Decomposing the Dirac Lagrangian into Weyl Spinors

[latentcorpse]

If we take the the Dirac Lagrangian and decompose into Weyl spinors we find

[itex]\mathcal{L} = \bar{\psi} ( i \gamma^\mu \partial_\mu - m ) \psi = i U^\dagger_- \sigma^\mu \partial_\mu u_- + i u^\dagger_+ \bar{\sigma}^\mu \partial_\mu u_+ - m(u^\dagger_+ u_- + u^\dagger_- u_+ ) =0[/itex]

So far I have that since [itex]\psi= \begin{pmatrix} u_+ \\ u_- \end{pmatrix}[/itex] and [itex]\bar{\psi} = \psi^\dagger \gamma^0[/itex],

[itex]\mathcal{L} = \begin{pmatrix} u^\dagger_+ & u^\dagger_- \end{pmatrix} \gamma^0 ( i \gamma^\mu \partial_\mu - m ) \begin{pmatrix} u_+ \\ u_- \end{pmatrix}[/itex]
[itex]= i u^\dagger_+ \gamma^0 \gamma^\mu \partial_\mu u_+ + i u^\dagger_- \gamma^0 \gamma^\mu \partial_\mu u_- - m ( u^\dagger_+ u_+ + u^\dagger_- u_- ) [/itex]

But I can't get anywhere near the answer from here...


Oh and we define [itex]\sigma^\mu = (1, \sigma_i) , \bar{\sigma}^\mu = ( 1 , - \sigma^i )[/itex]

Thanks for any help.

 

[mark726]

Thank you for your question. To continue from where you left off, we can use the properties of the gamma matrices to simplify the expression further. Firstly, we can use the fact that $\gamma^0 \gamma^\mu = \bar{\sigma}^\mu \sigma^0$ and $\gamma^0 \gamma^\mu = \sigma^\mu \bar{\sigma}^0$ to rewrite the expression as:

$\mathcal{L} = i u^\dagger_+ \sigma^0 \bar{\sigma}^\mu \partial_\mu u_+ + i u^\dagger_- \sigma^0 \sigma^\mu \partial_\mu u_- - m ( u^\dagger_+ u_+ + u^\dagger_- u_- ) $

Next, we can use the fact that $\sigma^0 = \bar{\sigma}^0 = \mathbb{1}$ to simplify the expression even further:

$\mathcal{L} = i u^\dagger_+ \bar{\sigma}^\mu \partial_\mu u_+ + i u^\dagger_- \sigma^\mu \partial_\mu u_- - m ( u^\dagger_+ u_+ + u^\dagger_- u_- ) $

Finally, we can use the definition of the sigma matrices given in the forum post to write the expression in terms of the Weyl spinors:

$\mathcal{L} = i u^\dagger_+ \begin{pmatrix} 1 & 0 \\ 0 & -\sigma^i \end{pmatrix} \partial_\mu u_+ + i u^\dagger_- \begin{pmatrix} 1 & 0 \\ 0 & \sigma^i \end{pmatrix} \partial_\mu u_- - m ( u^\dagger_+ u_+ + u^\dagger_- u_- ) $

I hope this helps you continue with your calculations. Let me know if you have any further questions. Good luck!