,0,0,0)$. Generate $x,y,z$ and then compute $t=\sqrt{x^2+y^2+z^2}$; let $s=s+(t,x,y,z)$. Do this $n−1$ times. Let the $n$-th vector be $−s$." $\endgroup$
    – Todd Trimble ♦
    Feb 14, 2018 at 22:22

1 Answer 1

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The answer, thanks to @AndreasBlass: randomly generate the first n-2 null vectors, then find a generically non-null vector that cancels the sum of those vectors. Then decompose this vector into a sum over two null vectors.

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  • $\begingroup$ If you are re-posting someone else's comment (in this case @AndreasBlass's) as an answer, then it is appropriate to make your post community wiki so that you don't earn reputation from it. I think that you can do this yourself simply by editing the post and checking a box, but anyway I have flagged the post to have a moderator do it for you. $\endgroup$
    – LSpice
    Feb 14, 2018 at 23:03
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    $\begingroup$ You should also mention here (even though it's evident from the comments) that AlexM provided a big piece of the solution. $\endgroup$
    – Andreas Blass
    Feb 14, 2018 at 23:09

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