Vrftsjtryk

I have to show that the following mapping of momenta is surjective. The mapping ${p_i^{mu},p_j^{mu},p_k^{mu}}rightarrow{tilde{p}_{ij}^{mu},tilde{p}_k^{mu}}$ is given by

$$
tilde{p}_k^{mu}=frac{1}{1-y_{ij,k}}p_k^{mu}
$$

$$
tilde{p}_{ij}^{mu}=p_i^{mu}+p_j^{mu}-frac{y_{ij,k}}{1-y_{ij,k}}p_k^{mu}.
$$

with $y_{ij,k}=frac{p_ip_j}{p_ip_j+p_ip_k+p_jp_k}$.

In other words: Do we cover the whole phase space for the tilde quantities under the mapping from ordinary momenta to tilde momenta?

Furthermore momentum conservation and on-shellness is guaranteed.

$$
tilde{p}_{ij}^{mu}+tilde{p}_k^{mu}=p_i^{mu}+p_j^{mu}+p_k^{mu}
$$

$$
p_i^2=p_j^2=p_k^2=tilde{p}_{ij}^2=tilde{p}_k^2=0
$$

To me it seems to be a bit strange, since we have 3 momenta (9 degerees of freedom) that we map to 2 momenta (with 6 degrees of freedom).

In addition to that it was mentioned that the mapping is also injective. How is this possible?

I can provide more physical background if required.

Ok, I am dumping a few illustrative notes to flesh out that the number of parameters (d.o.f.) is 6 before and after the map.

Given the arbitrariness of your basis choice and orientation, we may as well pick our axes to simplify them, the triangular pyramid found by the spacelike pieces of $p_k,p_i,p_j$,
$$p_k=s(1,1,0,0); qquad p_i= r(1,cos theta, sin theta, 0); qquad p_j=u (1,cosphi coschi, sin phi coschi, sinchi).
$$
To put all 3 space vectors on a plane, pick $chi=0$, so the volume of the space pyramid collapses to zero.

Your total momentum is then
$$
(s+r+u, s+rcostheta + u cosphicoschi, rsintheta + u coschi sin phi,usin chi ),
$$
so when all 3 spatial vectors are on a plane, the sum is on a plane.

The scale factors for $p_k$ are
$$
frac{y}{1-y} = frac {ru(1-costheta cosphicoschi -coschi sinphi sintheta )}{s(r+u-rcostheta -ucosphi coschi ) },
$$
and
$$
frac{1}{1-y}= \ ({sr (1-cos theta)+su(1-cos phicoschi)+ru(1-costheta cosphi coschi -coschisinphi sintheta )}) \
/~({s(r+u-rcostheta -ucosphi coschi ) }).
$$
For $chi=0$ the "sum vector" tilts away from the "original" one , $tilde p_k$. I think we need not do any spherical trigonometry, but I am unfamiliar with the hyperbolic geometry involved.

In any case, we may directly verify the before and after sufficiency of the 6 parameters. Given a null tilded vector, we might, in principle solve for these parameters, and plug them into 3 null untilded vectors... I am not seriously suggesting doing this... just a notional fantasy on our bijection. The extra 3 d.o.f. we might think they'd have, are the three 0 entries, above, and they are just a peculiarity of the coordinate axes' choices.

To handle the strangeness, we might look at $theta=chi=pi/2$, $s=r=u=1$,
so the original triplet momenta space parts become the orthogonal axes $hat x, hat y, hat z$, and the equivalent doublet of tilded output light-like vectors $tilde p _k=frac{3}{2}(1,hat x)$, and $tilde p _{ij}= (frac {3}{2}, hat y+hat z-hat x /2) $. Aren't they doing the same job?

In any case, if we wanted to hack around with hyperbolic geometry, given your total momentum $P=tilde P$, distinctly non-null, we have
$$
y=1-frac{2Pcdot p_k}{P^2} =frac{p_icdot p_j}{tilde p_kcdot tilde p_{ij}} ~,qquad 2Pcdot tilde p_k = P^2 ,\
frac{1}{1-y}= frac{P^2}{2Pcdot p_k}, qquad frac{y}{1-y}=frac{P^2-2Pcdot p_k}{2Pcdot p_k} ~~.
$$