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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?

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  • You do know that helicity coincides with chirality only in the massless limit? And that the mass term connects the two chiralities? Why should the helicity operator you wrote not depend on the representation of γ s? – Cosmas Zachos May 08 '21 at 22:28
  • Linked. – Cosmas Zachos May 08 '21 at 22:48
  • @Cosmas I am new to this subject. Could you elaborate a little more please? For instance, does your comment imply that the plane waves mentioned in my post do not satisfy Dirac's equation? – MathMath May 08 '21 at 23:07
  • I think a mainstream text like Peskin & Schroeder would help you. Expanding solutions in helicity eigenstates is a bit unconventional. I really don't want to go into your formalism. – Cosmas Zachos May 08 '21 at 23:15
  • @Cosmas, this das taken from a book. The idea was actually to use these plane waves together with the ones associated with negative energies and show by using the superposition principle that their fourier transform is a general solution for Dirac's equation. The final expression displayed in the usual expression found in texts like peskin&Schroeder. – MathMath May 08 '21 at 23:36
  • This is why I thought these should satisfy Dirac's equation. – MathMath May 08 '21 at 23:36
  • I agree a positive helicity state may be a solution of Dirac's equation, but I am reluctant to get enmeshed in basis changes... – Cosmas Zachos May 08 '21 at 23:51

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