Help Understanding Dirac Equation in Notes

[latentcorpse]

I have a very simple question about the Dirac equation that I just cannot see the answer to.

In these notes:

http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf

In equation 4.115, I keep getting

$u( \vec{p} ) = \begin{pmatrix} \sqrt{p \cdot \sigma} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \\ \sqrt{ p \cdot \bar{\sigma}} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} \sqrt{E - p \sigma^3} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \\ \sqrt{E + p \sigma^3} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{pmatrix}$

So why does $p^3 = p \sigma^3$ or is there a typo?

I would have thought that since $p^\mu = ( E,0,0,p)$, we should get $p^3=p$, no?

Thanks for any help!

 

[dextercioby]

There's something dubious about this guy's notation. What's the sqrt of a matrix equal to ? I would recommend you find a derivation of the free-particle solution in any other book.

Check Gross F. Relativistic quantum mechanics and field theory (Wiley, 1999)(635s), Page 123 or D.J.Griffiths' Introduction to Particle Physics text.