Partial completeness relation for Dirac spinors in studying trace techniques to obtain matrix elements, I came across a problem when we treat scattering of neutrinos on protons. Indeed, since those neutrinos are supposedly created in a weak decay, they are all left-handed. This means that, when taking the sum of all the spins, we do not sum on the two spins for the neutrino.
Usually, we use the relation $\sum_s u^s\overline{u}^s= \gamma^{\mu}p_\mu+m$ where s stands for the spin state (up or down). This can be proven either by explicitely solving Dirac's equation using the Pauli representation of the matrice, either by using the representation-free method illustrated here :
https://arxiv.org/pdf/physics/0703214.pdf .
Now, I'm trying to see if there is a representation free notation for $u^{up}\overline{u}^{up}$. I'm ok with deriving it with a representation dependent method, but I'd like a result that is representation free to be able to use it with the trace techniques.
I have found some more information : if we take $\frac{S_z+1}{2}(\gamma^{\mu}p_{\mu}+m$ then it is equal to $u^{up}\overline{u}^{up}$ because it's action on $u^s,v^s$ for $s=(up,down)$ is the same. So the problem that remains is to express $S_z$ as a function of the gamma matrices, if that's possible.
 A: Here is the answer. We will use from the paper linked that $\overline{u}^su^{s'} = 2m\delta^{ss'}$ and $\overline{u}^sv^{s'} = 0$. We also know that the operator $\gamma^{\mu}p_{\mu}+m$ "selects" particle spinors in the sense that $(\gamma^{\mu}p_{\mu}+m)u^s = 2mu^s$ and $(\gamma^{\mu}p_{\mu}+m)v^s = 0$. 
We know also, using the relations from the paper that $u^{up}\overline{u}^{up}$ yields 0 on every spinor except $u^{up}$, for which it yields $2m$.
We find then that all we have to do is first select the particle spinor, and then select the spin up part of it. This is simply done by concatening the two operators, yielding : 
$u^{up}\overline{u}^{up} = \frac{\Sigma_z+1}{2}(\gamma^{\mu}p_{\mu}+m)$. Here $\Sigma_z$ is the extended pauli matrix, defined by $\Sigma_z u^{up} = u^{up}$, $\Sigma_z u^{down} = -u^{down}$ and inversely for antiparticle spinors. All is left to do is to express this in terms of the gamma matrices only. This is simply done by noticing $\Sigma_z = i\gamma^1\gamma^2$.
Hence, the answer is : $u^{up}\overline{u}^{up} = \frac{i\gamma^1\gamma^2+1}{2}(\gamma^{\mu}p_{\mu}+m)$. And it can be verified, as expected that $u^{up}\overline{u}^{up}+u^{down}\overline{u}^{down} = (\gamma^{\mu}p_{\mu}+m)$
