QFT Dirac Chiral Equations of Motion

[orentago]

Homework Statement

From Mandl and Shaw (exercise 4.5):

Deduce the equations of motion for the fields:

$$\psi_L(x)\equiv{1 \over 2} (1-\gamma_5)\psi(x)$$
$$\psi_R(x)\equiv{1 \over 2} (1+\gamma_5)\psi(x)$$

for non-vanishing mass, and show that they decouple in the limit m=0. Hence show that the Lagrangian density

$$L(x)=\mathrm{i} \hbar c \overline{\psi}_L(x) \gamma^\mu \partial_\mu \psi_L(x)$$

describes zero-mass fermions with negative helicity only, and zero-mass antifermions with positive helicity only.

Homework Equations

Lagrangian density for Dirac field:

$$L=c\overline{\psi}\left[ \mathrm{i}\hbar\gamma^\mu \partial_\mu -mc\right] \psi(x)$$

Equations of motion:

$${\partial L \over \partial \psi} - {\partial \over \partial x^\mu} \left(\partial L \over \partial \psi_{,\mu} \right)=0$$

The Attempt at a Solution

I'm not exactly sure where to begin, partly because I don't understand the wording of the question. Do I simply swap $$\psi$$ for $$\psi_L$$ and $$\psi_R$$ into the Lagrangian above and sub this into the equations of motion as per normal, or should I swap $$\psi(x)$$ for $$\psi_L(x)+\psi_R(x)$$?

For the second part I'll have to use the helicity operator I expect, but I'll cross that bridge when i come to it.

 

[orentago]

I'm still stuck on this. I tried to go for the former of the two approaches mentioned above. I let $$\psi\rightarrow\psi_L$$ and $$\overline{\psi}\rightarrow\overline{\psi}_L$$, then:

$$L=c\overline{\psi}_L\left(\mathrm{i}\hbar\gamma^\mu\partial_\mu-mc\right)\psi_L$$

Splitting this into two terms and tackling individually:

$$c\overline{\psi}_L\mathrm{i}\hbar\gamma^\mu\partial_\mu\psi_L=c\left(1-\gamma_5\right)\overline{\psi}\mathrm{i}\hbar\gamma^\mu\partial_\mu\left(1-\gamma_5\right)\psi$$

$$c\overline{\psi}_L mc\psi_L=c\left(1-\gamma_5\right)\overline{\psi} mc \left(1-\gamma_5\right)\psi$$

Then I expand the brackets and use the anticommutation relations $$\left[\gamma_5,\gamma^\mu\right]_{+}=0$$ and $$\left[\overline{\psi},\gamma^\mu\right]_{+}=0$$ to get:

$$c\left(1-\gamma_5\right)\overline{\psi}\mathrm{i}\hbar\gamma^\mu\partial_\mu\left(1-\gamma_5\right)\psi={1 \over 2}\mathrm{i}\hbar c \left(1-\gamma_5\right)\overline{\psi}\gamma^\mu\partial_\mu \psi$$

and

$$c\left(1-\gamma_5\right)\overline{\psi} mc \left(1-\gamma_5\right)\psi=0$$

So

$$L={1 \over 2}\mathrm{i}\hbar c \left(1-\gamma_5\right)\overline{\psi}\gamma^\mu\partial_\mu \psi$$

Substituting this into the equations of motion and doing some rearrangement gives:

$${1 \over 2} \mathrm{i}\hbar c\left(1-\gamma_5\right)\gamma^\mu \partial_\mu\overline{\psi}=0$$

A similar process for $$\psi_R$$ gives:

$$L={1 \over 2}\mathrm{i}\hbar c \left(1+\gamma_5\right)\overline{\psi}\gamma^\mu\partial_\mu \psi$$

with equations of motion:

$${1 \over 2} \mathrm{i}\hbar c\left(1+\gamma_5\right)\gamma^\mu \partial_\mu\overline{\psi}=0$$

This leaves me a bit confused. I'm pretty sure I've gone wrong somewhere, as the two equations don't decouple in the zero-mass limit. Can anyone see where I've gone wrong?

EDIT: In fact I'm fairly sure they're not coupled at all!

 

[orentago]

Does anyone have any hints for this, or should I have another stab and post my findings?