Covariant projection method - Meson bound states

Question

I have seen many papers that discuss the production or decay of mesons ( quark bound states ) to make use of the covariant projection method where the product $\upsilon\bar{u}$ of the quark spinors that are about to create the mesons is replaced by some dirac operators:

I have seen it in two forms: $$1)\ \frac{1}{\sqrt{2}}\not{\epsilon}\dfrac{\not{P}+M}{2}$$ and $$2)\ \frac{1}{\sqrt{8(\frac{M}{2})^{3}}}\left(\frac{\not{P}}{2}-\not{q}-\frac{M}{2}\right)\gamma^{i}\left(\frac{\not{P}}{2}+\not{q}+\frac{M}{2}\right)$$ Where $P$ is the four momentum of the meson, q is the relative momentum of the quarks and M is the mass of the meson.

Are they different and if yes what different purpose do they serve?\ Moreover I am trying to show the first. To be exact I want to show (equation (2.1a) here) : $$\upsilon(\uparrow)\bar{u}(\uparrow)=\frac{1}{\sqrt{2}}\not{\epsilon}(\uparrow)\dfrac{\not{P}+M}{2}$$ If I write the spinors as $\upsilon(\uparrow)=\left(\begin{array}{c} \sqrt{p\cdot\sigma}\left(\begin{array}{c} 0\\ 1 \end{array}\right)\\ -\sqrt{p\cdot\bar{\sigma}}\left(\begin{array}{c} 0\\ 1 \end{array}\right) \end{array}\right)$ , $u^{\dagger}=\left(\begin{array}{cc} \left(\begin{array}{c} 1\\ 0 \end{array}\right)\sqrt{p\cdot\sigma} & ,\left(\begin{array}{c} 1\\ 0 \end{array}\right)\sqrt{p\cdot\bar{\sigma}}\end{array}\right)$ I can't get past the square roots.

Also is it $\epsilon^{\mu}(\uparrow)=(0,1,\imath,0)$ ?

Thank you! Any hint would be appreciated.

This post imported from StackExchange Physics at 2015-08-27 17:49 (UTC), posted by SE-user Lefteris