# How to convert a singlet current in electroweak theory to a doublet current?

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.

• 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! Dec 25, 2021 at 23:40
• Dec 26, 2021 at 15:06

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 $$\psi_{\nu_e}$$. $$\bar{\psi}_e \frac{1+ \gamma_5}{2} \gamma^{\alpha} \frac{1- \gamma_5}{2} \psi_e = \begin{pmatrix} \bar{\psi}_{\nu_e} \frac{1+ \gamma_5}{2} \gamma^{\alpha} & \bar{\psi}_e \frac{1+ \gamma_5}{2} \gamma^{\alpha} \end{pmatrix} \frac{1- \gamma_5}{2} \begin{pmatrix} 0 \\ \psi_e \end{pmatrix}= \begin{pmatrix} \bar{\psi}_{\nu_e} & \bar{\psi_e} \end{pmatrix} \frac{1+ \gamma_5}{2} \gamma^{\alpha} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \frac{1- \gamma_5}{2} \begin{pmatrix} \psi_{\nu_e} \\ \psi_e \end{pmatrix} = \bar{L}_e \gamma^{\alpha} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} L_e$$