Sakurai credits B. L. van der Waerden 1932 a pretty derivation of Dirac equation from two-component wave functions. First decompose E^2-p^2=m^2 as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

(i \hbar {\partial \over \partial x_0} + {\bf \sigma} . i \hbar \nabla)

(i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla)

\phi= (mc)^2 \phi

[/tex]

This phi has two components, but it is a second order equation, so another two components are needed (say, the first derivative of phi) to fully specify a solution. Instead, we define

[tex]

\phi^R\equiv{1 \over mc}

(i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla) \phi

[/tex]

and [tex]\phi^L\equiv\phi[/tex]. Then we have

[tex]

i \hbar ({\bf \sigma} . \nabla - {\partial \over \partial x_0} ) \phi^L= - m c \phi^R

[/tex]

[tex]

i \hbar (-{\bf \sigma} . \nabla - {\partial \over \partial x_0} ) \phi^R= - m c \phi^L

[/tex]

Now you see the trick. These are the usual left and right handed two-component spinors; if you define

[tex]

\psi=

\begin{pmatrix}{\phi^R + \phi^L \cr \phi^R - \phi^L}

\end{pmatrix}

[/tex]

then the equation for the four component spinor [tex]\psi[/tex] is just Dirac equation!

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# Waerden's Dirac

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