Weyl Equation

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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.
 

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