I'm currently working through Chapter 13 of Wald's General Relativity and spinors are being a little illusive to me. The question is pretty much: Using the Klein-Gordon equation in the form: $$\partial_{A'_{1}A}\phi^{A_{1}...A_{n}} = \frac{m}{\sqrt{2}}\alpha_{A'_{1}}^{A_{2}...A_{n}}$$ and $$\partial^{A'_{1}A_{1}}\alpha_{A'_{1}}^{A_{2}...A_{n}} = -\frac{m}{\sqrt{2}}\phi^{A_{1}...A_{n}}$$ for n = 2s, $\alpha_{A'_{1}}^{A_{2}...A_{n}}$ represents an auxiliary variable. Take s = $\frac{1}{2}$ and the pair of spinors $(\phi^{A},\alpha_{A'}) $ taking the components to be $\psi^{0},\psi^{1},\psi^{2},\psi^{3}$ respectively, show this boils down to the Dirac Equation shown in most textbooks. It gives in the question that you should choose the spinor basis $(o^{A},i^{A})$ so that $o_{A}i^{A}=1$, which to me says that use the basis elements associated with $(o^{A},i^{A})$ are $(\frac{1}{\sqrt{2}}I,\frac{1}{\sqrt{2}}\sigma_{x},\frac{-1}{\sqrt{2}}\sigma_{y},\frac{1}{\sqrt{2}}\sigma_{z})$. ($I$ is the 2x2 identity, and $\sigma^{i}$ are Pauli matrices.)
My approach is to set up the spacetime so that the basis vectors have the y component reversed, hence we have all the Pauli matrices of the same sign when we use that 'conversion' tensor $\sigma^{a}_{AA'}$ and then simply substituting into the equations (and putting into bases), I have something along the lines of $$I_{\Lambda\Lambda'}\partial_{t}\phi^{\Lambda} - \sigma^{a}_{\Lambda\Lambda'}\partial_{a}\phi^{\Lambda} - m\alpha_{\Lambda'} = 0$$ and $$I_{\Lambda}^{\Lambda'}\partial_{t}\alpha_{\Lambda'} - \sigma^{a\Lambda'}_{\Lambda}\partial_{a}\alpha_{\Lambda'} + m\phi_{\Lambda} = 0$$ You can use $\epsilon_{\Sigma\Omega}$ to raise and lower the indices as you want to get the component form and a = {x,y,z}. My logic was to effectively 'squish' the two equations together so that we can have a complex vector for $\psi^{\mu}$ as the components of $(\phi^{A},\alpha_{A'})$ and also some 4x4 matrix that represents the operators acting on components of $\psi$. (In this, I've taken $\phi^{\Lambda} = (\psi^{0},\psi^{1})$ and $\sigma_{\Lambda'} = (\array{\psi^{2}\\\psi^{3}})$) But when I write out in components, it doesn't seem to match up to any form of the Dirac equation. This problem would be rectified if the t derivatives were acting on $\alpha$ instead of $\phi$.
I'm yet to study relativistic quantum mechanics so I don't actually know the Dirac equation at all. Clearly these equations are reminiscent of the Dirac equation but they also not exactly the same. I feel that my conversion to a basis is wrong and hence why the time derivatives are acting on the wrong elements of $\psi^{\mu}$. I do apologise for my tensor notation though, I'm very inexperienced with LaTex and using Stack Exchange.
