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## Homework Statement

I'm stuck at my particle physics exercise about 4-component chiral fields.

The following problem is given: "Derive the expression for the QED Lagrangian in terms of the four component right-handed and left-handed Dirac fields ##\Psi_R(x)## and ##\Psi_L(x)##, respectively."

What I understand is that I need to get from the QED Lagrangian:

$$\mathscr{L}_{QED}=-\frac{1}{4} F_{\rho \sigma} F^{\rho \sigma}+\bar{\Psi} i \gamma^{\mu} \partial \Psi - m \bar{\Psi} \Psi - e A_\mu \bar{\Psi} \gamma^{\mu} \Psi$$

to here:

$$\mathscr{L}_{QED}=-\frac{1}{4} F_{\rho \sigma} F^{\rho \sigma}+\bar{\Psi}_L i \gamma^{\mu} \partial \Psi_L + \bar{\Psi}_R i \gamma^{\mu} \partial \Psi_R - m \bar{\Psi}_L \Psi_R - m \bar{\Psi}_R \Psi_L - e A_\mu [ \bar{\Psi}_L \gamma^{\mu} \Psi_L + \bar{\Psi}_R \gamma^{\mu} \Psi_R ]$$

## Homework Equations

The operators ##\Psi## can be written in terms of the left- and right handed fields: \begin{equation}\Psi= \Psi_L +\Psi_R\end{equation}

The left- and right handed fields can be expressed through the projectors ##P_L## and ##P_R## so that:

\begin{equation}\Psi_L=P_L \Psi \quad \Psi_R=P_R \Psi\end{equation}

The projectors are:

$$P_L=\frac{1}{2}(1-\gamma^5) \quad P_R=\frac{1}{2}(1+\gamma^5)$$

With the properties:

$$P_{L,R}^2=P_{L,R} \quad P_L P_R=0 \quad P_L + P_R = 1$$

All properties of the ##\gamma##-matrices are given too.

## The Attempt at a Solution

My first step was to take the second term of the lagrangian and rewrite it with help of equation (1) and (2):

$$\bar{\Psi} (P_L + P_R) i \gamma^{\mu} \partial (P_L +P_R) \Psi$$

Then I tried it with resolving the statement between the ##\Psi##'s and looking where it leads, but I ended up with long calculation and a nonsense statement in the end.

Then I realized that the parenthesis are scalars, so i could write:

$$\bar{\Psi} (P_L + P_R)(P_L +P_R) i \gamma^{\mu} \partial \Psi$$

With the properties of the projectors I then get:

$$\bar{\Psi} (P_L + P_R)^2 i \gamma^{\mu} \partial \Psi$$

$$\bar{\Psi} (P_L^2 + P_R^2) i \gamma^{\mu} \partial \Psi$$

$$\bar{\Psi} P_L^2 i \gamma^{\mu} \partial \Psi +\bar{\Psi} P_R^2i \gamma^{\mu} \partial \Psi$$

As the projectors do not commute with the ##\gamma##-matrices (I checked that myself), I can't get the projectors to the other ##\Psi##'s on the right side, which would lead to the wanted end result.

I think that I am missing something important or very trivial. Maybe I need another approach for the problem. Could someone help me there?

Thanks in advance.