Suppose $\sigma_{1},\sigma_{2}$ and $\sigma_{3}$ are the Pauli matrices. Given a momentum ${\bf{p}}$, we define the helicity operator: $$ h = \frac{1}{2}\begin{pmatrix} {\bf{\sigma}}\cdot {\bf{\hat{p}}} & 0 \\ 0 & {\bf{\sigma}}\cdot {\bf{\hat{p}}} \end{pmatrix} $$ where $\sigma \cdot {\bf{\hat{p}}} = \sum_{\mu=0}^{3}\sigma^{\mu}\frac{p_{\mu}}{|{\bf{p}}|}$. Let $|\pm \rangle$ be the eigenvectors of $\sigma \cdot {\bf{\hat{p}}}$ with eigenvalues $\pm 1$. Consider the following plane waves: $$\psi_{+} = \frac{1}{\sqrt{2}}e^{i(-Et+{\bf{p}}\cdot {\bf{x}})}\binom{e^{-\frac{\theta}{2}}|+\rangle}{e^{\frac{\theta}{2}}|+\rangle} \quad \mbox{and} \quad \psi_{-} = \frac{1}{\sqrt{2}}e^{i(Et-{\bf{p}}\cdot{\bf{x}})}\binom{e^{\frac{\theta}{2}}|-\rangle}{e^{-\frac{\theta}{2}}|-\rangle}$$ Then $\psi_{+}$ is an eigenstate of $h$ with eigenvalue $1/2$ while $\psi_{-}$ is an eigenstate of $h$ with eigenvalue $-1/2$. Here, $\theta$ is a known parameter.
Question: Shouldn't $\psi_{\pm}$ be solutions of the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi = 0$? They seem not to since, according to my calculations: $$i\gamma^{0}\partial_{0}\psi_{\pm} = \frac{1}{\sqrt{2}} E ^{i(-Et+{\bf{p}}\cdot{\bf{x}})}\binom{e^{\pm \frac{\theta}{2}}|\pm\rangle}{e^{\mp \frac{\theta}{2}}|\pm\rangle}$$ and, analogously: $$i \gamma^{\mu}\partial_{\mu}\psi_{\pm} = \frac{1}{\sqrt{2}}|{\bf{p}}|e^{i(-Et+{\bf{p}}\cdot{\bf{x}})}\binom{e^{\pm \frac{\theta}{2}}|\pm\rangle}{-e^{\mp \frac{\theta}{2}}|\pm\rangle}$$ while the last term of the equation reads: $$m \psi_{\pm} = m \binom{e^{\mp \frac{\theta}{2}}|\pm\rangle}{e^{\pm \frac{\theta}{2}}|\pm\rangle}$$ and the sum is clearly nonzero. To do such calculations I used the Weyl representation of the Dirac matrices: $$\gamma^{0} = \begin{pmatrix} 0 & I \\ I & 0 \end{pmatrix} \quad \mbox{and}\quad \gamma^{\mu} = \begin{pmatrix} 0 & \sigma^{\mu} \\ -\sigma^{\mu} & 0 \end{pmatrix}$$ Am I doing something wrong? Shouldn't the plane wave solutions above given in terms of eigenstates of $h$ be solutions to the Dirac equation? Moreover, should it depend on the particular representation of the gamma matrices?
