In Peskin and Schroeder's Quantum Field Theory, there is an identity of Pauli matrices which is connected to the Fierz identity, (equation 3.77)
$$(\sigma^{\mu})_{\alpha\beta}(\sigma_\mu)_{\gamma\delta}=2\epsilon_{\alpha\gamma}\epsilon_{\beta\delta}.\tag{3.77}$$
The author explains that
One can understand the identity by noting that the indices $\alpha,\gamma$ transform in the Lorentz representation of $\Psi_{L}$, while $\beta,\delta$ transform in the separate representation of $\Psi_{R}$, and the whole quantity must be a Lorentz invariant.
How can one see $\alpha,\gamma$ and $\beta,\delta$ transform in different representation?
Best Answer
The Pauli matrices are invariant tensors that couple left and right-handed spinors. These spinors transform in different representations of the Lorentz group (as you mentioned) and hence are usually denoted with different indices. This is trivial to see in two component notation, however if you are not familiar with this notation this can also be seen from a four-component Lagrangian:
$$ \bar{\psi} \gamma _\mu \psi = \psi^\dagger\gamma^0\gamma^\mu\psi=\left( \begin{array}{cc}\psi_R^* & \psi_L^* \end{array} \right) \left( \begin{array}{cc} 0 & \sigma ^\mu \\ \bar{\sigma} ^\mu & 0 \end{array} \right) \left( \begin{array}{c} \psi _L \\ \psi _R \end{array} \right) = \psi_R^*\sigma^\mu\psi_R +\psi_L^* \bar{\sigma} ^\mu \psi _L. $$ One can then show that $\psi_L^*$ ($\psi_R^*$) transforms as a right-handed (left-handed) spinor. Clearly a $ \sigma ^\mu $ field then connects a $ \psi _R $ field with a $ \psi _L $ field. We can write these contractions more explicitly by denoting the left-handed representation indices by greek indices and right-handed reprentation indices with dotted greek indices: $$ \psi^*_{L \, \dot{\alpha}} \left(\sigma^\mu\right)^{\dot{\alpha}}_{\phantom {\alpha}\alpha} \psi_L ^\alpha $$
Note: you might be tempted to think of $\psi _R $ and $\psi_L$ not as separate fields, but just fields with projectors acting on them. This makes this whole topic very confusing and I would urge to get comfortable thinking in terms of two component fields as the fundamental objects making up fermions.