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I am referring to an aspect of Glashow-Salam-Weinberg theory of leptons. More specifically a step in the book 'Gauge Theory of Weak Interactions' by Walter Greiner page 147.

We can write the electromagnetic current as $J^{(e) \alpha}_{EM} = \bar{\psi}_e \gamma^{\alpha} \psi_e,\\$ it is then straightforward to show $J^{(e) \alpha}_{EM}=\bar{\psi}_e \frac{1 + \gamma_5}{2} \gamma^{\alpha} \frac{1- \gamma_5}{2} \psi_e + \bar{\psi}_e \frac{1- \gamma_5}{2} \gamma^{\alpha} \frac{1+ \gamma_5}{2} \psi_e,\\$ the next step is where I get stuck, he states, $J^{(e) \alpha}_{EM}=\bar{L}_e \gamma^{\alpha} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} L_e + \bar{R}_e \gamma^{\alpha} R_e,\\$ where $R_e = ( \frac{1 + \gamma_5}{2}) \psi_e,\,\, L_e = \frac{1- \gamma_5}{2} \begin{pmatrix} \psi_{\nu_e} \\ \psi_e \end{pmatrix}\\$ I understand how to get to the right handed part, just use the fact that $\bar{\psi}_e=\psi_e^{\dagger} \gamma_0$ and the fact that $\gamma_5$ anti commute with $\gamma^{\alpha}$. However, the left handed part confuses me, partially because I am confused on the dimensions of the quantities. Is $\gamma^{\alpha}$ a 4x4 matrix? is $\begin{pmatrix} \psi_{\nu_e} \\ \psi_e \end{pmatrix}$ an 8x1 column matrix since $\psi_e, \psi_{\nu_e}$ are Dirac spinors (if so this is even more confusing)? What basis is $\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ in. What basis is $\begin{pmatrix} \psi_{\nu_e} \\ \psi_{e} \end{pmatrix}$ in? Any help in answering these subquestions and justifying this step is greatly appreciated.

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    $\begingroup$ Study tensor products. Understand that 4x4 γ matrices and 2x2 weak isomatrices tensor to 8x8 matrices, effectively, but in practice you never need to write them as such! Write down all indices! $\endgroup$ Dec 25, 2021 at 23:40
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    $\begingroup$ Practice. $\endgroup$ Dec 26, 2021 at 15:06

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With the help of the comments and a friend I found a solution. The key to this is understanding that $\gamma$ and the doublets are in different spaces which explains the notation of the author. Thus $\gamma$ has the properties of a scalar when acting on doublet related objects, but not when it acts on say a Dirac spinor $\psi_e$ or $\psi_{\nu_e}$. $\bar{\psi}_e \frac{1+ \gamma_5}{2} \gamma^{\alpha} \frac{1- \gamma_5}{2} \psi_e = \begin{pmatrix} \bar{\psi}_{\nu_e} \frac{1+ \gamma_5}{2} \gamma^{\alpha} & \bar{\psi}_e \frac{1+ \gamma_5}{2} \gamma^{\alpha} \end{pmatrix} \frac{1- \gamma_5}{2} \begin{pmatrix} 0 \\ \psi_e \end{pmatrix}= \begin{pmatrix} \bar{\psi}_{\nu_e} & \bar{\psi_e} \end{pmatrix} \frac{1+ \gamma_5}{2} \gamma^{\alpha} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \frac{1- \gamma_5}{2} \begin{pmatrix} \psi_{\nu_e} \\ \psi_e \end{pmatrix} = \bar{L}_e \gamma^{\alpha} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} L_e$

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