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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?

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2 Answers 2

up vote 6 down vote accepted

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:

For a system with the half-integral sum of the spins of particles, in an arbitrary electric field, all the levels must be doubly degenerate, and complex conjugate spinors correspond to two different states with the same energy.

For details, see § 60, L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Non-relativistic Theory.

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.

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This is called a Majorana representation, in your convention for the gamma matrices, you make sure that all the gamma matrices are pure imaginary.

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