# Relation for Dirac-spinors of different helicities

Assume that we have massless spin-1/2 particles. The Dirac-spinor is the solution of the Dirac equation:

$$p^\mu \gamma_\mu u_\pm(p) = 0, \quad p^2 = 0$$

The subscripts $\pm$ denote two different solutions, belonging to two different helicities. Is it possible to find a representation for the $\gamma_\mu$ so that the following relation holds true?

$$u_+ = \left( u_- \right)^*$$

I know this possible for (polarization) vectors, i.e. one may choose $\epsilon^\mu_+ = \left( \epsilon^\mu_- \right)^*$, but I guess it is not for spinors (in the weyl representation it obviously is not). So here is my

Question: Is it possible to find a representation of the Dirac-Gamma-matrices so that the spinors of different helicities are related by complex conjugation?

For massless particles, helicity coincides with chirality thus you ask to find the basis such that $$\psi_{\pm}=\left( \psi_{\mp}\right) ^{\star},\quad\gamma_{5}\psi_{\pm}% =\pm\psi_{\pm}.$$ Using the decomposition of hermitian operator: $$\left( \gamma_{5}\right) _{ij}=\left( \psi_{+}\right) _{i}\left( \psi _{+}^{\star}\right) _{j}-\left( \psi_{-}\right) _{i}\left( \psi_{-}% ^{\star}\right) _{j}=\left( \psi_{+}\right) _{i}\left( \psi_{-}\right) _{j}-\left( \psi_{-}\right) _{i}\left( \psi_{+}\right) _{j},$$ we find that $\gamma_{5}$ should be an antisymmetric matrix. Since $\gamma_{5}$ is a hermitian operator it implies that all components of $\gamma_{5}$ should be pure imaginary. In Majorana basis all $\gamma$-matrices are pure imaginary and since $$\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3},$$ it means that $\gamma_{5}$ is also pure imaginary (and thus antisymmetric): $$\gamma_{5}=\left( \begin{array} [c]{cc} \sigma_{2} & \\ & -\sigma_{2}% \end{array} \right) .$$
Update. Off topic. From my personal point of view, I would never use the basis of the polarization wave function such that $\psi_{j,m}^{\star}=\psi_{j,-m}$. The reason is the following: sometimes it is extremely convenient (see explanation below) to associate $\psi^{\star}$ with the «time-revesed» wave function as it is suggested by the Schrödinger equation and the anti-unitary nature of the operation of time reversal. The polarization wave function of a particle of spin $s>0$ is a contravariant rank $2s$ symmetric spinor: $$\psi_{s,\sigma}=\phi^{i_{1}..i_{2s}},$$ in whose indices $1$ occurs $s+\sigma$ times and $2$ $s-\sigma$ times. The complex conjugation leads to the covariant spinor: $$\phi_{i_{1}\ldots i_{2s}}^{\left( rev\right) }=\left( \phi^{i_{1}..i_{2s}% }\right) ^{\star}.$$ Using the antisymmetric tensor $\epsilon^{ij}$ (the metric tensor for $SU\left( 2\right)$) one can construct again a covariant spinor, so that the sign changes as many times as there are twos among the indices: $$\psi_{s,-\sigma}^{\left( rev\right) }=T\left( \psi_{s,\sigma}\right) =\psi_{s,\sigma}^{\star}\left( -1\right) ^{s-\sigma}.$$ For example, when the time-reversal operation is repeated: $$T^{2}\left( \psi_{s,\sigma}\right) =\left( -1\right) ^{2s}\psi_{s,\sigma},$$ that immediately leads to the well-known Kramers' theorem:
Therefore a (polarization) wave function with is usually normalized by the condition: $$\psi_{j,m}^{\star}=\psi_{j,-m}\left( -1\right) ^{s-\sigma}.\quad\quad(1)$$ For example, for $s=1$ the polarization vectors have the form: $$\varepsilon_{\pm1}=\frac{\mp i}{\sqrt{2}}\left( 1,\pm i,0\right) ,\quad\varepsilon_{0}=i\left( 0,0,1\right) ,$$ see also Eqs. (28.7-28.9) in L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Non-relativistic Theory. However, some people find the normalization (1) too complicated.