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In QFT we define the projection operators:

\begin{equation}

P_{\pm} = \frac{1}{2} ( 1 \pm \gamma^5)

\end{equation}

and define the left- and right-handed parts of the Dirac spinor as:

\begin{align}

\psi_R & = P_+ \psi \\

\psi_L & = P_- \psi

\end{align}

I was wondering if the left- and right-handed parts of the Dirac spinor are dependent on the representation of the gamma matrices (for instant Dirac representation, Weyl representation or Majorana representation)?

For instance, a general solution of the Dirac equation (following the book "Symmetry and the Standard Model" by Matthew Robinson who works with the signature (-,+,+,+)) is given by:

\begin{equation}

\psi(x) = a v(\mathbf{p}) e^{i p_\mu x^\mu} + b u(\mathbf{p}) e^{- i p_\mu x^\mu} \label{15.1}

\end{equation}

Then, it is not too difficult to show that for a particle at restusing the Weyl representationthis results in:

\begin{equation}

v_L=-v_R

\end{equation}

and:

\begin{equation}

u_L=u_R

\end{equation}

Subsequently, he derives that in the Dirac representation the above two equations also hold. However, I do not agree with his derivation and his result, and according to my derivation the above two equations do not hold in the Dirac representation. So my question is: do the above two equationsmusthold for any representation (i.e. is my derivation definitely wrong)?

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# Is chirality dependent on the representation of the gamma matrices?

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