Understanding the kinetic term in the action expression for Majorana Fermion (Peskin and Schroeder QFT 3.4)

Question

In the action term for Majorana fermion as given in Peskin and Schroeder's question 3.4, the first term is :

$\int d^4x \space\chi^\dagger i \bar{\mathbb{\sigma}} \cdot \partial\chi$

which can be written as

$\int d^4x \space \chi^\dagger i \bar{\mathbb{\sigma}}^\mu \partial_\mu \chi$

where $\bar{\mathbb{\sigma}}^\mu = (1,-\vec{\sigma})$

Now, $\chi$ is a two components column and $\chi ^\dagger$ is a 2 component row. But $\partial_{\mu}$ is a four derivative, so it has four component.

a) So mathematically speaking, what kind of object is $\partial_{\mu} \chi$?

Then, going one step further, there is $\bar{\mathbb{\sigma}}^\mu$ which is a 4 components row, where each component is a 2 by 2 matrix.

b) Can someone show me what then is the overall term $\chi^\dagger i \bar{\mathbb{\sigma}}^\mu \partial_\mu \chi$?

This sort of mathematical manipulation is still not my strong suit, and I would really appreciate some help.

P.S. $\chi(x)$ is defined as a 2-component field.

Answer 1

Here is the action written with explicit sums over the indices

\begin{equation} S = \int {\rm d}^4 x \sum_{\dot{\alpha}=1}^2 \sum_{\beta=1}^2 \sum_{\mu=0}^3 \chi^\dagger_{\dot{\alpha}} i \sigma_{\dot{\alpha} \beta}^\mu \partial_\mu \chi_\beta \end{equation}

$\chi$ is a two component spinor and $\partial_\mu \chi$ is the gradient of a spinor.

At a deeper level, in this expression, $\alpha,\beta$ indices label components of objects that transform in the spinor representation of the Lorentz group, while the $\mu$ index labels an object that transforms in the fundamental representation.