A Dirac spinor $\Psi=\left(\begin{array}{c}\chi_\alpha\\\psi^\dot{\alpha}\end{array} \right)$ can be expanded in the following way:$\Psi=\int \frac{d^3p}{(2\pi)^3}\sqrt{\frac{1}{2E_p}} \sum_s\left( u_s(p)a_{s p} \exp(-ipx) + v_s(p) b^{\dagger}_{s p} \exp(ipx) \right)$.
In particular we have $v(p) = u(p)^c$ where $x^c$ means the charge conjugated expression of $x$.
I was always wondering how a Majorana spinor would be expanded in a similar way. Actually, I found that this expansion is a exercise in the book of Peskin & Schroeder, exercise 3.4(e). Furthermore I even found solutions on the internet "published" by Zhong-Zhi Xianyou. He states that a 2-dim. Majorana spinor could be expanded in creation and annihilation operators in the following way ($\xi_s$ must be the spin-components of some 2-dim. spinor)
$\chi(x) =\int \frac{d^3p}{(2\pi)^3}\sqrt{\frac{p\cdot \sigma}{2E_p}} \sum_s\left[\xi_s a_{s p}\exp(-ipx) + (-i\sigma^2)\xi^{*}_s a^{\dagger}_{s p}\exp(ipx)\right]$
However, I cannot understand it. Actually, I have the impression that $\xi_s$ and $(-i\sigma^2)\xi^{*}_s$ don't transform in the same way under Lorentz-transformations. $\xi\equiv \xi_\alpha$ seems to transform like normal Weyl-spinor, whereas $(-i\sigma^2)\xi^{*}_s \equiv \xi^\dot{\alpha}$ seems to transform like a dotted Weyl-spinor but are added up both in the same expression which would mix up the transformation properties.
According to the appendix E in A.Zee's book a charge-conjugated 4-Dirac spinor has the following form: $\Psi^c=\left(\begin{array}{c}\psi_\alpha\\\chi^\dot{\alpha}\end{array} \right)$ If I apply this representation on $u(p)$, then with $v(p)=u(p)^c$ I get the correct behaviour of $v(p)$ under Lorentz-transformations. Both upper components of $u(p)$ and $v(p)$ are undotted Weyl-spinors whereas the both lower components are both dotted Weyl-spinors. So for the expansion of the Dirac-spinor there is no contradiction (of course not!). From this I was tempted to write down my own expression for $\chi(x)$, but finally I realized that $u$ and $v$ are still solutions of the Dirac-equation, therefore the relation for Majorana-spinors $v(p)$=$u^c(p)$=$u(p)$ is a wrong short-cut. So at this point I don't know to proceed further. Can somebody help me out? Thank you.